EPA-R2-72-023
May 1972
CYCLIC OPERATION OF PLATE COLUMNS
FOR GAS-LIQUID CONTACTING
by
James D. Dearth
Lawrence B. Evans
Edwin R. Gilliland
Department of Chemical Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
May 1, 1972
This report describes work carried out from October, 1971
through May, 1972 under Task Order No. 2 of EPA Contract
68-02-0018.
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PREFACE
This report describes the results of work carried out un-
der Task Order No. 2 of EPA Contract 68-02-0018. during the
period October, 1970 through May, 1972. This work comprises
doctoral thesis investigation of Mr. Dearth being carried
out under the supervision of Professors Gilliland and Evans.
When it is completed the results will be described in full
in Mr. Dearth's doctoral thesis.
A decision has been made to seek continued support for
completion of the work by means of an EPA grant rather than
through extension of the current contract. This report,
therefore, serves as an interim report to document the pro-
gress achieved to date under the contract and to support the
proposal for a grant.
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ABSTRACT
Previous experimental and theoretical work has indicat-
ed that the efficiency and capacity of gas-liquid contact-
ing equipment may be improved by controlled-cycling of gas
and liquid flows. However, difficulties encountered in the
design of large-scale systems, where hydraulic instabilities
and axial liquid mixing effects have been found to be severe,
has prevented wide-scale application of such systems to in-
dustrial separations. This report describes a proposed pro-
gram for obtaining needed information on tray hydraulics
and liquid mixing, and summarizes some initial data and ten-
tative conclusions for both steady-state and cyclic flows
of an air-water system in a three-tray section of a cyclic
absorber. Fluctuations in liquid-heads and interstage pres-
sure differences; entrainment of small air bubbles in the
downflowing liquid; and weepage of liquid due to liquid
"sloshing" in the gas flow period have been found to have
a pronounced effect upon column hydraulics.
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CONTENTS
Page
I. Introduction
A. General Description and Motivation 1
B. Literature Review 2
C. Current Problem Areas 5
II. Objectives
A. Short-range 7
1. Experimental 7
2. Computational 8
B. Long-range 8
III. Apparatus 10
IV. Experimental Approach
A. Hydraulics 12
B. Liquid Mixing 15
V. Results and Tentative Conclusions
A. Hydraulics 16
1. Steady-state Air Flow 16
2. Steady-state Liquid Flow 19
3. Variable-Venting Experiments 26
4. Vented Cyclic Results 30
5. Sealed Steady-State and Cyclic Operation 34
B. Liquid Mixing 36
C. Mathematical Simulation Models 37
1. Hydraulics 37
2. Mass-transfer Model with Mixing 40
VI. Direction of Current and Future Work 40
VII. Summary 42
VIII. Appendix 43
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A. Models for Axial Liquid Mixing in Cyclic Operation . ^3
1. Liauid-Phase Mixing from Evolution of Vertical ^3
Concentration-Distance Profile.
2. Overall Mixing Parameter from Measurements of ^5
Well-Mixed VFP Composition
B. Model for Gas and Liquid Flows Through a Perforated ^7
Plate.
C. Mathematical Model for Mass-Transfer and Licruid Mixing 51
in a Cyclic Absorber.
1. Gas Flow Period 51
2. Plug-Flow Conditions 52
3. Mixing-Model Equations 52
4. Periodicity Conditions 53
5. Average Exit-Gas Composition 53
6. Overall Column Efficiency -5^
7. Dimensionless Form 5^
D. Sample Calculations 59
1. Comparison of Experimental Slope of Liquid-head 59
vs. Flow Rate with Hagen-Poiseuille Law.
2. Calculation of Flowrate for Critical Weber Number. 60
3. Effect of Licruid Aeration on Velocity-Head 60
4. Initial Rate of Pressure Rise Due to Air Flow 6l
into a Sealed Column Section
5. Estimate of Liquid Residence-Time and Ratio of 6l
Flow-Period Lengths.
E. Nomenclature. 63
F. Literature Cited. 69
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I. Introduction
A. General Description and Motivation
Increased process efficiency and equipment capacity are
among the benefits which can result from controlled cycling
of many traditionally steady-state chemical engineering oper-
ations. Sufficient studies, both experimental and theoreti-
cal, have been conducted to demonstrate that benefits can
accrue from imposing a repetitive discontinuity upon an oth-
erwise continuous operation. Because of the widespread use
of gas-liquid contacting equipment in industry for distilla-
tion or gas absorption applications, it is felt that the intro-
duction of controlled cycling into this area may have a sub-
stantial impact upon the economics of any process which uti-
lizes these unit operations for purification or product re-
covery. In particular, the perfection of a high-capacity,
high-efficiency tail-gas scrubber design might help to re-
duce the economic burden imposed on many industries by the
advent of stricter state and federal pollutant emissions stan-
dards .
This investigation is concerned with the design of
staged gas-liquid contacting equipment for cycled operation.
The configuration currently under consideration resembles a
conventional sieve-tray column, but lacks the weirs and down-
comers associated with normal crossflow operation. Each oper-
ating cycle consists of two distinct phase-flow periods. Dur-
ing the vapor flow period gas is allowed to flow upward for
a short period of time, at a rate sufficient to maintain a
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pool of liquid on each tray without leakage through the per-
forations. Mass transfer occurs across the interface between
the rising gas bubbles and the vigorously agitated liquid.
During the liquid flow period the gas flow is interrupted
and liquid is allowed to drain through the trays by a weep-
age mechanism, while fresh liquid is fed into the column.
Before proceeding to outline the objectives of the current
study and the organization of the research program, it would
be appropriate to review the historical development of con-
trolled-cycling and to summarize some of the important con-
clusions of previous investigators.
B. Literature Review
The concept of cycled operation was originated by Cannon
(!_) in 1952. At that time it was suggested that many exist-
ing mass-transfer operations might be substantially improved
by changing the method of contacting the flowing phases. The
first mention of an operating cycle was made four years later
in connection with a proposed liquid-extraction process which
employed alternate upflow and downflow of the light and heavy
phases (2). Cycled or "rmlsed" columns of this type were
found to have high separation efficiencies and capacities,
and many have since found their way into industrial use. Oth-
er cycled systems have been employed on a laboratory scale for
particle-size segregation, crystallization, and ion exchange
(1§.' ii' 12.) •
The first data on cycled gas-liquid systems were reported
by Gaska and Cannon (5) and McWhirter and Cannon (13) for the
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rectification of benzene-toluene and methylcyclohexane-n-
heptane mixtures in a 2-inch ID glass column. It was conclud-
ed from these early investigations that cycled distillation
columns could be operated for either high efficiency or high
capacity, depending upon the relative lengths of the gas and
liquid flow periods. The highest column efficiencies observed
were in the range 130 to 150%, indicating that each tray of a
cycled column had a separating ability superior to that of a
single "theoretical plate." In a subsequent investigation
McWhirter and Lloyd (14) confirmed the earlier results using
a 6-inch ID column containing five 3-inch "packed plates."
For operating cycles of 3 to 9 seconds' duration overall ef-
ficiencies were in the range 160 to 200%. The observed sepa-
rating ability was rationalized in terns of the greater aver-
age mass-transfer driving forces present in a cycled column.
A simple mathematical model of a cycled column confirmed this
hypothesis and predicted efficiences similar to those observed.
Theoretical analyses by Robinson and Engel (17) and
Sommerfeld, et al. (1_9_) established an exact analogy between
the time-axis concentration profiles in a cycled column and
the crossflow concentration gradients in conventional columns.
The latter case had been analyzed by Lewis (9) who concluded
that such gradients could produce substantial improvements in
the efficiency of bubble-cap trays. Analytical transient solu-
tions to McWhirter's model were developed by Chien, et al. (3)
for the simplest case of a linear equilibrium relation, and
iterative procedures were developed to handle more general non-
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linear expressions. An extensive simulation study was carried
out to determine the effects of local point efficiency, slopes
of the operating and equilibrium lines, amount of liquid dropped
during the liquid flow period, and other parameters, on both
overall column and individual plate efficiencies. The most
important conclusions may be summarized as follows:
1) For controlled cycling operation at total reflux
with negligible liquid mixing during the liquid flow
period, the overall column efficiency attains its max-
imum value when the contents of one tray are dropped
per cycle.
2) Liquid mixing during the liquid flow period tends
to degrade the efficiency improvement obtained by cy-
cling. In the limit of perfect mixing there is no im-
provement.
3) The fractional increase of column efficiency be-
tween noncycled and cycled operation is greatest when
the individual point efficiencies are high.
4) The overall column efficiency increases to an
asymptotic limit as the total number of stages is in-
creased. This limiting behavior of overall efficiency
was noted by Horn (8) , who developed an asymptotic
solution using z-transform techniques.
Schrodt, et al. (18) carried out a series of experiments
using a 12-inch ID cycled column containing 15 perforated
trays without downcomers. It was concluded that the cyclic
unit was superior in flexibility and throughput character-
istics to the same unit in conventional dual-flow operation;
however, the cycled column efficiencies were, at best, slightly
lower than for conventional-flow. One reason for this was
that the column operation was hydrodynamically unstable. It
was noted that the existence of dynamic flow lags often caused
the lower trays to drain completely before the liquid had be-
gun to flow from the upper trays. Gerster and Scull (6\, in
studying the cyclic desorption of ammonia from aqueous solu-
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tion into an air stream, observed a different type of insta-
bility in the form of excessive liquid accumulations on the
lower trays. Despite unimpeded liquid mixing, efficiencies
as much as 40 percent higher than those obtainable in conven-
tional service were reported. A model for liquid mixing was
proposed and incorporated into the existing model for mass
transfer. The data were found to be in good agreement with
the model predictions, and values of a single "mixing param-
eter" were estimated.
To gain insight into the nature of the hydraulics of a
cycled column Wade, et al. (21) formulated a reasonably de-
tailed mathematical model of a cycled distillation column.
Using several empirical relations for gas and liquid flow
rates, they found that it was possible to alleviate the prob-
lem of liquid-flow lags by strategic manipulation of isola-
tion valves in the lines to the condenser and reboiler. The
shortcomings of the model and of the method of solution were
as follows:
1) The hydraulic relations used were completely
empirical, with no experimental justification.
2) The solutions obtained were for single cycles
or small numbers of cycles. No attempt was made
to select initial conditions to satisfy the perio-
dicity conditions of pseudo-steady cycled operation.
3) For convenience in the solution of the model
equations, perfect mixing of the liquid phase was
assumed at all times.
C. Current Problem Areas
Thus far, a general lack of understanding of the tray
hydraulics and liquid-mixing phenomena which occur in full
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scale cyclic equipment has made it virtually impossible to
apply the results obtained from simulation studies to the
design of multistage cyclic cascades. In the first place, the
existing models for cyclic mass-transfer presuppose a knowl-
edge of the liquid holdup on each tray. This quantity is de-
termined by the liquid head needed to drive a specified flow
rate of liquid down through the tray perforations against an
opposing gas-phase pressure difference. This flow configura-
tion has received little attention in previous literature,
since in a conventional crossflow tray weepage of liquid through
the holes is undesirable: in fact, much effort has been di-
rected towards the prediction of the minimum gas velocities
needed to prevent liquid flow through the holes. Existing
theories are unable to predict the magnitudes of liquid heads
and gas-ohase pressure differences, and the phenomenon of free
liquid drainage with no gas-phase pressure difference appears
not to be adequately described by simple orifice equations.
Yet liquid holdup is a predominant factor in determining
point contacting efficiency and cycle time.
There is additional experimental evidence which indicates
that hydraulic coupling among successive stages may result in
undesirable drainage-rate and holdup characteristics. Trays
near the top end of the column may run dry while excessive
amounts of liquid may accumulate on the lower trays. Quali-
tative explanations for these phenomena have been suggested,
but no quantitative techniques have been developed to predict
the occurrence of hydraulic instabilities and to indicate
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methods for alleviating their undesirable effects upon col-
umn performance by proper design.
The results of a ground-breaking study of axial liquid
mixing have suggested an effective one-parameter mixing model
which can be easily incorporated into the existing overall
simulation scheme for cyclic absorption. It remains to cor-
relate the extent of mixing against flowrates, liquid heads,
and geometric parameters in order to indicate column design
configurations for minimizing the undesirable effects of liq-
uid mixing during the LFP.
II. Objectives
A. Short-Range.
The purpose of the current research program is to pro-
vide information necessary to the formulation of rational
criteria for the design of cyclic absorbers. The immediate
objectives of such a study are summarized below.
1. Experimental.
The goal of the experimental ohase of the
work is to provide information on the physical
processes which influence the hydraulic opera-
tion of cyclic columns. Once the important phe-
nomena are understood, the data obtained can be
used to test mathematical models and evaluate
parameters in the proposed correlations. The
specific objectives are as follows:
a. To develop correlations for predicting
gas and liquid flows as functions of liquid
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head and interstage pressure difference, for
various tray geometries.
b. To evaluate alternative column configura-
tions and modes of operation to gain an appre-
ciation of their relative advantages and limi-
tations .
c. To evaluate mixing parameters and correlate
them to hydraulic and geometric variables.
2. Computational.
The shortcomings of previous attmpts to simulate
hydraulics and mass-transfer under controlled-cyclic
conditions were due primarily to the lack of experi-
mental hydraulic data. It is anticipated that this
need will be adequately filled by the correlations
derived from the experimental phase of the work, and
that the following simulation studies may then be
carried out:
a. To simulate the hydraulic behavior of a
long cyclic cascade, noting any non-uniformi-
ties of flow or liquid-head distribution pre-
dicted.
b. To simulate the overall mass-transfer be-
havior of a cyclic absorber, to obtain values
of overall column efficiency as functions of
a set of column parameters specified by the
designer.
B. Long—range.
The ultimate objective of the research program is to
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answer the following set of design questions:
1. Is cyclic operation of absorption and distil-
lation systems technically feasible?
2. What are the optimal choices for the design
and operating parameters?
3. Is the "best" cyclic system economically com-
petitive with existing equipment?
The studies carried out to date seem already to have an-
swered the first question in the affirmative. It remains
to solve the optimal design problem and to interpret the
results in the light of similar experience for convention-
al columns.
The design model formulated in the initial stages of
the research program should be tested against experimental
results to assess its reliability. A reasonable method
for carrying out such tests is to select a gas-liquid sys-
tem and to conduct a series of small-scale experiments
to develop correlations for point efficiency. The effi-
ciency correlations, together with equilibrium data and
specifications of inlet and outlet gas concentrations and
gas flow rate, are fed into the mathematical model, and
an optimum set of design-variable values is generated.
The actual performance of this "optimal" column once con-
structed may then be compared with that predicted by the
model. The results of such a study would yield informa-
tion on both the shortcomings of the model and the actual
separations obtainable in a well-designed cyclic absorber.
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III. Apparatus.
The experimental apparatus shown in Figure 1 consists of
a three tray section of a 6-inch diameter Lucite column, with
associated gas and liquid supply systems. The trays are fabri-
cated from 1/4-inch thick plexiglas sheet with holes drilled
on a triangular pitch. A sightglass is provided on each of
the bottom two trays for measurement of the average clear liq-
uid head, and a manometer is provided for measurement of the
interstage pressure difference across each of the test trays.
The tray inserts currently in use contain 1/8-inch holes, with
a fractional perforated area of 10%. In order to insure uni-
form drainage, the bottom of each tray is sprayed with a fluo-
rinated hydrocarbon coating to make it non-wettable.
The gas used is air, supplied from a 125 psig oil-free
compressed air line, passed through a reducing pressure regu-
lator, humidified to 95% of saturation, and delivered to the
base of the column through 1-inch supply lines containing ori-
fice plates for flow metering. The liquid phases consist of
water and a dilute sodium chloride tracer solution pumped
from two identical 150-gallon polyethylene tanks through a set
of rotameters to the top of the column.
Cyclic control of flows is imposed by placing 3-way so-
lenoid valves in the lines, in close proximity to the column.
During their respective "off-cycles the air will be vented,
while water and tracer solution will be returned to their
storage tanks. This arrangement permits continuous metering
of flows even under conditions of cyclic operation. The sole-
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"Sliding Box" for Step Tracer Inputs
Rotameters
AT
Return
1 i
Rotameters
A
r
Bypass
}
Pneumatic cylinder
,
Water I (Tracer
Side
S1de
Return
1%-HP
Centrifugal
Pump
125 psig
Compressed
Air
Leads
to conduc-
tivity
bridge
and re
corder
Vent -e-
Orifice
Plate
Bypass
Sightglasses, 1%-HP
Manometers, Centrifugal
and flanifold Pump
4-kw. Heating
Element
Saturator
Drum
Straightening
Vanes
FIGURE 1
Drain
SKETCH OF PROPOSED APPARATUS
Column Scale: 2 mm = 1 inch
Gas and Liquid Supplies Not
to Scale.
Cxi
- Flow-Control Valve
- Pressure-Reducing Valve
- Solenoid Valve
(ODD 11/15/70)
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noid valves are operated by electrical signals from a cycle-
timer-relay control box, which permits independent variation
of the gas and liquid flow period lengths.
To permit optional venting of the column sections to at-
mospheric pressure, a set of solenoid valves is mounted at the
rear of the frame. Each valve is connected to a venting port
in a particular column section, and may be closed or opened
by manual switching or by cyclic relay control. A float valve
has been installed in the bottom of the column to seal the liq-
uid drain line and prevent the oscillations in the overflow
leg during the gas flow period.
A rather uniaue feature of the apparatus is the "sliding
box" arrangement provided for the alternate introduction of
water and salt solution during liquid-mixing experiments. The
box has two chambers, one containing water and the other, tra-
cer solution. The position of the box relative to the perfor-
ations in the top plate is controlled by the piston of a pneu-
matic cylinder, so situated that either chamber may be located
above the column. Salt concentrations in the liquid on a par-
ticular tray may be monitored by inserting 5 mm. glass conduc-
tivity probes through the side wall of the column. Fittings
allow the probes to be traversed radially in order to check
for concentration gradients in the radial direction.
IV. Experimental Approach.
A* Hydraulics.
In dealing with the complex phenomena associated with
the hydraulics of a sealed cyclic column, it has been
found advantageous to decompose the more difficult general
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problem into a sequence of individual sub-problems more
amenable to measurement and interpretation. The piece-
wise - steady nature of cyclic operation suggests that
one might reasonably group the phenomena of interest into
three main subdivisions as follows:
1. Steady-state gas flow through a non-flowing mass
of aerated liquid supported on a sieve tray.
2. Steady-state liquid drainage through the perfor-
ations of a sieve tray against an opposing gas-phase
pressure difference.
3. Departures from steady-state gas flow and liquid
drainage, resulting from on-off flow cycling.
Each of these sub-areas can, in turn, be broken down into
their component phenomena, as illustrated by the diagram
in Figure 2. Pressure drops for gas flow are, in general,
represented as the sum of a dry-plate component and a liq-
uid-head contribution. In an analogous fashion, steady-
state liquid heads can be thought of as being composed of
a liquid head for free drainage with no gas-phase pressure
difference, and an additional level difference to overcome
the effect of pressure buildup in the gas phase. Simi-
larly, the departures from steady-state hydraulics may
result both from phenomena which arise when the column
sections are vented during the liquid flow period, and
from other effects due exclusively to the decay of the
pressure gradient set up during the vapor flow period.
The independent variables and their approximate
ranges of variation may be summarized as follows:
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1. Gas Flow Rate: 0.5 - 1.5 ft3/sec (1 atm, 70°F)
2. Liquid Flow Rate: 1 - 30 gal/min.
3. Gas Flow Period: 0.5 - 10 sec.
4. Liquid Flow Period: 0.5 - 10 sec.
5. Tray Hole Diameter: 1/8 - 1/2 inch
6. Tray Free Area: 5 - 20 %
B. Liquid Mixing.
The values of liquid mixing parameters reported in
the literature have been deduced from mass-transfer ex-
periments by comparing the experimental results with
those obtained from the mathematical simulations modi-
fied to include a simple mixing model. The excellent
agreement reported between theory and experiment for a
constant mixing parameter value ic an encouraging sign
that the model selected may give a good representation
of the actual column behavior.
What is now needed is a correlation of mixing-param-
eter values as functions of liauid flow rate, liquid hold-
up, plate spacing, and tray geometry. For this purpose,
mass-transfer experiments would be both expensive and
time-consuming, and the results might contain errors and
uncertainties due to imperfect modelling of mass-transfer
phenomena and inaccurate values of input parameters such
as point efficiency. As an alternative it has been found
desirable to devise a means to test the mixing model in
the absence of mass-transfer, thus taking advantage of
the fact that mixing of liquid on the trays results only
from hydraulic phenomena.
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The experimental technique involves the introduction
into the test column of water and a dilute sodium chloride
tracer solution during alternate liquid flow periods of
cyclic operation. This alternating step-function concen-
tration input gives rise to salt-concentration gradients
in the vertical direction on each of the lower two test
trays during the liquid flow period, as well as to up-
and-down fluctuations in average tray licruid composition
in alternate cycles. The distance-gradients and time-vary-
ing fluctuations in average concentration can be monitored
by inserting conductivity probes into the tray liquid and
recording the changes in the voltage imbalance of a bridge
circuit which result from changes in the resistance of
the solution. The distance-gradient results can be used
to provide information on the detailed mixing process,'
while the average-concentration measurements permit the
straightforward evaluation of an overall parameter. The
interpretation of the results is discussed in Appendix
VIII.A.
V. Results and Tentative Conclusions.
A. Hydraulics.
In the following sections, the results for gas and
liquid flows through a 1/8-inch hole size, 10% free area
tray insert have been summarized. Tentative conclusions,
based on the existing information, have been formulated,
and suggestions for further study in each of several areas
have been made.
1. Steady-State Air Flow.
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One of the first experiments performed on the
completed column assembly was the measurement of
dry-plate pressure drops for the first set of tray
inserts. A large quantity of experimental data in
this area is available in the literature and a gen-
eral correlation of the data has been suggested by
McAllister et al.(ll). The usual approach is to ex-
press the pressure drop as a multiple of the velocity-
head of the gas flowing through the holes. The pro-
portionality constant which will be referred to as
a "discharge-loss coefficient," is assumed to be
equal to the product of a term representing the sum
of sudden contraction losses, sudden expansion losses,
and friction losses,multiplied by a function of the
plate thickness to hole diameter ratio. A comparison
of the data with this calculation is shown in Figure
3. The results have been plotted as discharge-loss
coefficient vs. Reynolds number. The calculated val-
ue of the coefficient, neglecting the friction term,
is 1.15. This is in excellent agreement with the
data at the high-Reynolds number asymptote. At Rey-
nolds' numbers below about 3000 the omitted friction-
loss term seems to become important, resulting in an
increase in the experimental coefficient. The scatter
in the data at the low Reynolds' number end is due
to the fact that in this region both gas flowrates
and pressure-drops were extremely small, resulting
in large percentage errors.
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-19-
All experimental observations of cyclic opera-
tion to date suggest that, with one possible excep-
tion, transients at the beginning of the gas flow
period are very fast. A fraction of a second after
resumption of gas flow, the column is hydraulically
at steady-state, and this steady-state is, for all
practical purposes identical to that which one finds
in a cross flow tray in the limit of zero liquid rate.
Hence, one may make use of all available correlations
for calculating wet-plate pressure drops from liquid
heads and dry-plate pressure drops. If the liquid
holdup on every plate is known at the end of the liq-
uid flow period, the pressure gradient through the
column during the gas flow period may be simply and
efficiently computed. The modelling difficulties
are associated almost entirely with liquid flow.
2. Steady-sJ:ate Liquid Flow.
Two limiting modes of operation are of particu-
lar interest, when one considers the relationships
governing liquid flow in a cyclic column. In the
first case, henceforth referred to as "vented" oper-
ation, the intertray spaces are connected via elec-
tric or pneumatic valves to a common manifold. At
the beginning of the liquid flow period these valves
are opened, permitting essentially instantaneous
equalization of gas-phase pressure across every tray.
As a result, liquid flow is influenced only by the
tray liquid head; no pressure buildups or fluctuations
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may occur in the gas phase. The opposite extreme
of "unvented" operation dispenses with the addition-
al valves and manifold and allows interstage pressure
differences to "seek their own levels." Pressures
remain at non-zero values even after the inflow of
air at the bottom is interrupted, and oscillations
in the gas phase may have a significant effect upon
liquid drainage. All stages are coupled together
through the gas phase, and a disturbance in liquid
head will be communicated up and down the column by
resulting pressure disturbances.
A mathematical representation of either of
these cases must make use of the fundamental steady-
state discharge relationship for liquid flow through
a perforated plate. This relationship is determined
by means of the following experiment:
With the pressure in all column sections main-
tained at one atmosphere, water is introduced through
a distributor pipe at the top of the column at a
specific volume flowrate. The liquid drains through
the perforations of each plate and falls through the
intertray space onto the tray below. Liquid heads
are established on the various stages and are meas-
ured, with respect to some horizontal datum plane,
on sightglasses.
The results are plotted as clear liquid head above
datum in inches of 1^0 vs. liquid volume flowrate
in gallons per minute. Such a plot is shown in Fig-
ure 4 for the 1/8-inch hole size, 10% free area tray
insert.
Although the data could be easily replotted as
discharge-loss coefficient vs. Reynolds number, in
analogy with the air-flow results, we will be content
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-22-
for the moment with this simple representation.
Note that at low flowrates the dependence is essen-
tially linear and that at flowrates above about 10
gallons per minute the curve becomes concave upwards.
All flowrates in the range of experimental meas-
urements correspond to Reynolds' numbers less than
2100, but because the entrance length required for
complete development of a parabolic velocity profile
is from one to two orders of magnitude larger than
the length of each short tube, the discharge charac-
teristics are dominated by entrance effects. A sim-
ple calculation (Appendix VIII.D.I) illustrates that
the slope of the linear portion of the experimental
curve is about 20 times what one would predict using
Poiseuille's law. The predominant effects are the
contraction of the entering jet and the shear of
the tube wall. A discharge-loss coefficient defined
for this case would be expected to have a value in
the range 1.5 to 2.7, with a weak inverse power-de-
pendence on Reynolds' number. This agrees well with
the data at the high-flowrate end of the curve.
If there is no wetting of the bottom of the tray,
liquid will drain as single drops until the inertia
of the liquid flowing through the holes is sufficient
to distort the drop shape and produce an elongated
jet. The critical Weber number for this drop to jet
transition is about 6.8. For the tray insert under
consideration, this critical value corresponds to a
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volume flowrate of 11.3 gallons/minute.CAppendix
VIII.D.2.)
Above this flowrate jets have been observed to
form at all holes; however, intermittent jetting has
also been observed at flowrates considerably below
this critical value. This seems to imply that local
hole velocities may be large enough to give rise to
jet formation, even though the average velocity, based
on the entire perforated area, is not. In effect,
we are saying that at any given time, only a frac-
tion of the available holes is passing liquid. The
data seem to indicate that this effective fraction
increases as flowrate is increased, thus tending
to moderate changes in the velocity head, and yield-
ing the nearly-linear dependence observed at low
flowrates. The sealing and unsealing of holes is
believed to be brought about by the fluctuations
of liquid head at various locations on the tray, with
liquid flow occurring at those perforations where
the instantaneous local liquid head is sufficient to
overcome surface tension.
A further complication results from the fact
that liquid drops or streams falling onto a free
liquid surface entrain air bubbles into the bulk
liquid phase. Some of these pass on through the
perforations with the downflowing liquid and enter
the gas space below. This phenomenon produces a
small steady flow of the gas phase down the column
-------
-24-
during the liquid flow period.
If there is good disengagement of gas from liq-
uid in the bottom of the column, then in the complete-
ly vented mode all air that passes down through the
bottom stage by this entrainment mechanism must
exit through the bottom venting port. Hence, to
measure the rate of air entrainment one needs to meas-
ure only the flow rate of air through the vent.
Since rotameters and soap-film meters were found to
be unsatisfactory flow-measuring devices due to ex-
cessive pressure drops, a scheme was devised for
mixing a small metered flowrate of CC>2 with the
main vent air stream. The air flowrate was calcu-
lated from the known CC>2 rate and the mole fraction
of C02 in the mixture, determined chromatographically.
The results are shown in Figure 5.
This plot shows air entrainment rate in cc./
sec. as a function of liquid flowrate in both gallons/
minute and cc./sec. The curve is seen to be "s"-
shaped, with an apparent point of inflection in the
vicinity of 8 gallons/minute. The interpretation of
such data is quite complicated, since at least two
competing processes are at work. Increasing the
liquid flowrate by itself probably tends to increase
the entrainment rate by increasing the total volume
of air bubbles in the liquid phase, as well as by in-
creasing the size of the largest bubble whose terminal
rise velocity will be exceeded. On the other hand,
the accompanying increase in liquid holdup makes it
-------
-25-
FLOW &ATE (c.c./sec.)
o /oo zoo
300
^so
lg ZOO
K.
§ /so
!
£ /OO
so
o
i i
I I I I
Sr£AD/ - Sr/t re
I I I J I
O
L/Q U/D
-------
-26-
more difficult for bubbles to penetrate to the tray
surface. The latter effect is more important at
high flowrates and probably accounts for the de-
creasing slope at the upper end of the curve. These
data indicate a 20 to 30 percent aeration of the
liquid flowing down through the holes. If the mix-
ture of air and water is a uniform froth of small
bubbles in the liquid it can be shown (Appendix VIII.
D.3.) that the pressure drop for flow through the
perforations is from 30 to 40 percent larger than
for the same volume rate of non-aerated liauid.
3. Variable-venting Experiments.
Another simple calculation (Appendix VIII.D.4)
shows that pumping air at a rate of 200 cc./sec. in-
to a sealed volume of 5.5 liters maintained at room
temperature results in an initial rate of pressure
change of 1 inch of water every 0.07 second. Thus,
the hydraulic conseauences of air entrainment can
be quite significant in an unvented column. One can
imagine an experiment in which the bottom vent line
is gradually closed. If air is carried down by the
liquid into this closed gas space the pressure in-
creases, the liquid head builds up, and as a result,
the rate of influx of air is diminished. At steady-
state the flowrate of air into the sealed section
must be balanced by an equivalent outflow, either by
entrainment of small bubbles in the exit stream or by
air bubbling up through the perforations.
Such variable-venting experiments have been car-
-------
-27-
ried out by adjusting a valve in the vent line from
the bottom section. Figures 6 and 7 show the effect
upon tray liquid head and interstage pressure differ-
ence of decreasing the air venting rate from its
maximum value to zero. Pressure difference and
clear liquid head in inches of water have been
plotted against air venting rate in cc./sec., for a
range of different liquid flow rates. When the vent
valve is fully-open the pressure difference is zero
and the liquid-head lies on an envelope corresponding
to the "s" shaped entrainment curve discussed pre-
viously. As the venting rate is reduced to zero the
pressure difference and liquid head increase, gradu-
ally at first, with a final sharp rise as the vent-
ing rate is decreased below about 15 cc./sec.
The difference between the clear liquid head
and the differential pressure constitutes a nei: pres-
sure difference favoring liquid flow. In Figure 8
values of this quantity are plotted against air vent-
ing rate. These data indicated that at a fixed liq-
uid flowrate the net difference decreases as the vent
is closed, even though the individual values of liq-
uid head and gas-phase pressure difference are in-
creasing. In effect, this says that it is "easier"
to force a given flowrate of liquid down through a
tray into a sealed vapor space than into a vented space,
whereas one would intuitively expect that the net
difference of head should be about the same in both
-------
2.2
o.z -
S^/f£oi
V
O
A""
O
a "
•
A
i.'fu'C/ Put' (tjpm)
4 .36
' S-5
7 . Z5
...... ..8.73. ...
/O.Z
// ,6S
/•3. /
/OO
A /x?
/so
&Are
2.00
(c.c.,
I
10
00
I
-------
-29-
OJ -
Syrneo i>
V
O
A
O
D
«
A
Liquid Rote. Cop?»)
4.36
S-.B.
7.2S
8 .73
/0 .2,
/ / .63-
/ 3 . /
O
/CO
ZOO
3 CO
./sic.)
8
-------
-30-
cases.
Two factors seem to be at work here. First, as
the vent valve is closed and liquid head is allowed
to build up, the fraction of air bubbles in the liquid
draining through the holes is cut down dramatically.
As pointed out earlier, decreasing air entrainment
decreases the velocity-head of the flowing gas-in-
liquid dispersion, so that one might expect to ob-
serve a drop in the required pressure difference.
Second, when the column section is sealed, pressure
fluctuations in the gas phase increase in magnitude.
It can be shown that such fluctuations act to over-
come the stabilizing influence of surface tension,
and allow larger flow rates of liquid with the same
average net pressure difference. More air-entrain-
ment data are needed at different flowrates and liq-
uid heads in order to draw quantitative conclusions
about the relative importance of these factors (Ap-
pendix VIII.B.). It is anticipated that experiments
with different tray inserts will provide the necessary
information.
4. Vented Cyclic Results.
One of the reasons for investigating the clear
liquid head vs. liquid flow relationship at steady-
state was that the results were expected to be appli-
cable to a vented cyclic column. It was hoped that
the removal of gas-phase pressure differences would
stabilize liquid flow, so that liquid holdups would
-------
-31-
be about the same on all stages and eaual to the
values predicted from steady-state data. In fact,
although the expected stabilization of flow was
achieved, clear liquid heads measured in controlled-
cyclic operation were found to be consistently lower
than those measured at steady-state. A comparison
of results for the two cases is shown in Figure 9.
The plot shows clear liquid head in inches of water
as a function of liquid flow rate in gallons per min-
ute. The solid circles denote steady-state results,
while the open symbols represent cyclic data at var-
ious liquid flow period lengths, ranging from
1.2 to 9.6 seconds. The gas flow period and air flow
rate were held constant at 5-4 seconds and .9 ft /sec.,
respectively.
The clear liquid heads measured for the cyclic
case are observed to be always less than the corres-
ponding values at steady-state, regardless of liquid
flow rate or length of liauid flow period. The effect
of liquid flow period length is most pronounced at
the high liquid rates where the shorter liquid flow
periods give rise to smaller clear liauid heads. The
near-coincidence of the curves for liquid flow periods
of 5.4,7.8, and 9.6 seconds seems to indicate that
as LFP length is increased indefinitely an asymptotic
upper limiting liauid head is approached at a given
flowrate.
There is good evidence to support the contention
that weepage during the vapor flow period may be part-
-------
-32-
-------
-33-
ly responsible for the observed discrepancies. Grad-
ual shifts in the sightglass levels have been noted
during the course of a cycle, indicating a decrease
of liquid level during the vapor flow period. At
large holdups or gas flowrates a side to side oscil-
lation of the froth has been found to produce brief
spurts of liquid flow near the side walls. If liq-
uid flow occurs during part of the VFP, the actual
flowrate per unit time will be reduced from the value
obtained by assuming flow only in the LFP, and this
reduction in average flowrate will be reflected in
a smaller value of clear liquid head. Since the
total liquid weepage per cycle is equal to the aver-
age weep rate multiplied by the gas flow period
length, the correction for weepage is largest when
the VFP/LFP ratio is large, and it is under these
conditions that the maximum discrepancies in liquid
holdup are observed. However, the weepage hypothesis
does not seem to account for the differences between
the cyclic and steady-state results at low flowrates.
At present time experiments are being undertaken
to measure the effect of varying LFP time, VFP time,
and the VFP/LFP ratio, in order to determine the ex-
tent of their interactions. It would also be instruc-
tive to eliminate liquid oscillations by appropriate
baffling and to observe the effect upon the clear
liquid head results. It is believed that physically
-------
-34-
meaningful corrections can be applied to the
steady-state results to predict the hydraulic be-
havior in vented cyclic operation.
5. Sealed Steady-State and Cyclic Operation
Although no reliable quantitative studies have
been made either at steady-state or in the controlled-
cyclic mode with all column sections sealed, some
qualitative observations are available. It has been
noted that although the liquid holdups and interstage
pressure differences on the lower trays were similar
in magnitude, those on the top plate were always found
to be much larger, sometimes by factors of three or
four. In the cyclic mode, this problem became extreme-
ly serious, and holdups of as much as 6 inches of
clear liauid have been observed on the top stage at
liquid flow rates around 2 gallons per minute.
It is feared that many of the problems encountered
are the result of liquid-flow end effects. At the
top of the column the liquid phase is introduced on-
to the highest tray through a perforated-pipe distri-
butor. The .liquid streams leaving the distributor
have velocities much higher than the drainage veloc-
ity from a tray insert, and this may result in a
greater entrainment of air bubbles into the gas space
below the top plate. It has already been mentioned
that column operation may be sensitive to the amount
of air entrained in the drain-line liquid. This is
undesirable, since drain-line configurations may dif-
-------
-35-
fer considerably from one unit to the next, yet it
is desired to make the hydraulic performance as uni-
form as possible. A solution suggested by Gerster
and Scull (6) was to vent the bottom section through
a valve. Clearly, if the rate of air flow through
the partly-opened valve is considerably larger than
that through the drain line, the buildup of pressure
is under the control of the operator and not at the
mercy of spurious effects due to column drain config-
uration. In an actual column the vent line would
probably be connected from the bottom section of the
column to the top.
Considerations of good column design seem also
to favor some sort of variable-venting scheme to
increase the liquid residence-time on a tray. Cal-
culations (Appendix VIII.C.5) show that for a com-
pletely-vented tray with 1/8-inch holes and 10%
free area the mean liquid residence time is around
0.6 second. Liquid flow period lengths of this or-
der are too small to be of much commercial interest,
so it is desirable to find some means to increase the
liquid holdup at a given liquid flow rate. This can
be accomplished by reducing the fractional perforated
area, but a reduction in free area will also increase
the pressure droo of the gas during the VFP. One
solution to this problem is to allow the pressure-
difference in the gas phase to oppose the drainage
of liquid through the holes; moreover, it may be
-------
-36-
possible to control tray liquid holdups throughout
the column by manipulation of the venting valve in
the bottom section.
Another problem with vented cyclic operation is
that to obtain a ratio of molal gas flow rate to molal
liquid flow rate of order unity, the gas flow period
must be twenty to fifty times as long as the liquid
flow period. With such large "£/^ ratios, it is
feared that weepage of liquid during the vapor flow
neriod would become prohibitive. This problem would
be made much less serious by partial venting to de-
crease the allowable LFP liquid rate.
B. Liquid Mixing.
Although no in-rig experiments on axial liquid mix-
ing have been conducted, the probes, bridge circuits,
and recorder have been tested by performing a simple..
stirred-tank residence-time study and observing the decay
of solution conductivity with time. The value of mean
residence-time determined from the recorded exponential
curve was found to be in excellent agreement with that
calculated from the known flowrate and tank volume.
Another experiment was conducted to assess the effect
of noise due to bubbles in the liquid phase upon the meas-
ured voltage signal. It was found that severe.high fre-
cruency fluctuations in voltage signal made measurements
quite difficult, so that filtering of the response was
considered necessary for accurate measurements. Accord-
ingly, a filter box was constructed, containing two iden-
-------
-37-
tical 2nd-order, low-pass, operational amplifier circuits
designed to attenuate the noise frequencies by a factor
of 100 with no appreciable influence upon the form of the
measurement signals. It was found that this arrangement
successfully eliminated bubble-noise, but that the time-
average electrical conductivity of the aerated liquid was
lower than that for clear liquid having the same salt con-
centration.
This dependence of solution conductivity upon degree
of aeration of the liquid phase is unfortunate in that it
necessitates careful calibration of the probes under the
same hydraulic conditions to be used in a particular run.
The voltage signals corresponding to the maximum and mini-
mum salt concentrations in the aerated licruid must be known
accurately in order to normalize the response signals
properly. On the other hand, the measurement of depar-
tures from non-aerated behavior gives an estimate of void-
age fractions which are difficult to measure accurately
by comparing froth height with clear liquid height.
C. Mathematical Simulation Models:
1- Hydraulics.
In order to facilitate the correlation of data
for liquid flow with an opposing gas-phase pressure
difference and non-negligible surface tension effects
a semi-empirical flow model has been developed. Pre-
vious investigators have suggested that the behavior
of a particular perforation on a sieve plate is deter-
-------
-38-
rained by the instantaneous net pressure difference
across it, and that fluctuations in gas-phase pres-
sures and liquid heads may result in bubbling of gas,
draining of liquid, or bridging at any instant of
time and any location. Prince and Chan (15) attempt-
ed to cast this quantitative picture into a very
simple mathematical representation, which assumed
that superimposed upon the mean pressure difference
was a small fluctuating pressure having the value
+ Afy at holes which were bubbling gas and - A.Pp at
holes which were draining liquid. The possibility
of bridging with no flow of either phase was neglect-
ed. The fraction of holes bubbling was then select-
ed to minimize the pressure-fluctuation,^^.
The principal shortcoming in this approach was
that it failed to include any information on the
physical processes causing the fluctuation. As a
result, it is found that an additional equation
must be supplied in order to predict gas and liquid
flow rates from the mean pressure difference. Also,
since bridging of holes is commonly observed in the
case of trays with perforation-diameters of 1/4-inch
or smaller, the neglect of this effect could intro-
duce serious errors.
These difficulties can be overcome by substitu-
ting for the minimization-of-pressure-fluctuation
condition, the physically realistic assumption that
pressure-fluctuation amplitudes are random quantities
-------
-39-
having some distribution of values about a zero mean.
If the spread of the distribution is sufficiently
large, it is possible to have finite probabilities
for each 6f the three possible states. From momen-
tum-balances for gas and liquid flow, one may deduce
"expectation values" for the flowrates if the param-
eters of the distribution are known, or, for a given
distribution shape, the parameters of the distribu-
tion may be deduced from data. The details of .the
model equations and their use are discussed in Appen-
dix VIII.B.
Thus far it has been virtually impossible to use
the model for data interpretation, not because of any
inherent shortcoming in the mathematical representa-
tion, but because the data obtained from the single
tray insert used have not been sufficiently conclusive
to permit evaluation of all necessary input quantities
As mentioned in the discussion of hydraulic results,
the actual discharge relationship has been obscured
by surface-tension effects at the low-flowrate end
and by air entrainment at all flowrates. For inter-
pretation of the variable-venting results, the depen-
dence of air entrainment rate upon both liquid flow-
rate and clear liquid head must be specified. For
a single tray insert these two variables are strong-
ly coupled and the individual dependences are prac-
tically impossible to determine unless the exact form
of the desired correlation is known.
-------
-40-
2. Mass-transfer Model with Mixing.
The complete mass-transfer simulation model of
Gerster and Scull ( 6 ) has been programmed for use
on an IBM 1130 computer. The program can handle
short (less than 6 stages) columns with computation
times on the order of 30 seconds; conversion for use
on larger,.faster machines, such as the IBM 371 sys-
tem, is straightforward and will essentially remove
all limitations on total core storage and solution
time.
The dimensionless differential and algebraic
equations are summarized in Appendix VIII.C., together
with some typical simulation results. The differen-
tial equations are integrated numerically using a 4-
th order Runge-Kutta algorithm, and stable cyclic sol-
utions are obtained by simulating the startup of a
column from an assumed set of initial conditions until
the periodicity conditions are satisfied within a cer-
tain tolerance. Much of the programming and all of
the simulation experiments were carried out by Miss
D. Jarmoluk, a graduate assistant working on the pro-
ject.
The program is prepared to generate results, once
correlations for hydraulic variables and mixing param-
eters have been developed.
VI. Direction of Current and Future Work.
As suggested in the discussion of results, it is anticipated
that repetitions of some of the experimental studies using trays
of different free area fraction will provide the needed infor-
-------
-41-
mation on the intrinsic head-vs.-liquid flowrate and air en-
trainment rate vs. liquid flow rate and head relations. Once
such information is available, the flow model can be brought
into play to assess the relative effects of liquid aeration
and pressure fluctuations upon the liquid heads and pressures
measured in partially-vented steady-state experiments.
Currently, a series of experiments is being conducted
to measure the quantitative effects of changing flow-period
lengths in vented cyclic operation. It is hoped that the weep-
age mechanism postulated to explain some of the observed de-
partures from steady-state hydraulics can be made more quanti-
tative, and that other contributing phenomena can be postulated
and verified by examination of the data.
The electronic circuits developed for the salt-tracer
mixing studies and the associated hardware for two-supply
cyclic operation await their first in-rig test. It will prob-
ably be found desirable to adjust the rates of air venting
from the various sections, in order to increase the liquid
residence time to around 2 seconds for experimental conven-
ience. It may be found that tracer experiments also give in-
formation on weepage rates and other hydraulic phenomena which
will be useful in interpreting the previous hydraulic data.
Once a model for column operation is formulated, it should
be tested by carrying out absorption experiments in a well-
designed cyclic column as suggested in the section on long-
term objectives. The results of such a study should be helpful
in suggesting alternate design configurations and modes of oper-
ation.
-------
-42-
VII. Summary.
Hydraulic experiments carried out in a 3-tray cyclic col-
umn, using air and water as the counterflowing phases, have
indicated the form of the gas- and liquid-phase discharge re-
lationships , the importance of air-bubble entrainment in the
liquid phase, and the effect of flow-period lengths in produc-
ing departures from steady-state liquid heads in vented cyclic
operation. Equipment for measurement of mixing parameters
has been assembled and tested and computer programs and sim-
ulation models have been developed for interpretation of ex-
perimental data and prediction of cyclic column performance.
It remains to complete the cyclic hydraulics and liquid mix-
ing studies, and to use different tray inserts to finalize
the steady-state hydraulic relationships.
-------
-43-
VIII. Appendix
A. Models for Axial Liquid Mixing in Cyclic^ Operation.
1. Liquid-phase Mixing from Evolution of Vertical
Concentration-Distance Profile.
If the mixing in the liquid phase (excluding the
falling liquid streams) is assumed to be described
by a "dispersion coefficient", then in analogy with
Pick's law,
(A.1-1)
in normalized coordinates. The boundary conditions
suggested by Danckwcrts apply:
1 • c - O a.£ & " O _> -£0*- a// J" .
3. = O
The infinite-series solution to this problem has the
form,
where,
•"• / -^ X" ^//. * ~~9 ~\~ I
,.1-4)
"^~~" ^ ~PC..
r
- / /fe
-------
/
= 777
/O
^ /
/
s^'
F/GURE //
-------
-45-
y is the n-th eigenvalue solving,
-
art, <-^- ~ ?*- / ' (A. 1-5)
and » ta-KT' . (A. 1-6)
The predicted functional form can be fitted, in
a least-squares sense, to data on concentration vs.
distance and time, by choice of the Peclet number,
Pe.
2. Overall Mixing Parameter from Measurements of
Well-Mixed VFP Liquid Composition.
Let, ( ZCeJ- = (& . (A. 2-1)
( ^ -
Then, for cycle fl,
\ =
J
Ct) , (A. 2-2)
—^ ~y J } = /'(/) . (A.2-3)
V % j? — x /
For periodicity,
: X/o (A. 2-4)
s
(A. 2-7)
Hence, fft , _ _^^_ (A. 2-8)
~
-------
-46-
The mixing-model equations are,
(A.2-9)
= O
(A.2-10)
/
O
Hence,
(A. 2-11)
The solution is,
*
f°}= ] , >, -C-&J r /-^/ (A.2-12)
J /- ]Te. Sor --~ < /
(A.2-13)
from which ^" can be determined if r > / °
The case £*- /-"^ means that the mixing situation
cannot be distinguished from plug flow.
-------
-47-
B. Model for Gas and Liquid_Flows Through a Perforated
Plate.
It has been suggested that "simultaneous" flows of
gas and liquid through a perforated plate at steady-state
may result from pressure and liquid-head fluctuations
which bias the holes positively or negatively. Thus, at
any given hole any one of three possible conditions may
exist at any given time :
1. Bubbling of gas up through the hole.
2. Draining of liquid down through the
hole .
3. Bridging, with no flow of either phase.
If we may assume for simplicity that the pressure
drop of a continuous medium flowing through a perforation
is a constant multiple of the velocity-head based on the
cross-sectional area of the hole, the following approxi-
mate steady-state momentum balances may be written:
1. For the gas phase,
, (AP6 -^ JL} + Aty >
2. For the liquid phase,
3. For bridging,
. (B-3)
-------
-48-
The basic assumption upon which the flow model
rests can be stated as follows :
The pressure-fluctuation,^)/^, is a random variable,
in the probabalistic sense, whose density function,
0~) will be assumed to have the following properties
1. A mean of zero.
2. An index of deviation from the mean, 0~.
3. Symmetry about the mean; i.e.,
Before proceeding to derive the relationshios for
average gas and liquid f lowrates , it is advantageous to
cast the preceeding momentum balances into a more symmetri-
cal form. Thus ,
(B-5)
Now, define,
7~P) + Afy J (B.6)
* (B-7)
and =- (B_8)
Clearly, then,
1. For gas bubbling, £ > }
2. For liquid draining, £ <"
3. For bridging, - y £ / ^
' *-* (B-9)
-------
-49-
From the basic assumption it is easily shown that the
density function for the "net pressure difference," o ,
has a mean of a0 , an index of deviation of C~ , and sym-
metry about the new mean. It may be represented as
The probability of each of the three types of flow
behavior may now be calculated.
1. Probability of gas bubbling = ^3 '/
r °
/ /e> Ccf- ** J (^1-^)^/9(2-2.^ /J <2 ; (B-17)
and,
-------
-50-
Now, introduce the new variables,
= JL - Z0 .
" (B-19)
> 3- ~ P — do •
-*-" J
Then,
.^
- J
for gas flow. For liquid flow, let
- 3 - 7
~" Xv xO. x)
,*>- 7 (B-21)
= — f — A. £> *
whence it can be shown that
(fi-22)
If we let,
' (B-23)
(B-24)
\U&/ = T (. F ~ xC0y j
u
and,
(B-25)
One point of hydraulic data consists of experimental val-
ues for the quantities ,
Since we have two relations to determine <3~ , the "spread"
of the pressure fluctuation, we may select a
-------
-51-
Since S0 is a difficult quantity to measure experimentally,
however, the recommended procedure is to solve the two equa-
tions exactly for O~ and o0 , knowing \ && / and \t4^/-
The agreement of measured and calculated oa-values consti-
tutes a test of the predictive power of the model (using
the chosen density-function) while 0~is a single empirical
parameter which can be correlated.
The "flowrate correlation" is completely specified
by,
1. A probability density function, i& .
2. An empirical correlation for O~, the deviation
from the mean pressure-difference.
C. Mathematical Model for Mass-Transfer and Liquid Mixing
in a Cyclic Absorber.
The mathematical model equations originally suggested
by McWhirter and Lloyd (14) and modified by Gerster and
Scull (6) to account for the effects of axial liquid mix-
ing are summarized for a general stage as follows:
1. Gas Flow Period.
Assuming constant and equal liquid holdups, H,
and equal Murphree point efficiencies, -M , on all
stages; a constant molal gas flowrate, G* ; and a. di-
lute gas phase with Henry's law constant, m; an un-
steady state component material balance on the i-th
stage (see Figure 10) gives,
- J^/Vz/. - */ • ~)
// r«-' ^ (c-i)
-------
-52-
The composition of the gas stream leaving the stage
is related to the composition of the gas stream enter-
ing the stage and to the tray liquid composition by
the Murphree efficiency,
-/ (C-2)
*,=
7 x, x; - *ss-s '
2. Plug-Flow Conditions .
If during the liquid flow period a fraction
-------
-53-
a nd an overall balance on the entire tray holdup
gives for the average tray liquid composition,
(C-5)
The composition of liauid leaving the plug-flow
section at any time is,
, X+ <"?& J ^ ^ •
where , _ (/- irj ^
A. =
^.
4. Periodicity^ Conditions.
Compositions at the end of the LFP are related
to initial VFP compositions by,
%*o ~ X+ C ^1 ) . (C-7)
5 . Average Exit-Gas Composition.
In order to calculate the number of theoretical
stages to accomplish the given separation, it is
necessary to compute the average exit-gas composition.
For the plug-flow case, an overall solute mater-
ial balance for all flows during one cycle gives,
(C-9)
-------
-54-
and k and 1 are previously-defined integers,
For the generalized mixing model case,
where, -^ C7!) is defined by,
(C-ll)
6. Overall Column Efficiency.
Colburn ' s analytical solution for the number of
theoretical plates required to accomplish a given
separation is, (C-12)
The overall efficiency of the cyclic column is,
then,
=
7. Dimensionless Form.
The foregoing equations may be cast into dimen
sionless form by means of the following transforma
tions :
/<
0 =
/ £•/ & s
(C-15)
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For gas flow:
-*1 -
J
>'
For plug flow:
=/
&r -
For generalized mixing
>• -
/ .
(C-17)
(018)
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For average exit gas composition
in plug flow,
where,
or, for generalized mixing, (C-19)
where - (c"20)
with
>
For overall column efficiency
and
(C-22)
The overall column efficiency,
is a function of 5 dimensionless parameters. It has
become customary to illustrate this relationship for
plug-flow by plotting £0 as a function of & , yield-
ing a curve with cusp-like maxima at the first N posi-
tive integral values of y . Such a plot is shown
in Figure 12 , for /^= O . For values of the mix-
ing parameter f> O , the solution curves break off
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-57-
^oo
/90\-
/80 - ~
-n~ /.o
A * /.O
-------
.OOO/
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-59-
from the plug-flow solution at fi = f/ — fj and show
flatter and less well-defined maxima at lower values
of overall efficiency.
Colburn's equation is normally plotted as log
normalized 'exit-gas composition vs. log( 1+ number
of theoretical trays) , with absorption factor as a
parameter. A similar plot for the cyclic case illus
trates the improvement in separation. (Figure 13.)
D. Sample Calculations .
1. Comparison of Experimental Slope of Liquid-Head
vs. Flow Rate with Hagen-Poiseuille Law:
_
Q =
D - 1/8 inch = 1/96 ft.
L = 1/4 inch = 1/48 ft.
AH = Ah (inches) /12 ft.
pL- 62.4 lbs/ft.2
g = 32.2 ft/sec2
^L= 6.72 x 10~4 lb/(ft.sec)
Q = q (gallons/min.) x 2.228 x 10~3 ft3/sec
Therefore ,
0}
^ ' 0
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2. Calculation of Flowrate for Critical Weber Number
ThUS'
NOW,
N = 230 holes
D = 1/96 ft.
g = 32.2 ft/sec2
0~= .00495 Ib/ft,
jO± =62.4 Ib/ft3
So that, C
Scr/'t - I*. 3J-& tSe cw£
The reported value of >4^ cr/£ = ^.8 ; hence,
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-61-
/ 2- (D.3-5)
or, a 33% increase in pressure drop over that for
clear licruid.
4. Initial Rate of Pressure Rise Due to Air Flow
Into a Sealed Column Section.
Assuming isothermal pressure increase,
(D.4-1)
' . (D.4-2)
If the gas enters at the same conditions,
(D.4-3)
2. ) (D.4-4)
For a given small ^/^
P = 1 atm. = 407.0 inches of H->O
AP = 1 inch of H20
V0 = 5500 cm3
vin =200 cm3/sec.
/ / /
.'. A -6 = ( yoj-( 200
Initial pressure increase is 1 inch of 1^0 in .068
sec.
5. Estimate of Liquid Residence-Time and Ratio of
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-62-
Flow-Period Lengths .
a. The residence-time of liquid on a tray may
be estimated from the measured values of liquid
flow rate and clear liquid height as follows :
A 7-rcy . (D . 5-1 )
^ (D.5-2)
s
Test case: AH » 1 inch H20 = 1/12 ft H,0
Qlia " 13 9Pm - .0290 ft3/sec
ATray " 0.196 ft2
b. To estimate the order of magnitude of the
'7G/'?2. ratio for vented cyclic operation, we
make the following assumptions:
(1) Dry-plate pressure drop ^clear liquid
head ^liquid velocity head.
.-. f& U& * ~ ^ ^.^ . (D.5-3)
(2) Ratio of average molal gas and liquid
flowrates of unity.
/" X^L £>£* *^2j
/ ^ - (D.5-4)
From which,
~- ^ ^~ (^ ^) (D'5-5)
But, ''
(D.5-6)
(D.5-7)
With „ i i_/,t_ ,
29 Ib/lb-mole
ML * 18 Ib/lb-mole
PL - 62.4 lb/ft3
PG « .075 Ib/ft3
-72
-------
-63-
E. Nomenclature:
1. Appendix: A.I.
A = n-th coefficient term of series expansion, Eqn.
*" (A.1-3)
C = tracer concentration normalized on [0,1].
n, - subscript for terms in infinite-series solution
Pe = Peclet number
4^ = n-th eigenvalue, solution of Eqn. (.1-5)
$Z> = phase angle for n-th eigenvalue, defined by
Eqn. (A.1-6)
0 - dimensionsless time variable
J> = dimensionless length variable
2. Appendix:
R = reduced concentration ratio, defined by Eqn. (A.2-7)
% - average tracer concentration at any time
Z/f - tracer concentration in LFP#1 feed.
ytif = tracer concentration in LFP#2 feed.
•%j0 = tracer concentration at start of LFP#1
Kza = tracer concentration at start of LFP#2
ft"'= well-mixed fraction
& = fractional tray holdup dropped per cycle
& = dimensionless time
= reduced average tracer concentration
f = reduced tracer concentration in well-mixed cell.
p - reduced tracer concentration in stream leaving
plug-flow section.
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-64-
3. Appendix: B.
C/g = gas-phase discharge-loss coefficient
C^ = liquid-phase discharge-loss coefficient
h = clear liquid head on tray
A Pj = randomly-distributed pressure-fluctuation
APg = gas-phase pressure difference across tray
•A Ps = oressure difference due to surface-tension
required to form a drop or bubble.
T p = tray thickness
U^ = instantaneous gas velocity through hole
U^ = instantaneous liquid velocity through hole
= average gas velocity through holes
= average liquid velocity through holes
y = reduced average gas velocity
= reduced average liquid velocity
= dimensionless dummy intearation variable,
defined by Eq'n (B-19) and Eq'n (B-21)
/ = probability that a hole will bubble gas
£ = probability that a hole will drain liquid
= probability that a hole will bridge
= modified surface-tension term, defined by
Eq'n (B-8) .
= modified randomly-distributed net pressure
difference defined by Eq'n. (B-6) .
= modified average net pressure-difference,
defined by eq'n (B-7)
^ = reduced net randomly-distributed pressure-
difference
A0 = reduced average net pressure difference
= gas density
= liquid density
CT= index of deviation of random fluctuating
pressure from the mean
-------
p
-65-
= integration limit defined by Eq ' n (B-19) and
Eq'n (B-21)
= reduced surface-tension term
Appendix: C.
A = absorption factor defined by Eq'n (C- )
E0 = overall column efficiency
G = average molal gas flowrate for entire cycle
G = actual molal gas flowrate during the VFP
H = molal tray liquid holdup
k = lower-bounding integer for
1 = upner-bounding integer for
*• = average molal liquid flow rate for entire
cycle
m = Henry's law constant in mole -fraction units
N = number of stages in cyclic colujn
NC = number of theoretical stages in a conventional
column for same separation.
t = time variable
t/e= residence-time in plug-flow section
,2£g = average bottom liquid composition variable
-2%c = average tray composition in LFP on stage i
•^fc = tray liquid composition in VFP on stage i
s£*'o-= initial VFP liquid composition on stage i
/5**= composition of well-stirred cell in LFP on
stage i
2^/5^= composition of liquid stream from plug-flow
section in LFP on stage i
%.-r= entering liquid composition
&k = gas composition in entering stream
3f^ = gas composition in stream leaving stage i
& o
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-66-
= fraction of tray holdup assumed well-mixed
in LFP
= fraction of a tray holdup dropped per cycle
"^ =VFP length
1 = LFP length
' = reduced time variable
.= reduced bottom liquid-composition variable
j^j = reduced VFP tray liquid composition on stage i
j= reduced LFP average liquid composition on
stage i
ff/^ = reduced LFP well-mixed liquid composition
^ on stage i
^^ = reduced composition on liquid from plug-
flow section during LFP on stage i
= reduced entering liquid composition
& & = reduced gas compos: tion in entering stream
c^ = reduced gas composition in stream leaving
<-/ stage i
a"0(jf- - reduced average exit-gas composition
variable.
^1 = Murphree point vapor efficiency
5. Appendix: D.I.
D = hole diameter
g = gravitational acceleration
Ah = clear liquid head, in. of 1^0
-------
-67-
Appendix: D.2.
D = hole diameter
g = gravitational acceleration
N = number of perforations
/£= critical liquid rate for drop to jet
transition, gallons/min.
= critical liquid rate, ft-Vsec.
= critical liquid velocity
er/'A = critical Weber number
= liquid density
O~' = air-water surface tension
Appendix: D.3.
/r = flowrate of air through perforations by
entrainment in the liquid phase.
s^o^ liquid flow rate
= velocity of aerated liquid through holes
.= velocity of clear liquid through holes
= volume fraction of gas in gas-liquid mixture
= density of aerated liquid
= density of gas phase
= density of clear liquid
8. Appendix: D.4.
n = number of moles of gas in closed volume
P = absolute pressure of gas in closed volume
P 0 = initial absolute gas pressure
AP = small pressure increment
R = ideal-gas law constant
t = time-variable
-------
-68-
-At = small time -increment
't* = rate of gas inflow
=5 volume of gas space
Appendix: D.5.
^Tray = column cross-sectional area
A H = clear liquid head
' G - molecular weight of gas
= molecular weight of liquid
. = licruid flow rate
^
= gas velocity through holes
=liquid velocity through holes
= density of gas
= density of liquid
<% = gas flow period length
?Z = liquid flow period length
^e. = approximate mean liquid residence time
10. Special Abbreviations:
VFP - vapor (gas) flow period
LFP - liquid flow period.
&
-------
-69-
F. Literature Cited.
(1) Cannon, M. R. , "A New Type of Distillation,
Absorption and Extraction Column," Oil Gas J.,
51_, 268 (1952).
(2) Cannon, M. R., "Here's a New Liquid Extractor,"
Oil Gas J. , 5_4 (Jan. 23), 68 (1956).
(3) Chien, H. H., J. T. Sommerfeld, V. N. Schrodt, and
P. E. Parisot, "Study of Controlled Cyclic Distilla-
tion: II. Analytical Transient Solutions and Asymp-
totic Plate Efficiencies," Separation Science^ 1, 281
(1966)
(4) Garcia, A., and A. R. Bayne, "Gas Flow Through Sub-
merged Orifices," S. B. Thesis, Massachusetts Insti-
tute of Technology, 1965.
(5) Gaska, R. A., and M. R. Cannon, "Controlled Cycling
Distillation in Sieve and Screen Plate Towers,"
Ind. Eng. Chem., 53, 630 (1961).
(6) Gerster, J. A., and H. M. Scull, "Performance of Tray
Columns Operated in the Cycling Mode," ATChE J., 16,
108 (1970). '
(7) Holland, C. D. , "Multicomponent Distillation,"
Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1963.
(8) Horn, F. J. M., "Periodic Countercurrent Processes,"
Ind. Eng. Chem. , Process Design Develop. , 6_, 30 (1967)
(9) Lewis, W. K., Jr., "Rectification of Binary Mixtures,"
Ind. Eng. Chem., 28, 399 (1936).
(10) Lin, R. C., "Periodic Processes in Chemical Engineer-
ing," Ph.D. Thesis, Rice University, 1966.
(11) McAllister, R. A., P. H. McGinnis, and C. A. Plank,
"Perforated-Plate Performance," Chem. Eng. Scij. ,
9_, 25 (1958) .
(12) McKay, D. L., and H. W. Goard, "Crystal Purification
by Cyclic Solids Movement," Ind. Eng. Chem., Process
Design De ve1op. , 6_, 16 (1967}T
(13) McWhirter, J. R., and M. R. Cannon, "Controlled
Cycling Distillation in a Packed-Plate Column,"
Ind. Eng. Chem., 53, 632 (1961).
(14) McWhirter, J. R., and W. A. Lloyd, "Controlled
Cycling in Distillation and Extraction," Chem.
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-70-
Eng. Progr., 59, 58 (1963).
(15) Prince, R. G. H., and B. K. C. Chan, "The Seal
Point of Perforated Distillation Plates," Trans.
Instn. Chem. Engrs. (Australia), 4^3, T49 (1965).
(16) Robertson, D. C., and A. J. Engel, "Particle Separation
by Controlled Cycling," Ind. Eng. Chem., Process
Design Develop., 6_, 2 (1967).
(17) Robinson, R.G., and A. J. Engel, "An Analysis of
Controlled-Cycling Mass-Transfer Operations,"
Ind. Eng. Chem., 59^, 22 (1967).
4
(18) Schrodt, V. N., J. T. Sommerfeld, 0. R. Martin,
P. E. Parisot, and H. H. Chien, "Plant-Scale
Study of Controlled Cyclic Distillation," Chem.
Eng. Sci., 2£, 759 (1967).
(19) Sommerfeld, J. T., V. N. Schrodt, P. E. Parisot, and
H. H. Chien, ''Studies of Controlled Cyclic Distillation:
I. Computer Simulations and the Analogy with Conventional
Operation," Separation Science, 1^, 245 (1966).
(20) Tan, K. S., "A Study of the Dynamics of Cyclic
Fixed-Bed Ion Exchange," Ph.D. Thesis, University
of Toronto, 1969.
(21) Wade, H. L., C. H. Jones, T. B. Rooney, and
L. B. Evans, "Cyclic Distillation Control,"
Chem. Eng. Progr., 65, 40 (1969).
(22) ZelfeU, E., "Zur Stabilitat von Siebboden," Chem.
Ing. (Tech.', 3!9_, 433 (1967).'
(23) Zenz; F. A., "How to Calculate Capacities of
Perforated Plates," Plates," Petroleum Refiner,
3_3_ (2) , 99 (1954).
(24) Zenz, F. A., L. Stone, and M. Crane, "Find Sieve
Tray Weepage Rates," Hydrocarbon Processing, 46,
138 (1967).
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