LEANj WATER POLLUTION CONTROL RESEARCH SERIES #	17020DZ011/70
CARBON COLUMN OPERATION
IN WASTE WATER TREATMENT


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WATER POLLUTION CONTROL RESEARCH SERIES
The Water Pollution Control Research Series describes
the results and progress in the control and abatement
of pollution in our Nation's waters. They provide a
central source of information on the research, develop-
ment, and demonstration activities in the Water Quality
Office, Environmental Protection Agency, through inhouse
research and grants and contracts with Federal, State,
and local agencies, research institutions, and industrial
organizations.
Inquiries pertaining to Water Pollution Control Research
Reports should be directed to the Head, Project Reports
System, Office of Research and Development, Water Quality
Office, Environmental Protection Agency, Room 1108,
Washington, D.C. 20242.

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CARBON COLUMN OPERATION IN WASTE WATER TREATMENT
by
Syracuse University
Syracuse, New York 13210
for the
WATER QUALITY OFFICE
ENVIRONMENTAL PROTECTION AGENCY
Project #17020 DZO
November 1970

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EPA Review Notice
This report has been reviewed by the Wat^icatioTl
Quality Office, EPA, and approved for pub
Approval does not signify that the conten
necessarily reflect the views and po iciej
the Environmental Protection Agency> nor
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ABSTRACT
A mathematical model has been devised to simulate the adsorption and
filtration of waste water in an isothermal column packed with granular
activated carbon. The adsorption process is considered to be controlled
by a combination of liquid phase diffusion and intraparticle diffusion
which can be approximated by a solid phase rate expression based upon
Glueckauf's linear driving force. The filtration rate equation is assumed
to be the same as that of filtration of clay suspension in a carbon bed,
which was investigated experimentally as a companion study of this work
In addition to adsorption and filtration, the effects of column backwashing
and carbon regeneration are included in the model. A newly developed
algorithm (discussed in detail in Part IV of this final report) is used
for the numerical integration of the pertinent characteristic normal
hyperbolic equations. With this algorithm, an industrial column of 20 feet
height operating a one-hundred day period can be simulated with less than
10 minutes of IBM 360/50 computer time giving two decimals or better
accuracy.
Based on this model, a simulation program is prepared and coded in
FORTRAN IV to be run on the IBM 360/50 level G compiler. A uniqufc feature
of this program is the clear separation of calculation framework and model
for the column behavior. Thus, it is possible, for example, to adapt the
present program to a variety rate expression and adsorption isotherm, which
are not considered in the present model. This is especially important in
view of the incompleteness and uncertainty about our understanding of the
carbon contacting process in waste treatment and the likely new discovery
to be made in the future.
This report was submitted in fulfillment of Project Number 17020DZ0
under the partial sponsorship of the Water Quality Office, Environmental
Protection Agency.

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TABLE OF CONTENTS
Page
Table of Contents 				i
List of Tables 		ii
List of Figures 		ii
Chapter I - Introduction 				1
Chapter II - Carbon Column Operation in Waste Water Treatment -
A General Consideration 		3
Chapter III - Formulation of Model 		6
(III-A) Modelling Consideration - Waste Water System 		6
(III-B) Modelling Consideration - Packed Column 		7
(III-C) Adsorption Effects 		9
(III-D) Biochemical Effects 		11
(III-E) Filtration Effects 		12
Chapter IV - Mathematical Equations Describing the Itynamic Behavior
of Column Operation 		13
(IV-A) Basic Equations 		13
(IV-B) Adsorption Rate 		13
(IV-C) Biochemical Reaction 		15
(IV-D) Filtration Rate 		l6
(IV-E) Interaction Between Adsorption and Filtration 		IT
(IV-F) Effect of Backwashing		IT
(IV-G) Effect of Regeneration 		18
(IV-H) Summary of Equations and Associated Conditions ....	19
Chapter V - Simulation Program 	 		2h
(V-A) Computation Algorithm 	 		2k
(V-B) Program Structure 		25
Chapter VI - Simulation Results 			33
(VI-A) Effect of Increment Size 		33
(VI-B) Comparison with Filtration Results 		^0
(VI-C) Effect of Solid Diffusion and Liquid Phase Mass
Transfer on Adsorption 		^2
(VI-D) Adsorption and Filtration Profiles Over a Hundred
Day Period 		^3
(VI-E) Effects of Regeneration and Number of Sections Over
Extended Periods 		M
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TABLE OF CONTENTS (Cont'd.)
Page
Chapter VII - Discussion, Conclusions and Recommendations 		1+7
(VII-A) Usage of the Simulation Program with Model I 		1+7
(VII-B) Short-Term Improvements for Model I 		1+9
(VII-C) Long-Term Simulation Improvements 		50
Literature Cited 		51
Nomenclature 		55
Appendix A - Fortran Listing of Simulation Program 		57
Appendix B - Development of Algorithm for Numerical Computation of
Semi-Linear Hyperbole Equations 	 69
LIST OF TABLES
Table 5-1 List of Subroutines of the Simulation Programs 		28
Table 5-2 List of Variables and Parameters 		31
Table 6-1 Model Input Parameter in Simulation Tests 		3^
Table 6-2 Test on Increment Size (Adsorption Increments)
Data Set (l) 	 38
Table 6-3 Test on Increment Size (Filtration Increments)
Run Sequence (2) 	 39
Table 6-1+ Comparison with Filtration Results 	 1+1
Table 6-5 Backwash Cycles 	 1+1+
Table 6-6 Effect of Number of Reactions 	 1+5
LIST OF FIGURES
Fig. 5-1 Computer Simulation Flow Chart - Main Program
Fig. 5-2	Computer Simulation Flow Chart - Calculation
Subroutines
Fig. 6-1	Effect of Solid Diffusion Coefficient on C
Fig. 6-2	Effect of Solid Diffusion Coefficient on Q
Fig. 6-3	Effect of Liquid-Phase Mass Transfer Coefficient on C
Fig. 6-1+	C-Profiles over 100 Days
Fig. 6-5	Q-Profiles over 100 Days
Fig. 6-6	Y-Profiles over 1+ Days
Fig. 6-7	S-Profiles over 1+ Days
Fig. 6-8	P-Profiles over 1+ Days
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I. INTRODUCTION
The use of activated carbon for the removal of contaminants has been
known for a long time and its use has been traced back to almost three
thousand years ago [H-3]. In terms of water purification, activated carbon
has "been used primarily for the removal of odor and color even though some
pioneer work was carried out as early as 1931 [G-l, H-2, B-l] in which the
granular activated carbon was used for sewage treatment.
The presence of excessive refractory material in lakes and rivers
caused much anxiety on the part of the public during the early part of
i960. The concern prompted renewed interest in the use of activated carbon
for the removal of organic contaminants. Weber and Morris [W1] conducted
batch experiments on the adsorption of ABS (Alkyl Benzene Sulfate) onto
granular activated carbon and feasibly studies involving column operation
with secondary effluent were made by Joyce and Sukenik [J-l] and Bishop
et al [B-3]. At approximately the same time, large scale pilot plant and
demonstration plant work of treating waste water with granular activated
carbon were initiated (and still in progress) at several places, notably
Lake Tahoe [C-9, C-10] and Pamona of Los Angeles County Sanitation District
[E-l, P-3] under the auspices of the Federal Water Quality Administration.
The application of carbon treatment in these studies is considered as a
critical step in the tertiary treatment process for the removal of organic
contaminants. This removal is deemed essential for complete water
renovation as well as necessary preparation for further treatment such as
reverse osmosis.
A most significant development in the application of carbon treatment
for waste water has taken place recently. In their demonstration plant
work at Washington, D.C., a new process named IPC (Independent Physical
Chemical Treatment) was developed, in which the raw sewage is treated in
a process consisting of clarification, filtration, ion exchange and carbon
adsorption. The quality of the IPC process is found to be superior to
that obtained from conventional-tertiary treatment. The IPC treatment gives
better removal of contaminants of every category. In addition, there are
two distinct advantages: (l) The land requirement of the treatment plant
based on IPC process is far less than that of the conventional plant.
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This is most important for some metropolitan areas where available land is
already scarce, (2) The IPC process can be operated at fairly low
temperatures while the conventional plant which requires biological treat-
ment is inevitably temperature-dependent, The development of IPC would,
therefore, be most important in the area such as Alaska and the Artie
region where the waste disposal problem poses the most severe challenge to
the regions' future development. Beyond these applications, carbon adsorp-
tion can be applied for industrial waste treatment. Experimental
evidences abound in literature which indicates the successful removal of
organic substance such as phenols, fatty acids, insecticides and pesticides
by carbon adsorption.
In spite of the very promising feature of the carbon process for
waste treatment, many problems remain to be solved before its application
can be put in general practice. One of the major concerns is its relatively
high cost, which has been estimated to be varying from 10 ^ 15$ per thousand
gallons as a tertiary treatment step [CT] Other than the possibility of
obtaining carbon at a lower cost through the improvement of its manufacturing,
a significant reduction of the cost can be achieved by the improvement of
the process itself. This requires further pilot plant work as well as
process simulation.
The object of the present study is to devise a mathematical model
which will simulate the dynamic behavior of the carbon column in waste
water treatment. Much of the basic information about the carbon contact
process was obtained as parts of the overall work in carbon column
operation, and are described in detail in Parts II and III of this report.
In terms of its ultimate purpose, a model, when perfectly constructed,
should provide a rational basis for design optimization and process
control. Such a claim, however, cannot be made to the present model.
Rather, the present model is intended (l) to show the efficacy of modelling
by indicating what experiments are urgently needed and (2) to examine, by
confrontation with adequate data what modifications are required to make
the model more reliable. Only when these have been accomplished, can
meaningful design studies be undertaken.
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II. CARBON COLUMN OPERATION IN WASTE
WATER TREATMENT - A GENERAL CONSIDERATION
For the removal of organic contaminants from waste water column
operation with granulated activated carbon represents only one of the
many possible contacting processes. Other possibilities include the
fluidized bed process considered by Weber and his collegues [W-U, K-l]f
but at present the economics of such a system appear less promising than
a fixed-bed process. Such fluidized columns tend to be unstable and
difficult to control. They require higher capital and operating costs.
Furthermore, the usual advantages of fluidized bed operation will probably
not be realized in this system. Due to agitation and particle motion, the
transport of adsorbate to the particle surface will be facilitated in a
fluidized bed, However, a significant increase in mass transfer rates
will only result if the liquid side resistance is rate-limiting. For
solid/liquid adsorption systems, and, in particular, for activated carbon/
water/organics, many experiments indicate that intraparticle diffusion is
the limiting rate process [E-l, A-2, S-2, W-3, D-2, M-l, S-l, W-2].
Weber and Keinath [W-H, K-l], do not agree with this conclusion. Weber
himself has recently compared the performance of fixed and expanded beds
in a pilot plant at Ewing-Lawrence, New Jersey [W-5], Although the
difference was hot large (about 2.5$) over a four month period, the fixed
beds gave consistently purer effluents. One advantage of fluidization
cannot be denied: baekwashing of the beds is obviated [W-5], However,
this may well "be at the expense of effective filtration.
Another type of contacting process involves the use of powdered
carbon. In general, the powdered carbon is first mixed with waste water
in a large tank, then filtered and discarded. If recent pilot plant
studies on a regeneration process for powdered carbon prove successful,
then usage of such a powder may become economically attractive. However,
efficient regeneration of granular carbon is already a proven process [C9]
Furthermore, continuous treatment of waste water in a column packed with
active carbon makes more efficient use of the carbon. Fresh waste water
first contacts the most nearly spent carbon particles, thus allowing a
closer approach to its adsorption capacity. Using downflow contactors,
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the pressures required to pump waste water through granular beds at
flowrates of 2 to 10 gpm/sq. ft. are quite reasonable, whereas the pressure
head needed to force water through a bed packed with powdered carbon is
impracticably high.
One of the first large scale demonstrations of granular activated
carbon columns in the U.S. was in 1930 [fi-2], The carbon columns used
were generally additions to existing equipment, often simply a sand filter
refilled with carbon granules. In contrast, recent proposals by Hager and
Rizzo [H-l], by Weber [W-5], and by Zuckerman and Molof [Z-l] would center
the whole treatment process around the active carbon unit. The secondary
stage of treatment would be eliminated in favor of a more efficient
coagulation/clarification step. This would deliberately put a heavier
load on the carbon in the form of TOC and suspended solids. The carbon
particles can act as a filter for such suspended solids. In fact, reports
from the 7.5 mgd South Tahoe PUD plant [C-10] and by Cooper and Hager [CU]
show that whether filtration is intended or not, the large throughput of
waste water always involves some suspended matter, and causes clogging of
the column. Such clogging (which may easily double or treble the pressure
drop across the column) is removed by regular backwashing.
There are a number of unique features regarding the waste water-carbon
contacting process. The exact concentration of each and every undesirable
contaminant which is present in waste water is unknown. Rather, the
concentration of these contaminants are described by a gross quantity such
as TOC (total organic demand) or COD (chemical oxygen demand). Even for
a given location, there are seasonal as well as daily variations about the
contaminant concentration. "Furthermore, these concentrations are usually
very small quatities varying from 20 30 ppm (parts per million) for
secondary effluent and approximately 50 ^ TO ppm for primary effluent. It
should be pointed out that the accuracy of one of the more commonly used
apparatuses for TOC determination [Beckman Model 915 Total Organic Carbon
Analyzer] only has an accuracy of  1 ppm.
Within the column, the process is also distinctly different from
conventional adsorption operation, According to some authors [C-7] the
so-called carbon loading (grams of organic contaminants removed per gram of
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carbon applied) is found to approach or even exceed that of the maximum
adsorption capacity of carbon granules from the pertinent adsorption
isotherm. The difference cannot simply be explained in terms of experimen-
tal error. The carbon granules also appear not completely saturated even
after prolonged usage. These together with other evidence [B3] strongly
suggest the possible existence of biological reactions which take place
within the granules. These biological actions decompose the contaminants
adsorbate and act as a regenerative process for carbon particles. The
nature of these biological actions is totally unknown. Some evidence
suggests its being aerobic, but an anerobic type of action has also been
observed.
Besides the adsorption and biological actions, carbon column also
functions as a deep filter bed. The retention of the suspended matters
usually takes place at the first few feet of the column with an accompanying
increase in hydraulic load. The rapid build-up in pressure drop requires
frequent backwashing of the column as attested by the demonstration plant
work at Lake Tahoe, Pomona, etc. It has been found that for proper carbon
column operation, backwashing of the leading column is required daily
while the regeneration of carbon granules is made once in several months.
The brief account on carbon column operation in waste treatment
clearly indicates that any realistic modelling for the carbon column
should consider all these three functions - adsorption, biological
degradation and filtration, and proper accounts should be provided for
their interactions. In addition, the large volume of waste water to be
treated and the relative long contact time necessary for almost complete
contaminant removal requires the use of columns with relatively small
height to diameter ratios (probably in the range of 1.5:1 to 2.5:1).
For columns with dimensions such as these, many non-ideal behaviors of
packed beds such as inhomogeneity of packing, channeling pressure of deal
pockets with the bed are likely to occur. For any realistic model to be
established, it is necessary to include these features into the model even
though these may not be important for the preliminary version of this
model to be discussed in the following.
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III. FORMULATION OF MODEL
(III-A) Modelling Consideration - Waste Water System
Since the function of the carbon column process is the removal of
undesirable contaminants, the characterization of these contaminants
becomes necessary before a quantitative description of the removal process
can be formulated. The large variety of contaminants which are likely to
be present in waste water and their minute amounts, make it impossible
for individual identification and traditionally these wastes can only be
represented by gross overall concentration variables such as BOD, COD, or
TOC which is used in the present modelling work. The use of TOC (total
organic carbon) for characterizing waste concentration is a convenient one
because of the commerically available instrument [Beckman TOC Analyzer]
which can be used to obtain TOC values from a sample waste water with
reasonable ease. Some earlier work, however, has employed the use of COD
(chemical oxygen demand). It is also necessary to make a distinction
between the two mechanisms responsible for the removal of contaminants
from waste water in the adsorption process and the filtration process.
For this investigation, we shall assume that the TOC of a waste water
sample represents the sum of the DOC (dissolved organic carbon) and SOC
(suspended organic carbon). The former is removed by adsorption into the
carbon granule while the latter is removed by deep bed filtration.
The use of a single gross quantity, DOC, for the description of an
adsorption process is tantamount to the approximation of a multi-solute
system by a pseudo single solute system. The validity of this assumption
can be only tested through experimental confirmation.* At best, this can
only be considered as an approximation.
As part of our overall program on the study of carbon column operation,
experimental results on batch adsorption of waste water (secondary effluent)/
granular activated carbon were obtained and interpreted on the basis of a
pseudo single-solute assumption. The experimental results, to a degree,
substantiate the validity of this assumption and the numerical values of
IT"		'	-   "
Experimental confirmation is obtained if one can obtain experimentally a
consistent adsorption isotherm based upon DOS, and a successful interpretation
of adsorption kinetic data of waste water/activated carbon consistent with
the pseudo single solute assumption.
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the pertinent parameters obtained experimentally are used in the column
simulation, The detailed description of experimental work is given in
Part II of this report.
(III-B) Modelling Considerations - Packed Columns
The chemical reactor, and, in particular, the packed tubular reactor,
has been a subject of central concern to chemical engineers for many years.
Comprehensive reviews of the state-of-the-art in the early 1960's have
been given by Beek [B-2] for the design of steady-state, nonisothermal,
catalytic reactors and by Wilhelm [W-6] for the transient problem as well.
In 1970, Hlavacek [H5] summarized the steady-state aspects, and Paris
and Stevens [P-2] examined the methods and assumptions used for time-
dependent problems. We select from these four reviews two major trends
of the last decade: an increased interest in the Deans-Lapidus finite-stage
or mixing-cell model [D-l, M-2] and a demand for practical computation
times. The former of these regrettably turns out to be incompatible with
the latter.
In principle, a packed tubular reactor has at least four independent
variables: axial distance from the reactor inlet, radial distance from the
reactor wall, radial distance measured from the surface to the centre of
each porous granule, and time. Now, the brutal facts of present-day
compuation are as follows:
a)	A useful simulation program, to be run many hundreds of times,
must not require more than a few minutes of computer time.
b)	For virtually all processes of industrial size, on most available
computers, this time constraint means that no more than two
independent variables can be dealt with.
It will be shown that the two radial distances are the logical variables
to neglect. This implies the negligence of radial dispersion of the
column and the approximation of intraparticle diffusion by somewhat
simpler expression.
First, consider the mixing-cell model previously mentioned. The
original idea of simulating a packed column by a finite number of
perfectly-stirred tanks in series is generally attributed to Kramers and
Alberda in 1953 [K2] in connection with tracer studies of longitudinal
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dispersion in fixed beds. Deans and Lapidus [D-l] extended this concept
to include radial dispersion "by postulating a two-dimensional, staggered
array of tanks.* Agreement with experimental data is quite good if the
size of each tank (t>oth thickness and depth) is of the order of one
particle diameter. Dispersion parallel to the axis is simulated by
complete mixing within each tank, and normal to the axis by splitting
each CSTR** effluent into two streams which become influents to the
adjacent tanks below. Backmixing, in the sense of downstream tajiks
affecting upstream tanks, cannot occur. This model is perhaps the best
physical representation available for packed beds, and could be made even
better by the incorporation of non-ideal stirred tank features such as
incomplete mixing (Cholette, Cloutier and others [C-2, C-3t F-l]), and dead
space interactions [C-5, C-6[. Levich and others [B1+, L-l] have already
used this latter idea to model ispersion in a porous medium.
Realistic as it may be, the cell model takes far too much computer
time to be used for transient calculations. McGuire and Lapidus [M-2]
have carried out such calculations for the stability of a nonisothermal
packed bed reactor with a diffusion equation describing intraparticle
effects.*** For a reactor 5 particles wide by 15 particles deep it
required 3 hours of IBM-7090 time to simulate a flow of 13 reactor
residence time units. A carbon granule size of 8 x 30 mesh would give
about 500 particles per foot, so that a typical industrial waste water
treatment column (based on Cover's 1970 design [C8]) might involve
23,000 x 13,000 particles over at least 1000 bed residence times. Not only
is the estimated calculation time ridiculous, but even with the removal
of the heat transfer effects, the intraparticle effects, and the radial
effects, and making allowance for much larger time steps, the estimate
still amounts to hundreds of hours!
As another example, Feick and Quon [F-2] have simulated the transient
behavior of a packed bed reactor with axial and radial dispersion using
Every tank has an annular shape, with the same annular thickness and depth.
*
CSTR = Continuous Stirred Tank Reactor
**#
This is a partial differential equation, whereas each CSTR is governed
by an ordinary differential equation.
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the more orthodox homogeneous dispersion model and a modified, alternating
direction, explicit, finite difference procedure. Their reactor was 10
catalyst particles wide by 50 catalyst particles deep, and the simulation
lasted for 2.5 reactor residence times. The computation on an IBM 360/67
took between 3 minutes and 90 minutes, depending on whether an intraparticle
diffusion equation was used or not. This is an improvement of perhaps a
factor of 10 over the cell model, chiefly on account of the larger spatial
grid, but such a model is nevertheless entirely impractical for carbon
column simulation. Because of these considerations Paris and Stevens
[P2] stated that most industrial-scale packed columns operating in a
transient mode cannot be simulated with anything more complex than a plug
flow model.
Having decided this on computational grounds alone, it should be added
that the plug flow model happens to be a reasonably sound physical
representation of both adsorption and filtration in fixed beds. Carbon
columns are usually operated at a particle Reynolds number of about 2,
which implies an axial Peclet number of about 2 from Wilhelm's correlation
[W-6]. The axial dispersion will be small, and Beek [B-2] recommends that
it be ignored under similar circumstances. Carberry and Wendel [C-l]
estimate that there will be no significant axial effects beyond 50 particle
diameters, Since the carbon columns operate isothermally, it is also a
good assumption that the concentration profile has no radial variations,
A note of caution is needed here, however. Since the length-to-diameter
ratio of industrial columns is only about two or even less, complete radial
mixing will not occur. If the flow distribution near the column inlet is
particularly uneven, or if the packing of the granules has large voidage
variations*, then radial concentration gradients could arise. This possible
complication, however, will not be considered in the present work.
(ill-C) Adsorption Process
It is felt that the adsorption of DOC is controlled by at most two
mechanisms: transfer through the liquid phase to the exterior of the
granule and intraparticle diffusion, Other mechanisms such as surface
*
Haughey and Beveridge [H-U] have recently reviewed the structural
properties of packed beds.
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attachment is ruled out. Furthermore, the intraparticle diffusion is
believed to "be the dominant mechanism in view of the very low values of
diffusion coefficients evaluated experimentally,* The liquid side transfer
is only important where the carbon granule is fresh or under certain
special conditions.**
If a unit volume of the packed bed is taken as a basis, the liquid-side
transfer rate can be written as
3t
= <>6 If""* *	(III"1>
where is the bulk density of granular carbon, of the solid phase
concentration of the adsorbate (DOC), k^ is defined as the liquid side
mass transfer coefficient, The function f^(cc*) represents the proper
driving force, c* represents an effective liquid phase concentration of
DOC adjacent to the granule surface.
Mass transfer within the granule apparently takes place by combination
of force and surface diffusion mechanisms [D-2], A proper treatment of the
pertinent diffusion equation, in general, requires the numerical integration
in a time-space grid.*** As mentioned earlier, the practical demand on the
econony of computer time rules out the consideration of more than two
independent variables for carbon column modelling, Approximation must,
therefore, be used to represent this intraparticle diffusion by a simple
rate expression. Hence,
f= kg  f2(q,q*)	(HI-2)
where k0 is the solid phase transfer coefficient and f^ represents the
proper driving force, q* is the solid phase concentration adjacent to the
granule's exterior surface.
*					
See discussion in (VI-C).
For example, the result of filtration may effect the coating of the
carbon particle with a layer of slime and thus greatly increase the mass
transfer resistance in the liquid side.
***
Space, in this case, refers to that within the carbon granule.
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Combining Equations (ill-l) and (III-2), one has

(III-3)
If the equilibrium condition is maintained and the solid-liquid interphase
(i.e. granule's exterior surface), one has
carbon expressed in terms of dissolved total organic carbon.
(III-D) Biochemical Effects
This perhaps is the least-known aspect of carbon column operation.
Biological reactions can, in principle, affect the TOC level for each of
the four concentration variables. Such reactions can be described rather
crudely as follows:
[Dissolved Oxygen] + [Biodegradable TOC] + [Bacteria] = [Oxidation Product]
Since the oxidation product usually includes CO^ only, the net effect
is to decrease the TOC level. Most of the rate data for such reactions
come from. BOD studies on closed systems. Both first and second order rate
expressions have been suggested [R-l, Y-l] and possess roughly equivalent
predictive values.
The biodegradable TOC are present in both the dissolved organic
contaminants as well as the suspended matters. Since both types of
substances can be either in liquid or solid phase, four distinct reactions
are possible. This includes the degradation of suspended organic carbon
(SOC) in liquid phase, of SOC in solid phase, of dissolved organic carbon
(DOC) in liquid phase and of DOC adsorbed in solid phase. The last kind
of reaction perhaps is not likely to occur if we consider the relative size
of the carbon particle pore and the average bacteria. The first three
q# = f ^(c*)
(III-U)
where f^ is the adsorption isotherm of the system  waste water-activated
+ [More Bacteria]
kinds of reaction rates designated as R , R and R can be assumed to be
(c-c ), {0o} [Bacteria]
(HI-5)
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- s2 = \ f6
(y-yn).	[Bacteria]~j -	(III-6)
" Ra = ke3 fT
~{o-o ), {0o} [Bacteria]"!	(HI-7)
n	d	I
where the subscript n denotes the non-biodegradable part of the
contaminant, and a the volume fraction of the suspended matters deposited
on to the carbon bed. The k's are rate constants. The term dnR of
6	1 c
Equation (6) accounts for the possible increase in suspended matters due to
the degradation of DOC.
(III-E) Filtration Effects
To complete the description of the dynamic behavior of a carbon column,
the filtration rate needs to be given. In general, on a volume basis, the
rate of deposition, of the filtration process is given as
S1 = K^cr.y)	(III-8)
where Kq is often referred to as the impediment modulus or the filtration
coefficient, The relationship between the deposition rate, the possible
biological degradation of deposited suspended matters (or R^) and the
rate of change of deposited matter in the carbon column, ~ , is given as
ot
- h	f111-?1
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IV. MATHEMATICAL EQUATIONS DESCRIBING THE
DYNAMIC BEHAVIOR OF COLUMN OPERATION
(IV-A) Basic Equations
Based upon the plug flow and other assumptions stated in (III-B), the
continuity equations describing the adsorption of dissolved TOC and the
filtration of suspended matters can be written, respectively, to be:
For Adsorption
11 If + E If + B If " Rc - 
For Filtration
u!f+ e It* si - ss- 0	(IV-2>
where
z = axial distance measured from the column inlet
t = time
u = superficial velocity
e = fraction of interstitial voids in packing
and meanings of the symbols are the same as before.
Equations (IV-1) and (IV-2) are the basis for column simulation. The
specification of certain terms that appear in these two equations, however,
are needed. This will be discussed in the following section.
(IV-B) Adsorption Rates
The equations describing the adsorption rates are given by Equations
(III-1), (III-2) and (III-U)
f.(c,c*)	(III-1)
PB 1

q* = f3(c*)	(III-14)
Equation (III-U) represents the adsorption isotherm of the system of
waste water/granular activated carbon on the basis of single-solution
-13-

-------
assumptions and the representation of organic contaminant concentration
with DOC (dissolved organic carbon). The adsorption isotherm data was
obtained as part of our study and the experimental work was described in
Part II. In general, the Freundlich's adsorption isotherm was found to
represent the result reasonably well, i.e.
b2
q* = bL  c*	(IV-3)
Based on one series of experimental work, it was found from least
square fit that
= 5.3217 x 10l6
b2 = 2.321+3
On the other hand, from, a computational point of view, it would be desirable
that the exponent, b2, should be an integer. A less accurate fit of the
data yields
q* = 1.3lt51 x 109  c*2	(IV-1|)
which gives a minimized sum of squares of .00089 in comparison to 0.00017 of
Equation (lV-3). This difference is obviously not too serious and
Equation (IV-H) is used in the simulation program.
The liquid film mass transfer coefficient, k^, can be estimated from
the well established correlation, provided the granules' exterior surface
remains clean. To make a distinction between the mass transfer coefficients
corresponding to different surface conditions, the mass transfer coefficient
for a clean surface is designated to be k . The actual mass transfer

coefficient k may or may not be the same of k . This is to be discussed
X/	Xi
under Section (IV-E).	0
From Perry's Handbook, [P-U], we have
k0 = 2.12 (D u )1/'2 d"3/2	(IV-5)
i.	to	p
o
-lit-

-------
By examining the diffusivities of various organic molocules in water, it is
2	"5	_i
estimated that k shall lie between 10 ^ 10 hr.
xo
To consider the intraparticle diffusion properly, the pertinent
diffusion equation should "be solved. This, however, is not practical as
pointed out earlier. Instead, a simplified approximation in terras of a
solid phase mass transfer coefficient will be used. In connection with
this concept, two types of driving forces have been suggested. They are:
|& = kg(q*-q)	(IV-6)
according to Glueckauf's linear driving force [P-l], and
f& = kg(q2 - q2)/(2q - qQ)	(iV-T)
according to Vermeulen [V-3]. When qQ is the initial solid phase
concentration, Equation (IV-6) is used because of its simpler form. The
solid phase mass transfer coefficient, k , is related to the intraparticle
s
diffusion coefficient, D , by the following expression [V2]:
3
ka = 15	(iv-8)
Experimental work has been carried out to evaluate Dg based on pore
diffusion and solid diffusion model (see Part II). The studies indicate
-10 -Q 2
that D lies between 10 ^ 10 y cm /sec,
s
(IV-C) Biochemical Reaction
At the present time, it is not possible to formulate even in the most
crude form, the functional form of f,., f^ and f^ [see Equations (III-5) -
(III-7)]. Consequently, the biochemical effect cannot be considered in our
model. A few qualitative comments may be in order. Generally speaking,
both Rq and R are perhaps of secondary importance. The data on blank
waste water decay reported in Part II of this report seems to substantiate
this point of speculation, The term Rff is more important and the decay of
deposited organic matter perhaps explains what was observed by Bishop and
co-workers [B-3].
-15-

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A quantitative expression of Rq obtained from direct experiment is not
available, and quite likely, will remain to "be so for sometime to come.
The culprit of this is that some of the important parameters in biological
action such as dissolved oxygen concentration, bacteria species and
population are not fixed quantities but of random nature. The determination,
therefore, will require some ingenious experimentation.*
Because of the lack of basic data, the biological terms will be
neglected in the preliminary work. The importance of these terms can be
discussed indirectly by comparing the simulated result with negligible
biological effect with actual performance data when these effects are known
to be pronounced. It may be possible that through such trial and error
procedure, a crude expression of may be obtained.
(IV-D) Filtration Rate
Most of the previous work on the filtration studies use sand filters
as "synthetic waste". Consequently, we have carried out experimental work
of the filtration of clay suspensions through a carbon bed (see Part III
of the report). The empirical rate was found to be
S1 = Kf^o.y) = (Kq)  fu(a)  y	(IV-9)
f.(a) = 	i		(IV-10)
U (1 + yo)a
Assuming that values K , y and n obtained from clean beds are
applicable for actual column operation when the granular carbon is used
over a prolonged period of time. Another important consideration is the
estimation of the increase of pressure drop during filtration, which
determines the frequency of backwashing. Our work on pressure drop, as
reported in Part III, suggested the following simple expression relating
the increase of pressure drop with the amount of particle deposition.
- 1 + 5"	(IV-11)
* 
It may be necessary to radioactively tag the suspended matters and,
therefore, to be able to monitor what happens to the deposited matter.
-16-

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(IV-E) Interactions Between Adsorption and Filtration
The original adsorption and filtration models suggested in Sections
(III-C) and (III-E) are independent of each other. Interactions can occur
from at least two sources. The first is the biochemical Equations (III-5),
(III-6) and (III-7). It is possible that substrate in the form of
dissolved matter may be in competition with the suspended solids for
available oxygen. Furthermore, oxidation of dissolved TOC could conceivably
produce additional filterable material.
The filtration process may affect the adsorption in another way,
because the deposited solids tend to form a sheath around the granules, thus
hindering the liquid-side mass transfer. To simulate this, the liquid-side
mass transfer coefficient is made a function of retention,
= k fg(
-------
physical criterion is the maximum head-loss that the pumping system can
reasonably maintain. This will differ from installation to installation;
a typical figure from Pomona is one psi per foot of carbon. The present
model is defined either to "backwash at regular intervals, or to backwash
automatically when a certain pressure drop maximum is reached,
With regard to (b), not much is known about the physics of backwahsing.
It is the filtration problem in reverse: to find the effluent distribution
and retained solids distribution when an initially dirty bed is washed with
relatively clean water.
The columns are, of course, backwashed in the reverse direction to
the normal flow. The flow rate is usually high enough to expand the bed
by 30# to 50# of its normal height. The extent of axial mixing (of
granules) is unknown, but it may be quite small. It is clear from the
Lake Tahoe reports [C-9] that the backwashing is never complete. Heavily
coated granules retain a thin layer of slime from backwash to backwash
and this may eventually prevent further adsorption, even though the carbon
is not fully saturated. The opposite effect has also been suggested [P-3],
in which the slime layer participates in biological oxidations and increases
the apparent adsorptive capacity of the carbon.
For lack of better information, our model assumes that backwashing
removes a constant fraction of the retained solids (the remaining fraction
is called the slime residue factor (SRF) and has been set at 0.05 throughout
most of the initial simulation runs. Thus, the slime profile has a similar
shape to that of the retained solids, but a smaller magnitude. It is
assumed that the backwashing does not alter the distribution of adsorbed
TOC (q) in the column, and in Model I only the first section of the column
may be washed.
(IV-G) Effect of Regeneration
The main questions about regeneration concern the quality of the
treated carbon granules. Is the adsorption capacity unchanged? Are the
filtration parameters the same? And so on. As far as adsorption is
concerned, the carbon appears to have undiminished capacity*, and in our
model the same assumption is amde for all the other parameters, including
1			1~
There is maybe some loss of capacity on the first few cycles, but this
levels out.
-18-

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filtration.
It should be clear that the regeneration process itself is not simulated
(unlike bachwashing). All that is done in our model is to remove the first
section of the column, and add a new section of fresh carbon after the last
section. The criteria for this operation are very similar to those for
backwashing. Regeneration may take place either at fixed intervals, or when
the effluent TOC concentration from the last section of the column passes a
certain specified limit, whichever condition arises first. The "effluent
TOC" in this case refers to the combined effects of dissolved organic
pollutants and suspended solids. This means that a "suspended solids
conversion factor" is needed to express volumes per volume of suspended
solids as grams of TOC per unit volume of waste water (SSCF ^ 1.0 gm/cc).
In the case of backwahsing and regeneration being called for at the same
time, regeneration will have the priority.
(IV-H) Summary of Equations and Associated Conditions
When all these assumptions stated above are invoked, the system of
equations describing the column behavior is found to be:
ufr+ e If+ \ (1 - v)(c -c*> *0
(IV-11+)
o
it= V4" -5)
(IV-15)
(IV-16)
<1* = b1c*1
2
(IV-17)
k (l - rda)(c - c*) = 0Bks(q.* - q.)
(IV-18)
0
0
(IV-19)
(1 + ya)n
-19-

-------
3a
at
Koy
(1 + Yo)
n
(IV-20)
~ = (|f) (1 + a)	(IV-21)
aZ 9z o
The concentrations c, q, y and a are made aimensionless in the following
manner:
C = c/c	(IV-22)
o
Q = l/(V^)	(IV-23)
Y = y/yQ	(IV-2U)
S = 0	(IV-25)
where the subscript "o" indicates an input concentration to the first
section of the column at the start of the simulation.	With these trans-
formations, Equations (IV-1T) and (IV-18) become
Q* = (C*)2	(IV-26)
and
Q* = Q + N2fQ(C - C)	(IV-2T)
where N2 is defined below, and
fQ = 1 - rdS	(IV-28)
By solving Equations (IV-26) and (IV-27), it is found that
C* = ~ N2fQ + (N^f| + U(Q + N2f8C))l/2}	(IV-29)
where the positive root has been chosen. The equations are then converted
to dimensionless characteristic normal form by the following transformation:
T = X (t - ez/u)	(IV-30)
s
Z = z/L	(IV-31)
where L is the total length of column sections in series. An additional
assumption is made that the mean superficial flow velocity and the
voidage remain constant at their initial values uq and eQ. The resulting
-20-

-------
system is
dC
dZ = - Nx(l - rdS)(C - C*)	(TV-32)
|| = N2(l - rdS)(C - C*)	(IV-33)
where C# is taken from Equation (IV-29), and
 = _ N 	-~
dZ	3 (1 + yS)n
# = - No 			r	(iv-34)
38 - N, 	1			{IV-35)
dT ' U (1 + YS)n
where
i - (>0 (1 + 8S>	(IV-36>
N = L k. /u	(IV-37)
-L	X/ o
o
n2 = k /	(iv"38)
O
N0 = L K /u	(IV-39)
i	oo
N, = y K /k	(IV-UO)
1+ Jo o s
The scale has "been selected so that for a reasonable set	of parameters
all of the diraensionless variables and groups will he of	ord^r unity.
The boundary and initial conditions used with Model	I are:
Q(Z,0) = 0	(IV-U1)
C(0,T) = 1	(IV-U2)
S(ZtO) = 0	(IV-U3)
Y(0,T) = 1	(IV-UU)
-21-

-------
As discussed previously, these associated conditions represent the reactor
inlet concentrations, and the initial solid phase distributions. In the
general model, one might consider associated conditions
Q(Z,0) = f (Z)	(iv-1+5)
C(0,T) = f1Q(T)	(IV-U6)
S(Z,0) = fn(Z)	(IV-1+7)
Y{0,T) = f12(T)	(IV-1+8)
These generalized conditions would be quite easy to implement in the
simulation. Either in the case of Equations (IV-1+1) to (IVUU) or (IV)
to (IV-1+8), the missing conditions at the inlet and the tip of the plug
have to "be found by integration. These conditions are
~ { C(Z,0) } = - H1 { C(Z.O) - C* }	(iv1+9)
where
C* = | { - N2 + (Ng + ^N2C(Z,0))l/2 }
C(0,0) = 1
and
^ { r(z,o) } = - w3y(z,o)	(iv-50)
where
Y(0,0) = 1
and
~ { Q(0,T) } = N2 { 1 - rdS(0,T) } {1 - C*}	(IV-51)
where
Q(0,0) = 0
f8 = 1 - rdS(0,T)
-22-

-------
C"  2 C " N2f8 +
R4 * 11 ((0,T) + H2fg) 1/2!
and.
.	N,
If t s(o,t) } 		a		
(1 + yS(O.t) }n
(IV-52)
where
S(0,0)  0
It will te observed that only Equation (IV-U9) can be integrated analytically.
A similar set of initial and boundary conditions must be integrated after
every backwash and regeneration.
-23-

-------
CHAPTER V SIMULATION PROGRAM
For the convenience of subsequent discussion, we shall designate
the model which we have constructed in this work as Model I and hopefully
to continue this sequence of names as further improvements are made.
(V-A) Computation Algorithm. For Model I, Equations (iV-lU) - (lV-21)
provide the basis for the simulation of the dynamic behavior of carbon
columns. The adsorption and filtration aspects of the operation are
given by Equations (IV-32) - (IV-33) and Equations (lV-3^) - (IV-35),
respectively. Both set of equations, however, can be represented by
/
the general form
(Vz = *1 (z' t' Uls V
(U2)t = *2 (z, t, Ux, U2)	(V-2)
where and Ug are the dependent variables and z and t are the
independent variables. The subscript refers to partial differentiation.
For example, for adsorption, and U2 are C and Q, respectively.
Similarly for filtration, and U2 become Y and S.
Equations (V-l) and (V-2) are known as semi-linear hyperbolic equations
and they are frequently encountered in engineering applications. A
number of computation algorithms based on methods of characteristics have
been developed in the past for their numerical solution, (A-l), but were
found unsatisfactory for the present application. The principal objection
is the high computer time demand because of the large column size and
the long period of operation in actual carbon column application.
An extensive study was undertaken by Vanier [V-l] for the
development of new algorithms as part of the overall program.
A particular algorithm designated by Vanier as CN553 was selected for
the numerical integration of the adsorption and filtration equations.
This algorithm was developed based on Taylor's series expansion and
is of fourth order in local truncation error and third order in global
discretization error. It also has the additional advantage of requiring
less computation time in comparison to other algorithms. It should be
pointed out that the simulation program developed here is not restricted
-2k-

-------
to the use of this particular algorithm. If for some reason another
method of computation is preferred, it is only necessary to make a change
in the appropriate subroutine. A brief description of the algorithm is
given in Appendix B.
In addition to the adsorption and filtration equations, the initial
conditions of C and Y for T = 0 and Q and S at z = 0 have to be calculated.
This step, referred to as initialization, is carried out by the numerical
integration of Equations (lV-i+9) and (lV-50) with respect to z and
Equations (IV-51) and (IV-52) with respect to T. These integrations are
made with a fourth order Runge-Kutta method.
(V-B) Program Structure The simulation consists of a main program and
eight subroutines of which the method of characterization subroutines is
the most important. The computation algorithm, designated as CN553 (see
Part IV), is used. However, this can be changed with more accurate ones
(fourth order is better) if this is warranted. A list of the subroutines
and their functions are shown in Table 5-1.
The structure of the program is described by the flow charts in
Figures (5-1) and (5-2). The main program reads and prints all the input
data, and calls on ancillary processes such as backwashing and regeneration
whenever needed. It also calls the method of characteristics subroutine
(MOC) which thereafter controls the calculations. There is a clear sepa-
ration of calculation framework and model. The model equations are con-
tained in subroutine UZT and the equations for initial and boundary con-
ditions in subroutines UZ and UT. To experiment with a new model, it is
only necessary to change these three subroutines and the definitions of
scale factors and dimensionless groups in the main program. Since the
pressure equations are unlikely to be altered, these have been carried
out in subroutine MOC. Experimentation at a lower level to investigate
the effect of various parameters has been highly automated. The program
will process sequentially a number of data sets, each representing an
experiment with a new group of parameter values. Unchanged parameters
need not be repeated in adjacent data sets, and the input is carried out
with key words (example: CMAX = 7.0E-6)1 and an otherwise free format.
Standard FORTRAN notation: E-6 = 10 ^
-25-

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Complete details of the input data requirements, together with extensive
comments on basic assumptions, usage, and ways to extend the simulation
are given in the program listing in Appendix A. This program is written
in FORTRAN IV to run on the IBM 360/50 level G compiler.
The naming of variables and parameters corresponds closely to the
notation of the text and a complete listing is given in Table (5-2). For
example, RDKL corresponds to the retention degradation of k^ defined in
Equation (lV-13). The program itself lists all essential names together
with appropriate units1 in its output.
The variables are stored in matrices U(I,J), Ul(l,j), and U2(I,J)
and their derivates in DF(I,J), DFl(l,J), and DF2(I,J) as defined in the
listing. The bulk of the storage is in blank or labelled common in order
to avoid unnecessary address transfers. The total storage is quite small,
and is approximately thirty times the number of column slices considered
(NSLICE). Since one seldom needs to consider more than 100 slices, the
maximum requirements2 are 3,000 units (12K bytes, since single precision
is used). On a smaller computer, this storage requirement could be re-
duced by at least a factor of three if Stimberg's algorithm (STIMBERGCN)3
were used, and the pressure profile was not saved.
The program achieves its CPU time objectives provided that the con-
centration profiles are not too sharp. If backwashing or regeneration is
not called for more frequently than every ten time steps, the program can
compute 1+0 grid points per second of CPU time. This means that for a wide
range of parameters Model I can be applied to a 20 foot industrial column
and integrated over a 100 day period in less than 10 minutes of IBM 360/50
CPU time, giving two decimals or better of accuracy.
If it is desired to study the filtration or adsorption equations
separately, these phenomena can be uncoupled by specifying the unwanted
input concentration to be zero. In this case, the simulation program
will set the relevant scale factors, (N^, N^, or N^) equal to zero,
^e program accepts data in commonly used units, and makes appropriate
conversions for internal use.
2Not including the program itself.
3For description of this algorithm, see Ref. (V-l).
-26-

-------
so that in effect, only the desired phenomenon is integrated. The effects
of regeneration and backwashing can also "be "turned off", simply by
setting the control values CMAX, TREGEN, DPMAX, and TBA larger than their
greatest possible value during the simulation.
-27-

-------
TABLE 5-1
List of Subroutines of Simulation Programs
Code Name
MOC
BWASH
REGEN
UZT
zo
TO
uz
UT
Function
Solve plug flow equations by a
third order method of characteristics
and prints concentration profiles
Simulates the effect of backwashing
on the first section of the column
Remove the first section of the column
and place a fresh section at the end
Characteristic of normal ordinary-
differential equations (5) for
adsorption and filtration
Initializatir .1 along T = 0, and
after each backwash, integration by
a fourth order Runge-Kutta method
with respect to z (called by MOC)
Initialization at column inlet, z = 0
interpretation by a fourth order
Runge-Kutta method with respect to
T (called by MOC)
Special form of characteristic
normal equations in C and Y needed
along T = constant, (called by ZO)
Special form of characteristics normal
equations in Q, and S needed along
Z = 0 (called by TO)
-28-

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NO
YES
READ INPUT DATA SET:
COMMENTS, CARBON COLUNN SPECIFICATIONS
RATE PARAMETERS, ISOTHERM PARAMETERS,
CONTROL DATA, ETC.
YES
S TIME > TMAX?
(ISTOP = 1
NO
YES
REGENERATION
^NEEDED? ^
''THERE A NEW^
DATA SET TO BE
READ? ^
STOP
START
CALL SUB
MOC
INTEGRATION BY METHOD
OF CHARACTERISTICS
BACKWASHING: (ISTOP c 0)
SUB BWASH
DEPOSITED SOLIDS IN
FIRST SECTION OF COLUMN
ARE PARTIALLY REMOVED
INITIALIZE: CC = Q = 0, TIME = 0, ETC.)
CONVERT UNITS; CHECK DIMENSIONS;
PRINT ALL PARAMETERS
REGENERATION: (ISTOP = 2)
SUB REGEN
REMOVE FIRST SECTION OF COLUMN
(TRANSFER Q AND S IN STORAGE)
ADD FRESH CARBON AT END OF COLUMN
(Q = S = 0 IN LAST SECTION)
FIGURE (5-1) COMPUTER SIMULATION FLOW CHART: MAIN PROGRAM
-29-

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SUB MOC >
INITIALIZE C AND Y
ALONG T=0 AXIS
INCREMENT TIME BY ONE UNIT
CALCULATE BOUNDARY CONDITION
FOR Q AND S AT COLUMN
INLET AT NEW TIME
A
INTEGRATE FOR C, Q, Y, S OVER
ENTIRE COLUMN LENGTH, USING
3RD ORDER METHOD OF
CHARACTERISTICS ALGORITHM
V A
MODEL EQUATIONS:
SUB UZT
DC/DZ = 	
DQ/DT = 	
DY/DZ =	
DS/DT = 	
W
HAS
'BACKWASH TIME OR:
PRESSURE MAXIMUM
OCCURRED?,
YES
SUB zo:
Q AND S ARE
ASSUMED KNOWN.
INTEGRATION FOR
C AND Y W.R.T.
Z OVER ENTIRE
COLUMN BY
*+TH ORDER
R-K METHOD.
SUB UZ:
DC/DZ = ,
DY/DZ = ,
CAT T=Q)
CALCULATE TOC LEVEL AND
PRESSURE DISTRIBUTION
PRINTOUT ACCORDING TO CONTROLS
HAS
SIMULATION TIME
LIMIT BEEN
EACHED?
fSTOP
HAS
'REGENERATION T]
OR BREAKTHROUGH
OCCURRED?
SUB TO:
C AND Y ARE
ASSUMED KNOWN.
INTEGRATION FOR
Q AND S W.R.T.
T OVER ONE
TIME STEP BY
4TH ORDER
R-K METHOD.
I
SUB ut:
DQ/DT = .
DS/DT = .
CAT Z=0)
YES
ISTOP * 2
ISTOP = 0
RETURN
TO MAIN
PROGRAM
FIGURE (5-2) COMPUTER SIMULATION FLOW CHART: CALCULATION SUBROUTINES
-30-

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TABLE 5-2 List of Variables and Parameters
Physical Quantity	Honconductive in Text
bulk density	p
a
superficial velocity	uq
mean particle diameter	d^
void fraction of bed	e
o
total length, of column	L
number of sections	-
dissolved TOC inlet concentration	c
o
suspended solid inlet concentration	yQ
suspended solid conversion factor	-
for overall TOC values
liquid film mass transfer coefficient	k^
for clear surfaces	o
Freundlich isotherm parameters	b^ and bg
filtration coefficient	K
o
parameter relating a vith. change of	y
filtration coefficient
retention degradation of liquid	r^
film mass transfer
pressure gradient of fresh bed	^"fz^o
parameter relating pressure drop
increase vith solid retention	^
slime residue factor after backvashing	-
time for backvashing	-

-------
TABLE 5-2 List of Variables and Parameters (Cont'd.)
Physical Quantity
Nonconductive in Text
Computer Name
time for regeneration
simulation interval
maximum pressure drop
"breakthrough, concentration
position increment
time increment
dimensionless parameter of
governing equations
solid phase transfer coefficient
N1 N2
N3 N"
TRENG
TMAX
DPMAX
CMAX
X (1)
X (2)
N (1) N (2)
N (3) N (1*)
KS

-------
CHAPTER VI SIMULATION RESULTS
It shall be made clear at the outset that the results presented in
this section are part of the verification of Model I. They are not
intended for actual carbon column design, even though they can be utilized
for that purpose under careful provisions. One should be cautioned that
any design conclusions drawn from these tests must be tempered by the
knowledge that hardly any physical parameters involved in this model
have be n satisfactorily determined. Considerable experimental work
beyond those described in Parts II and III of this report is required.
One simply cannot expect significant results from very approximate
dat a 
As stated previously, the first objective of Model I is to indicate
what experiments are urgently needed, and the second is to examine,
by confrontation of adequate data, what modifications are required
to improve the Model. The purpose of this work, therefore, is not
merely a development of Model I which admittedly is a very crude model
based upon available data at present, but the provision of a computation
framework within a wide variety of models that can be examined without
undue effort.
A number of the simulation tests were made with Model I to examine
certain factors (such as increment size, magnitude of transfer
coefficient, etc.). A summary of the conditions of these simulations
are given in Table 6-1. The conclusion of these tests are given in the
following:
(VI-A) Effect of Increment Size
A preliminary step in the utilization of the simulation
program is the determination of increment sizes which are consistent with
accuracy and computer time requirements. In practice, the accuracy of
the simulation need not be appreciably greater than the accuracy of
typical experimental data. Since TOC measurements have an associate
uncertainty of at least  1 ppm, a computational error of  0.005 in
the dimensionless concentration can generally be tolerated.
-33-

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TABLE (6-1) MODEL INPUT PARAMETER IN SIMULATION TESTS
Text
Notation
Computer
Name
Units
(1)
(2)
Simulation Data Set Number
(3) (k) (5)
(6)
PB
PB
gm/cm^
0.39
0.39
0.39
0.39
0.39
0.39
%
UO
gpm/ft2
3.2
3.2
2.029
3.2
3.2
3.2
dp
DP
cm
0.06W
0.061*8
0.0591*
0.06U8
0.06^8
0.061*8
e
VOIDS

0.5
0.5
0.1*9
0.5
0.5
0.5
L
L
ft.
10.0
1*.0
U.265
O

O
r1
20.0
as shown
NS
NS

2
2
1
2
2
as shown
c
o
CO
gm/cm^
0.28E-.lt
0.1E-1*
0.0
0.28E-1+
O.UOE-U
O.UOE-l*

YO

0.0
0.5E-1+
0.882E-1+
0.0
0.10E-1+
0.10E-U
SSCF
SSCF
gm/cm^
1.0
1.0
1.0
1.0
1.0
1.0
D
s
DS
2 ,
cm /sec
0.25E-10
0.25E-1C
0.25E-10
as shown
0.1+0E-9
0. l*0E-9
*
o
KLO
hr-1
50.0
50.0
100
100.0
100
100
*1
Fl(l)
cm^/gm^
0.135^E 10
0.1351+E 10
0.131+5E 10
0.135UE 10
0.135l*E 10
0.135**E 10
b2
FI(2)

2.0
2.0
2.0
2; 0
2.0
2.0
K
o
FC(l)
hr"1
1*8,0
1*8.0
50.1+7
1*8.0
1*8.0
1*8.0
Y
FC (' 2)

67.0
67.0
30.1+6
67.O
67.0
67.0
n
FC(3)

2.5
2.5
5.095
2.5
2.5
2.5
rd
RDKL

10.0
10.0
10.0
10.0
10.0
10.0
(ap/3z)
o
PRD(l)
atm/cm
0.15E-2
0.152-2
0.3350E-3
0.15E-2
0.15E-2
0.15E-2
6
PRD(2)

283.0
283.0
1*61*. 1
283.0
283.0
283.0
SRF
SRF

0.05
0.05
0.05
0.05
0.05
0.05
(Continued)

-------
TABLE (6-1) MODEL INPUT PARAMETER IN SIMULATION TESTS (Cont'd.)
Text
Notation
Computer
Name
Units
(1)
(2)
Simulation
(3)
Data Set Number
(4) (5)
(6)
TBA
TBA
hr.
320.0
320.0
48.0
320.0
96.0
96.0
TREGEN
TREGEN
hr.
200.0
200.0
200.0
200.0
2300.0
9000.0
TMAX
TMAX
hr.
120.0
64.0
20.0
120.0
2400.0
as shown
DPMAX
DPMAX
atm
4.0
4.0
1.0
4.0
10.0
3.0
CMAX
CMAX
3
gm/cm
0.30E-4
0.30E-U
1.0
0.30E-4
0.60E-4
as shown
X1
X(l)
ft.
as shown
as shown
0.164
as shown
0.5
0.5
*2
X<2)
hr.
as shown
as shown
O.250
as shown
4.0
2.0
R1
N(l)

19.^82
T.7927
0.0

77.927
62.341
-2
N(2)

10.518
29.449
0.0

0.92028
0.92028
"3
N(3)

0.0
7.4810
13.228

37.405
29-924
N4
N(l+)

0.0
7.4650
11.634

0.93312E-1
0.93312E-1-
k
s
KS
hr-1
0.3215E-3
0.3215E-3
0.3826E-3

0.5144E-2
0.5144E-2

-------
There are two increments which have to be selected, a space step
(x^, in ft.) and a time step	in hr.). The effect of the increment
size chosen on the accuracy of the calculations will vary according
to the values of the physical parameters. In general, the more abrupt
the concentration profiles are (implying relatively large values of
D , k , and K ), the smaller the increments must be to attain a
s I '	o
specified accuracy. A change in an exponent such as "n" in the
filtration equations can have a pronounced effect on accuracy, and
the general picture is complex because of the large number of parameters
involved. Some specific criteria for Model I could be set up (in
graphical form), "but rather than this, a general procedure will be
given for finding appropriate increments.
The starting point of this procedure is a nominal increment size
which is thought to be adequate. If this is not available from previous
simulation experiments, the values x^ = 0.5 ft. and x^ = 1.0 hr. may be
used. A simulation sequence is then carried out on the computer in
which the increment sizes are varied systematically. All parameters used
in this test sequence should have their actual values, except for the
simulation time (TMAX), which need not be more than 50 hours. Experience
with Model I indicates that the largest errors occur in the initial
period of simulation, and after this point the accuracy is remarkably
well-maintained. It is not unusual for the accuracy to improve as
saturation conditions are approached.
The increments x^ and x^ should be varied by several factors of
two on either side of their nominal values. Thus, if the nominal
increments are denoted by (0.5 l)> the simulation sequence might consist
of nine runs as follows: (0.5, l) (0.5 2), (0.5, *0 (0.5, 0.5),
(Q.5, 0.25), (l, l), (2, l), (0.25, l), (0.125, l). If, on examining
the results of this run sequence, it is found that the nominal increments
were seriously in error, new values should be selected and a new run
sequence carried out. The desired accuracy has been reached when the
concentration variables at the same time and position in the column agree
to the requisite number of decimal places.
-36-

-------
An example of this procedure is given in Tables (6-2) and (6-3).
Some of the batch experiments performed in this laboratory (see Part II)
gave an average solid diffusion coefficient of 0.25E-10, and the
corresponding isotherm and bed parameters are shown in Table (6-1) under
simulation data set (l). The filtration effects were eliminated by-
setting yQ = 0.0, and the experiments in Table (6-2) -were carried out
starting from nominal increments of (0.5, *0  The results indicate
that to obtain a consistency error of  0.00005 in C and Q the
increments should be chosen so that x^ _ 1 ft. and x^ 32 hr. In
this example, Q preserved a relative accuracy of four significant
figures as well as the absolute accuracy stated above. No deterioration
in accuracy is observed between z = 1 ft. and z = 10 ft. or between
t = 32 hr. and t = $6 hr. To find increments for the filtration process,
the parameters in data set (2) of Table (6-1) were employed. The
adsorption phenomena were left operative, and the nominal pair of
increments (0.5, b) were chosen. The first experiments (omitted for
brevity) showed that these nominal values were unsatisfactory, and a
second sequence with nominal values (0.25, l) is shown in Table (6-3).
By comparing approximate Y and S values, it was found that for a
consistency error <_  .0005, the increments should be chosen so that
x^ <_ 0.125 ft. and x^  1.0 hr.
In these two examples, the filtration process is much more
efficient than the adsorption process, and, consequently, requires smaller
increments. When the two phenomena are coupled, it is generally necessary
to base the increment size on the more sensitive set of equations.
However, since the effective T0C carried by the suspended solids is
considerably less than that carried by dissolved substances, a somewhat
larger error can be accepted in the filtration equations provided that
the adsorption equations are not significantly "contaminated". For
Model I, the question of stability is subordinate to the question of
accuracy, because in all experiments carried out, adequate accuracy
ensured stability. The filtration equations in the example above will
become unstable and"blow up" if x2 >_ 16 hr. , but this is well outside
of the accuracy range found desirable. Mild oscillations can sometimes
be tolerated, as, for example, in the effluent of a long column over the
first 2k hours of operation. It has been found that such oscillations
-37-

-------
TABLE (6-2) TEST ON INCREMENT SIZE (Adsorption Increments) Data Set (l)
Expt.
K1
X2




No.
ft.
hr.
C(32,l)
C(32,10)
C(96.l)
C(96,10)
1
0.5
2.0
0.864357E 0
0.376708E 0
0.866U35E 0
0.381179E 0
2
0.5
4.0
0.86U357E 0
0.376707E 0
0.866U33E 0
0.38117^ 0
3
0.5
8.0
0.86U356E 0
0.376706E 0
0.866k32E 0
0.381172E 0
k
0.5
32-0
0.86U353E 0
0.376685E 0
0.866428E 0
0.381152E 0
5
0.25
i+.O
0.86U358E 0
0.376706E 0
0.866U33E 0
0.381175E 0
6
1.0
l+.o
0.86U3U1E 0
0. 376693E 0
0.866383E 0
0.381163E 0
7
2.0
l+.o

0.376U50E 0

0.380971E 0



Q(32,l)
Q( 32,10)
ft(96,l)
Q(96,10)
1
0.5
2.0
Q.659755E-2
0.13U955E-2
0.196619E-1
0.U05690E-2
2
0.5
U.O
0.659755E-2
0.13U956E-2
0.196619E-1
0.1t05689E-2
3
0.5
8.0
0.659756E-2
0.i3lt95ltE-2
0.196619E-1
0.H05686E-2
U
0.5
32.0
0.6573U1E-2
0.132665E-2
0.196383E-1
0.1+03U32E-2
5
0.25
1+.0
0.659757E-2
0.134955E-2
0.196620E-1
0.1+05688E-2
6
1.0
1+.0
0.6597^7E-2
0.13^9^5E-2
0.196609E-1
0.1+05660E-2
7
2.0
U.O

0.13^773E-2

0.405187E-2
Result: for consistency to - .00005, x^ <_ 1 ft.
 32 hrs.

-------
TABLE (6-3) TEST ON INCREMENT SIZE (Filtration Increments) Run Sequence (2)
Expt.
No.
xl
ft.
X2
hr.
Y(l6.l)
Y(l6,M
Y(6U,1)
Y(6k,k)
1
0.25
U.O
0.6383U8E 0
0.130llUE-l
0.85361t0E 0
0.321359E 0
2
0.25
2.0
0.63U081E 0
0.131668E-1
0.852676E 0
0.320606E 0
3
0.25
1.0
0.632232E 0
0.13218UE-1
0.852278e 0
0.3202U9E 0
1+
0.25
0.5
0.631635E 0
0.132211E-1
0.852129E 0
0.320069E 0
5
0.5
2.0
0.657553E 0
0.895232E-2
0.85871+8E 0
0.333212E 0
6
0.125
1.0
0.630680E 0
0.135576E-1
0.85183UE 0
0.319^15E 0
7
0.0625
1.0
0.630560E 0
0.135906E-1
	 *
	 *



S(l6.0)
S(l6,U)
s(6U,o)
s(6i+,ii)
1
0.25
4.0
0.1280U6E-1
0.163895E-3
0.2629^E-l
0.86l9^5E-2
2
0.25
2.0
0.133695E-1
0.176115E-3
0.266366E-I
0.862130E-2
3
0.25
1.0
0.136372E-1
0.180617E-3
0.268001E-1
0.861826E-2
k
0.25
0.5
0.137671E-1
0.181561E-3
0.268797E-1
0.861^75E-2
5
0.5
2.0
0.133695E-1
O.98966UE-I*
0.266366E-1
0.888225E-2
6
0.125
1.0
0.136372E-1
0.186690E-3
0.268001E-1
0.860659E-2
7
0.0625
1.0
0.136372E-1
0.187173E-3
*
	 *
* Computer run terminated inadvertently
Result: for consistency to  .0005, x1 <_ 0.125 ft.
<_ 1.0 hr.

-------
are damped out, and do not significantly affect the results of long
simulations. Similarly, if most of the filtration is taking place
near the column inlet, it may happen that the retention (S) becomes
negative (and very small) near the bottom of the column. This indicates
that the increments are too large, but it can sometimes be tolerated.
One condition that cannot be allowed is an oscillation near the column
inlet. If, for example, the second value of C or Y is lower than the
third value, then the space increment must be decreased.
(VI-B) Comparison with Filtration Results
As a test of the integration scheme and also of the
filtration equations in Model I, a run was made to simulate the
calculated results as reported in Part III of this report. These
results pertain to a column length 130 cm, with an integration
point every 5 cm. Due to the special form of the filtration equations
in Model I, they can be uncoupled and integrated independently of each
other. One integration is carried out (analytically) for S at the
column inlet, and then a second differential equation with Z as the
independent variable is solved with the Adams-Moulton predictor-
corrector algorithm to find Y at equally spaced points in the column.
The parameters used are shown in Table (6-1) as data set (3), and the
compared results are shown in Table (6-k). The results reported in
Part IV may be regarded as accurate to five significant figures.
However, the parameters used are known only to four significant
figures, so perfect agreement is not possible. The actual results
agree to three or four decimals, with the largest discrepany in Y
being 0.005 at z = 10 cm, t = 1 hr. Reductions in increment size
would doubtless improve the agreement, but only up to a certain
point. The pressure profiles from [M3] cannot be readily compared
with the simulation output, because in the experimental work the value
of 6 in Equation (IV-ll) was calculated using trapezoidal integration.
This is not adequate for dealing with sharp S-profiles , and causes
discrepancies of 10# to 20% when compared with the Simpson's rule
integration of the simulation program.
-1+0-

-------
TABLE (6-U) COMPARISON WITH FILTRATION RESULTS


T
= 1.0 hr.



T
= 6.0 hr.

z
cm
Y1
P
Y
G1
P
S

Y
P
Y
0
P
s
0.00
10.00
20.00
30.00
i+0.00
50.00
60.00
TO. 00
80.00
100.00
120.00
1.000
0.1+95
0.210
0.082
0.030
0.011
0.001+
0.001
0.001
0.000
0.000
1.000
0.500
0.213
0.082
0.030
0.011
0.001+
0.001
0.000
0.000
0.000
0.3U10E-2
0.1687E-2
0.7H6E-3
0.2781+E-3
0.103ltE-3
0.3773E-1+
0.1368E-1+
0.1+950E-5
0.1790E-5
0.2338E-6
0.3052E-7
0.3297E-2
0.1698E-2
0.7222E-3
0.2782E-3
0.1023E-3
0.3689E-1+
0.1321E-1+
0.1+72 3E-5
0^l687E-5
0.215^-6
0.2753E-7

1.000
0.765
0.527
0.313
0.156
0.067
0.026
0.010
0.001+
0.000
0.000
1.000
0.766
0.529
0.315
0.157
0.066
0.025
0.009
0.003
0.000
0.000
0.1117E-1
0.85U3E-2
0.588BE-2
O.3I+98E-2
0.17^2E-2
0.71+1+9E-3
0.2897L-3
0.1077^-3
0. 3933E--1t
0.5162E-5
0.6745E-6
O.llOl+E-l
0.8568E-2
0.5919E-2
0.3519E-2
0.171+1+E-2
O.73I+9E-3
0.2795E-3
0.1013E-3
0.3603E-U
0.1+1+8UE-5
0.5558E-6


T
= 9.5 hr.



T
= 16 hr.

0.00
10.00
20.00
30.00
1+0.00
50.00
60.00
70.00
80.00
100.00
120.00
1.000
0.828
Q.6kb
0.1+55
0.280
0.11+5
0.06U
0.025
0.009
0.001
0.000
1.000
0.829
0.61+6
0.1+57
0.282
0.11+5
0.063
0.021+
0.009
0.001
0.000
0.11+12E-1
0.1170E-1
0.9100E-2
0.61+32E-2
0.3955E-2
0.201+1+E-2
0.8982E-3
0.35^6E-3
0.1327E-3
0.1763E-1+
0.2308E-5
0. ll+OlE-1
0.1172E-1
0.9129E-2
0.61+59E-2
0.3968E-2
0.2038E-2
0.8813E-3
0.339^-3
0.1235E-3
0.15U6E-1+
0.1906E-5

1.000
0.881+
0.758
0.620
0.1+73
0.325
0.193
0.096
0.01+1
0.006
0.001
1.000
0.885
0.759
0.621
0.1+75
0.327
0.194
0.096
o.oi+o
0.005
0.000
0.1792E-1
O.1585E-I
0.1358E-1
0.1111E-1
0.81+8UE-2
0.5831E-2
0.31+51E-2
0.1713E-2
0.7302E-3
0.105UE-3
0.1395E-1+
0.1783E-1
O.1586E-I
0.1360E-1
0.llll+E-1
0.8505E-2
0.581+1E-2
0. 3l+^+5E-2
0.1691E-2
0.7059E-3
0.9600E-1+
0.1193E-1+
'The variables subscripted "p" are Payatakes' results.

-------
(VI-C) Effect of Solid Diffusion and Liquid-Phase Mass Transfer on
Adsorption
Having found suitable increment sizes for integration, Model I
was used to examine the effect of D and k0. It has been suggested
[W-U] that adsorption in packed columns might be controlled by the
liquid-side mass transfer resistance during the brief period after
startup (T ^ 0). Under the assumption of zero solid-side resistance
(Q - Q* k - ), it can be shown that for the Freundlich isotherm
s
t>2
q* = b1c*	(VI-1)
and no interaction effects (r^ = 0), the adsorption equations in
Model I (5*^5 5.^6) become
dP	1/b2
H = -	(C - Q )	(VI-2)
0
b -1
c/	1/b
(A-) 
-------
a)	that the calculated effluent from 10 ft. and 20 ft. columns
was considerably higher than reported from the industrial
pilot plants [E-l], and
b)	that the liquid-side mass transfer coefficient as calculated
was not the controlling resistance (k^ could he 100 times
as large without appreciably affecting0the effluent TOC
concentration).
The effect of the solid diffusion coefficient Dg was explored
using the simulation data set (U) in Table (6-1). Concentration
profiles for C are shown in Figure (6-1). Increasing values of D
_qs
make the C-profiles increasingly sharp, but above D = 1.6 x 10
2	1 S
cm /sec the liquid film resistance (k = 100 hr ) becomes limiting.
Figure (6-2) shows the corresponding solid-phase Q-profiles. As D
s
increases, it is necessary to decrease the integration increments.
The time-step x5 must be varied inversely with D , while the space-
c	S
step x1 remains fairly constant. This behavior is entirely expected
since D (or k ) is used to scale the time variable,
s	s
9
Taking D = 0.1| x 10 as a starting point, the liquid-phase mass
s
transfer coefficient was decreased to determine its effect. The results
are shown in Figure (6-3). The curves for k^ = 100 and k^ = 20 are
indistinguishable, but for k^ = 10 the liquid side resistance is
clearly the controlling one. Lower values of k^ inhibit the overall
adsorption almost entirely.	
(VI-D) Adsorption and Filtration Profiles Over a Hundred-Dav Period
The combined effects of filtration and adsorption were simulated
over a hundred-day period with the parameters in data set (5) of Table
(6-1). The concentration and pressure profiles for the first ten feet
are shown in Figures (6-U), (6-5), (6-6), (6-7), and (6-8) at four times.
The adsorption curves for C and Q are shown at intervals of about 33 days.
Regular backwashing of the first half of the column occurs every four
days, but no regeneration is allowed. If an adsorption zone is defined
by some arbitrary concentration range such as 0.2 <_ C <_ 0.5, it can be
observed from Figure (6-M that this zone expands in a nonlinear manner
as it moves down the column. At 2k hours this zone is only 1.2 ft. long,
-1+3-

-------
1.0
T- 112 HR.
1% = 100 HR.~'
NO FILTRATION
DATA SET (4)
0.9
0.8
0.7
0.6
C 0.5
Ds=0.IE-9 CM / SEC.
0.4
0.3
Ds=0.4E-9 CM. / SEC.
0.2
Dft= 6.4E-9 CM./SEC.
5.0
0
1.0
2.0
3.0
4.0
z (FT.)
FIG. (6-1) EFFECT OF SOLID DIFFUSION COEFFICIENT ON C

-------
0.40
NO FILTRATION
DATA SET (4)
Ds s 6.4 E-9 CM.2/SEC.. x2=0.5HR.
x, s 0.25 FT.
0.35
x, = 0.25 FT.
0.30
0.25 -
x, = 0.5 FT.
Q 0.20
0.15
0.10
lD8= 0.1 E-4 CM. /SEC., x2 = 8 HR.
0.05
0
5.0
1.0
2.0
4.0
3.0
z(FT.)
FIG. (6-2) EFFECT OF SOLID DIFFUSION COEFFICIENT ON Q

-------
kfc = 0.25 hr:
ke0 2 i hr:1
0.9
0.8
k, = 10 HR."1
0.7
0.6
C 0.5
0.4
k*0sIOO OR
20 HR.
0.3
T= 112 HR.
D = 0.4E-9 CM./ SEC.
NO FILTRATION
DATA SET (4)
0.2
5.0
3.0
0
1.0
2.0
4.0
z (FT.)
FIG. (6-3) EFFECT OF LIQUID-PHASE MASS TRANSFER
COEFFICIENT ON C

-------
DATA SET (5)
O.S
0.8
0.7
0.6
C 0.5
0.4
0.3
0.2
T = 792 HR.
T = 24 HR.
2 (FT.)
FIG. (6-4) C-PROFILES OVER 100 DAYS

-------
DATA SET (5)
0.9
0.8
0.7
0.6
Q 0.5
T = 2304 HR.
0.4
0.3
0.2
T = 24 HR.
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
0
z (FT.)
FIG. (6-5) Q-PROFI LES OVER 100 DAYS

-------
1.0
DATA SET (5)
0.9
0.8
0.7
0.6
Y 0.5
0.4
T = 96 HR.
0.3
Ts48 HR.
T= 24 HR.
0.2
z(FT)
FIG.(6-6) V- PROFI LES OVER 4 DAYS

-------
0.9
0.8
0. 7
0.5
0.-?
0.3
0.2
PF*0fILF c
^ 0\Zpa *
tR*DAYs
F'G (6'7) S.

-------
0.8
0.7
T = 96 HR.
0.6
T = 72 HR.
0.5
T= 48 HR.
0.3
0.2
DATA SET (5)
z (FT.)
FIG. (6-8) P- PROFILES OVER 4 DAYS

-------
140
DATA SET (6)
TOTAL LENGTH" 16 FT.
* 120
I 10
100
co
a: 90
u. 80
O 70
O
u_
ft 60
< 40
oc
LlJ
CL
o 30
20
8 10 12 14 16 18
0
2 4
6
LENGTH OF A SECTION (FT.)
FIG. (6-9) VARIATION OF COLUMN PERFORMANCE
WITH NUMBER OF SECTIONS

-------
"but at 2301+ hours it is 6.5 ft- long. The rear end of this zone moves
at an even pace down the column, but the front end accelerates. This
behavior agrees with the findings of English [E-l] for waste water in
carbon columns, and Vermeulen's predictions for unfavorable isotherms
[V-l].
The filtration and pressure profiles have been recorded for the
first backwash cycle only. Due to the simplicity of the backwash model,
the backwash cycles become completely regular after 38i+ hours as shown
below in Table (6-5):
TABLE (6-5) BACKWASH CYLES
Cycle
Time
S(T,0)
P(T,20)
No.
hr

atm
1
96
0.015095
1.2321+
2
192
0.01521+7
1.21+83
3
288
0.0152I+9
1.21+91
1+
38U
O.OI52U9
1.21+92
5
1+80
0.01521+9
1.21+92
The filtration parameters vised throughout this chapter have been taken
from Payatakes' and Mehter's* experiments with activated carbon/
clay suspensions. Real waste water may behave somewhat differently.
(VI-E) Effect of Regeneration and Number of Sections Over
Extended Periods
As an example of the sort of question that the simulation can be
used to answer, it was desired to investigate the relationship between
column length and quantity of carbon to be regenerated per unit time.
This point is important, because the capital costs required to build
waste water treatment plants are much, greater than the operational costs.
If a 10 foot carbon column could be used instead of a 20 or 30 foot
column, considerable additional regeneration might be feasible for an
overall lower cost.
The parameters in data set (6) of Table (61) were used to compare
a 10 foot column and a 20 foot column. Both columns had two sections, and
the breakthrough concentration was set at 3 ppm T0C. Operation was
*See Part III of this Report
-41+-

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simulated over a year, and both columns came to a pseudo-steady state
of operation after a few cycles. The larger column required the
regeneration of 10 ft. of carbon every 1252 hr., while the smaller
column required the regeneration of 5 ft. of carbon every kkk hr.
The smaller column thus has a regeneration rate which is h3% higher
than the larger column.
To see if this result depends on the breakthrough level chosen,
the maximum allowable concentration was reduced to 2 ppm, and the same
two columns were simulated over a half year period. This time, the
larger column required regeneration every 7^2 hr., and the smaller
column every 192 hr. This shows that under the tighter effluent
restrictions the smaller column needs a 93% higher regeneration rate
than the larger column.
As another example, the effect of the number of sections was
investigated. It is intuitively obvious that an infinite number of
sections (continuous regeneration) will make the most efficient use
of the carbon. Since this arrangement is probably impractical, it is
necessary to know the relative efficiencies of 2 sections or k sections
or 8 sections (note that the total length of column is assumed to be
constant). The parameters in run sequence (6) were again used, but this
time with a l6 ft. column over a three-month period.* The column was
first simulated with one section, then with two, then with four, then with
eight, and finally with sixteen sections. The results are shown in
Table (6-6).
TABLE (6-6) EFFECT OF NUMBER OF SECTIONS
No. of
Length of
Time between
Hr. of Operation
Sections
each Section
Regenerations
per ft. Carbon

(ft)
(hr)
Regenerated
1
16
578
33.0
2
8
518
61+.75
1+
k
360
90.0
8
2
216
108.0
16
1
118
118.0
The hours of operation per foot of carbon regenerated are plotted in
Figure (6-9) against the length of a section. An extrapolation to zero
*CMAX = 2 ppm

-------
length gives 128 hours per foot of carbon regenerated, which can "be taken
as the column's most efficient performance for this set of parameters.
This performance can "be alternatively expressed as 1010 gallons of waste
water treated per pound of fresh carbon.
-1+6-

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CHAPTER VII
DISCUSSION, CONCLUSIONS AND RECOMMENDATIONS
(VII-A) Usage of the Simulation Program with Model I
It has been shown that Model I simulates the behavior of a carbon
column in a reasonable manner. The results are qualitative rather than
quantitative, however, and certain requirements must be met before Model I
can be used for design. These requirements are:
(a)	The filtration and pressure drop parameters must be determined
experimentally for waste water suspensions rather than clay.
(b)	The waste water used in these tests should be standardized
in some way. The most important consideration is that
of coagulation. It appears that most advanced waste
water treatment systems of the future will include
coagulation with lime or alum followed by sedimentation
of mixed-media filtration prior to carbon adsorption.
Laboratory and pilot plant studies should thus be based
on a clarified primary or secondary effluent.
(c)	Further rate and equilibrium studies are needed for the
adsorption of TOC from clarified waste water. The
simulation studies have already shown that 0.25 E-10
is too small a value for the solid diffusion coefficient
(a more reasonable value would be D = 0.U E-9).
s
(d)	Experiments are needed to determine the suspended solids
conversion factor (SSCF) for typical "clarified" waste
water. One possible technique would be to take TOC and
turbidimeter readings for unfiltered waste water samples,
and repeat the measurements on samples which have been
passed through a millipore filter. The turbidimeter
reading may be related to the volumetric suspended
solids concentration, and the difference between the TOC
readings is due to the filterable solids.
(e)	A better estimate is needed for RDKL, but this is difficult
to obtain directly. Experiments on partially clogged beds
with large pressure gradients might be useful, but the
-1+7-

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parameter-fitting technique of section (VII-C) may
be necessary.
(f) For lengthy simulations the adsorption parameters probably
do not change much, but the same may not be true of
filtration. If the detailed effects of backwashing cannot
be readily discovered, then at least the filtration and
pressure drop parameters must be based on beds which have
been used and backwashed several times.
With some assurance on these points, Model I could become a
powerful tool for laboratory, pilot plant, and full-scale design studies.
The first of these has already been demonstrated; Model I has suggested
critical new experiments, checked the magnitude of certain rate
parameters and permitted the testing of simple backwash and regeneration
effects. Pilot plants and actual sewage treatment facilities can be
simulated before being constructed to estimate the length of carbon
column needed for given effluent purity, the frequency of backwashing
and regeneration, and the size of pump required to overcome the head
loss. The effect of flow rate, number of sections, carbon and waste
water properties can all be conveniently explored.
Model I, however, assumes no major biological effects. If such
effects occur, Model I should still provide a basis for further
calculations. A particular installation may pre-chlorinate its waste
water and attempt to use carbon column adsorption without biological
interference. In this case, a comparison of Model I simulations with
actual plant data might reveal abnormal conditions at an early stage.
Waste water facilities are often quite automated and unattended. A
simulation program such as Model I could monitor periodically sampled
data and test the plant performance. If the waste water quality
changes appreciably (seasonal changes are quite large), then some of
the parameters for the simulation must be reevaluated. This can either
be done experimentally or by adding parameter-fitting techniques to the
simulation. Further uses of the simulation technique are mentioned
in the following sections.
-It 8-

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(VII-B) Short-Term Improvements for Model I
The need for certain critical experimental data in Model I has
already been discussed. There are several other desirable changes
which might be made to Model I, and most of these have programming
implications.
(a)	Some of the rate parameters could be evaluated as functions
of flow rate, carbon granule properties, and waste water
properties. Equation (IV-5) has already been recommended
for the estimation of k , and similar equations can
X
probably be found for the filtration effects. The smaller
the number of parameters that must be specified, the easier
it is to use the simulation.
(b)	Since pilot plants never maintain a steady input concentration,
the computer program for Model I should be able to accept
time-varying values of cq and y . One method for doing this
would be to read in the information at the start of the
simulation in triplets (cq, y , t). An interpolation
subroutine, (probably linear), could then be used to give
inlet concentrations at any desired time; this would be
called by subroutines UT and TO.
(c)	The solid-side mass transfer can probably be better
represented by Vermeulen's quadratic driving force
expression, as already mentioned in section (IV-B).
This requires reformulation of the model equations
(the result should be called Model II), and will
undoubtedly require longer CPU times.
(d)	The computer running times can be appreciably improved
by compiling the program on the FORTRAN IV level H
compiler (with OPT = 2) and making an object deck.
Present compilation times on the G level compiler
are about one minute, and can be largely obviated.
The H-compiler optimization should produce a program
which runs in about 30% less CPU time, but some de-bugging
probably has to be done before the logic is correct.
A version of the simulation program in FORTRAN II compatible
-1+9-

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with Smith's [5-3] executive program would also be
desirable.
(e) No economic factors have been included in the simulation.
If such factors are dealt with elsewhere, then the
simulation program should at least print out a detailed
analysis of all the backwash and regeneration events
that have occurred during a run.
(VII-C) Long-Term Simulation Improvements
Certain features of the simulation cannot be improved without the
acquisition of considerable quantities of new data, or extensive
reprogramming. These are as follows:
(a)	A sub-model is needed for th.e biological reactions along
the lines of section (III-D). Ingenuity must be used
in designing suitable experiments, because these effects
will be masked by the more obvious adsorption and
filtration effects.
(b)	The backwashing equations used in Model I are rather
naive. Some new theory and experiments are required.
(c)	When the simulation has become somewhat more sophisticated,
and the usual range of parameter values is well known, it
would be desirable to add nonlinear parameter-fitting
techniques to the simulation. At present, the program
is given estimates of the parameters, and asked to
simulate the column effluent. What is envisaged is the
inverse procedure, where pilot plant data is read into
the program and estimates of the parameters are called
for. The number of rate parameters found in this
manner cannot be reasonably be greater than four, due
to the low statistical quality of the data. The search
procedure should include an automatic modification of
integration increment size. This simulation could then
be used to analyze the behavior of carbon columns solely
by means of effluent data.
-50-

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LITERATURE CITED
[A1] ACRIVOS, A., "Method of Characteristics Technique Application
to Heat and Mass Transfer Problems", I&EC, 1+8, H, 703 (1956).
[A-2] ALLEN, J.B., R.S. JOYCE AND R.H. KASCH, "Process Design
Calculations for Adsorption from Liquids in Fixed Beds of
Greuiular Activated Carbon", WPCF Journal, 39, 2, 217 (1967).
[B-l] BAYLISS, J.R., "Elimination of Taste and Odor in Water",
McGraw-Hill Book Co., N.Y. (1935).
[B-2] BEEK, J., "Design of Packed Catalytic Reactors", "Advances
in Chem. Eng.", Vol. 3, 20*+, Acad. Press (1962).
[B3] BISHOP, D.F., L.S. MARSHALL, T.P. 0'FARRELL and R.B. DEAN,
"Activated Carbon for Waste Water Renovation: II. Removal
of Organic Materials by Clarification and Column Treatment",
1^9th Annual Meeting of American Chemical Society,
April 6 (1965).
[B-U] BUFFHAM, B.A. and L.G. GIBILARO, "The Analytical Solution
of the Deans-Levich Model for Dispersion in Porous Media",
CES, 23, 1399, (1968).
[C-l] CARBERRY, J.J. and M.M. WENDEL, "A Computer Model of the
Fixed Bed Catalytic Reactor: The Adiabatic and Quasi-Adiabatic
Cases", A.I.Ch.E. Journal, 9, 1, 129 (l96l).
[C-2] CHOLETTE, A. and L. CLOUTIER, "Mixing Efficiency Determinations
for Continuous Flow Systems", J. of Chem. Eng., 105, June (1959).
[C3] CHOLETTE, A., J. BLANCHET and L. CLOUTIER, "Performance of
Flew Reactors at Various Levels of Mixing", Can. J. of
Chem. Eng., 1, Feb. (i960).
[CU] COOPER, J.C. and D.G. HAGER, "Water Reclamation with Activated
Carbon", C.E.P., 62, 10, 85 (1966).
[C-5] CORRIGAN, T.E., H.R. LANDER, JR., R. SHAEFER and M.J. DEAN,
"A Two-Parameter Model for a Nonideal Flow Reactor",
A.I.Ch.E. Journal, 13, 5, 1029 (19^7).
[C-6] CORRIGAN, T.E. and W.0. BEAVERS, "Dead Space Interaction in
Continuous Stirred Tank Reactors", CES, 23, 1003 (1968).
[C7] COVER, A.E. and L.J. PIER0NI, "Appraisal of Granular Carbon
Contacting Phase I - Evaluation of the literature on the use
of granular carbon for tertiary waste water treatment
Phase II - Economic effect of design variables", Report for
the FWPCA, TWRC-11, May (1969).
-51-

-------
LITERATURE CITED (Cont'd.)
[C8] COVER, A.E., "Tertiaiy Waste Water Treatment Using Granular
Activated Carbon: Economic Effect of Design Variables", A.I.Ch.E.
Symposium on W.W. Treatment by A.C., Feb. 18 (1970).
[C-9] CULP, G. and A. SCHLECTA, "Plant Scale Regeneration of Granular
Activated Carbon", Final Progress Report, FWPCA, Feb. (1966).
[C-10] CULP, R.L., "Water Reclamation at South Tahoe Public Utilities
District", JAWWA, 86, Jan. (1968).
[D-l] DEANS, H.A. and L. LAPIDUS, "A Computational Model for
Predicting and Correlation of the Behavior of Fixed Bed
Reactors", Part I, A.I.Ch.E. Journal, 6, U, 656 (i960);
Part II, A.I.Ch.E. Journal, 6, U, 663 (i960).
[D2] DEDRICK, R.L. and R.B. BECKMANN, "Kinetics of Adsorption
by Activated Carbon from Dilute Aqueous Solution", Symposium
i+9, 'Research on Adsorption', A.I.Ch.E. 59th Annual Meeting,
Dec. (1966).
[E-l] ENGLISH, J., "Carbon Adsorption Project: Report of First Year
Activities (196k to 1965)", U.S. PHS, Los Angeles County-
Sanitation Districts, Sept. (1965).
[F-l] FAN, L.T., M.S.K. CHEN, Y.K. AHN and C.Y. WEN, "Mixing
Models with Varying Stage Size", Can. J. of Chem. Eng.,
Vf, llil (1969).
[F-2] FEICK, J. and D. QUON, "Mathematical Models for the Transient
Behavior of a Paced Bed Reactor", Can. J. of Chem. Eng., 1+8,
205 (1970).
[G-l] GOUDEY, R.F., "J. Am. Water Works Assoc. 23., 230 (1931).
[H1] HAGER, D.G. and RIZZO, J.L., "A Chemical and Physical
Wastewater Treatment Process", Preprint, 69th A.I.Ch.E.
Meeting, Feb. 18 (1970).
[K-2] HARRISON, L.B., "Use of Granular Activated Carbon at Bay City",
JAWWA, 32, 1+, 593 Cl9^0).
[H 3] HASSLER, J.W., "Active Carbon", Chemical Publishing Co., Inc.,
N.Y. (1951).
[E-k] HAUGHEY, D.P. and G.S.G. BEVERIDGE, "Structural Properties of
Packed Beds - A Review", Can. J. of Chem. Eng., ^7, 130 (1969).
[H51 HLAVACEK, V., "Aspects in Design of Packed Catalytic Reactors",
I&EC, 62, 7, 8 (1970).
[J-l] JOYCE, R.S. and V.A. SUKENIK, "Feasibility of Granular
Activated-Carbon Adsorption for Waste-Water Renovation 2",
U.S. Dept. of H.E.W. , AWTR-15, Oct. (1965) P.H.S. Pub. No.
999-WP-28.
-52-

-------
LITERATURE CITED (Cont'd.)
[K-l] KEINATH, T.M., "A Mathematical Model for Prediction of
Concentration-Time Profiles", Ph.D. Thesis, Univ. of
Michigan (1968).
[K-2] KRAMERS, H. and G. ALBERDA, "Frequency Response Analysis of
Continuous Flow Systems", CES, 2, 173 (1953).
[L-l] LEVICH, V.G., V.S. MARKIN and Yu.A. CHISMADZHEV, "On
Hydrodynainic Mixing in a Model of a Porous Medium with
Stagnant Zone", CES, 22, 1357 (1967).
[M-l] MAGTOTO, E.R., "Fixed-Bed Adsorption of Organic Water
Pollutants", Ph.D. Thesis, Univ. of Maryland (1966).
[M-2] MCGUIRE, M.L. and L. LAPIDUS, "On the Stability of a Detailed
Packed Bed Reactor", A.I.Ch.E. Journal, 11, 1, 85 (1965).
[M3] MEHTER, A.A., R.M. TURIAN and C. TIEN, "Filtration in Deep Beds
of Granular Activated Carbon", Syracuse University, Chem. Eng.
Dept., FWQA, Research Report 70-3, July (1970).
[P-l] PAN, C.-Y. and D. BASMADJIAN, "Constant-Pattern Adiabatic
Fixed-Bed Adsorption", CES, 22, 285 (1967)
[P-2] PARIS, J.R. and W.F. STEVENS, "Mathematical Models for a Packed-
Bed Chemical Reactor", Can. J. of Chem. Eng., U8, 100 (1970).
[P-3] PARKHURST, J.D., F.D. DRYDEN, G.N. MCDERMOTT and J. ENGLISH,
"Pomona Activated Carbon Pilot Plant", WPCF Journal, 39, 10,
Part 2, R70 (1967).
[P-1+] PERRY, J.H., "Chemical Engineer's Handbook", Fourth Edition,
McGraw-Hill (1963).
[R-l] REVELLE, C.S., LYNN, W.R. and RIVERA, M.A., "Bio-Oxidation
Kinetics and a Second-Order Equation Describing the BOD
Reaction", WPCF Journal, 37 12, 1679 (1965).
[S-l] SCHHUGER, M.J., H. JUENTGEN"and W. PETERS, "Adsorption of
Dissolved Substances on Activated Coke in Water. Kinetic
Measurements in a Shaking Reactor at Decreasing Concentration",
Chemie-Ingenieur Technik, ^0, 18, 903 (1968).
[S-2] SMITH, S.B., A.X. HILTGEN and A.J. JUH0LA, "Kinetics of Batch
Adsorption of Dichlorophenol on Activated Carbon", Chem. Eng.
Prog. Symposium Series, 55* 2U, 25 (1959)
[S-3] SMITH, R., R.G. EILERS and E.D. HALL, "Executive Digital
Computer Program for Preliminary Design of Wastewater
Treatment Systems", WP-20-lU, Cincinnati, Ohio FWPCA
(Aug., 1968).
-53-

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LITERATURE CITED (Cont'd.)
[V-l] VANIER, C.R., Ph.D. Thesis, Syracuse Univ., (1970).
[V2] VERMEULEN, T., "Separation by Adsorption Methods", Adv.
in Chem. Eng. II, (ed. Drew and Hooper), Chap. 5,
Academic Press, New York. (1958).
[V 3] VERMEULEN, T. and R.E. QUILICI, "Analytic Driving-Force Relation
for Pore-Diffusion Kinetics in Fixed-Bed Adsorption", I&EC
Fundamentals, 9, 1> 179 (1970).
[W-l] WEBER, W.J., JR. and J.C. MORRIS, "Kinetics of Adsorption on
Carbon From Solution", Journal of Sanitary Engineering Division,
ASCE, 89, SA2, 31 (1963).
[W-2] WEBER, W.J., JR. and R.R. RUMER, JR., "Intraparticle Transport
of Sulfonated Alkylbenzenes in a Porous Solid: Diffusion with
Non-Linear Adsorption", Water Resources Research, 1, 361 (1965).
[W-3] WEBER, W.J., JR. and J.P. GOULD, "Sorption of Organic Pesticides
from Aqueous Solution", Adv. Chem. Ser. 60, 280, Am. Chem. Soc.
Pub., Washington (1966).
[W-it] WEBER, W.J., JR. and T.M. KEINATH, "Mass Transfer Rates for Water
and Waste Pollutants in Fluidized Adsorbers", Paper 1+9F presented
at the 59th Annual A.I.Ch.E. Meeting, Detroit, Michigan,
December U-8 (19&6).
[W-5] WEBER, W.J., JR., C.B. HOPKINS and R. BLOOM, "Physicochemical
Treatment of Wastewater", WCPF Journal, U2, 1, 83 (1970).
[W-6] WILHELM, R.H., "Progress Towards the A Priori Design of
Chemical Reactors", Pure and App. Chem., 5, ^03 (1962).
[Y-l] YOUNG, J.C. and J.W. CLARK, "Second Order Equation for BOD",
Journal of Sanitary Engineering Division, ASCE, 91, No. SA1,
1+3 (1965).
[Z-l] ZUCKERMAN, M.M. and A.H. M0L0F, "High Quality Reuse Water by
Chemical-Physical Wastewater Treatment", WPCF Journal, U2, 3,
Part I, 1+37 (1969).
-5b-

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NOMENCLATURE
Us age
parameters in Freundlich isotherm, Eq. (IV-3)
concentration of DOC
column inlet value of c at t = 0
dimensionless DOC concentration, Eq. (IV-22)
mean granule diameter
conversion factor, Eq. (III-6)
effective solid-phase diffusion coefficient
liquid-phase diffusion coefficient of adsorbate
unknown functions
liquid-phase mass transfer coefficients
biological rate constants, Eqs. (III-5, III-6,
III-T)
filtration coefficient
solid-phase mass transfer coefficients
length of column
amount of DOC transferred to particle
dimensionless scale factors, Eqs. (lV-37 IV-38,
IV-39,	IV-UO)
pressure
solid-phase concentration of adsorbate
initial value of q
dimensionless solid-phase adsorbate
concentration, Eq. (IV-23)
retention degradation of , Eq. (iv-28)
rates for biological reaction
rate of particle deposition of filtration
process
-55-

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NOMENCLATURE (Cont'd.)
S	dimensionless retention, Eq. (IV-25)
t	time, characteristic coordinate
T	dimensionless time, Eq. (lV-30)
u, uq	flow velocity
U,	dependent variable
x^, Xg	step sizes in z and t directions
y, yq	suspended solids (SS) concentration
Y	dimensionless SS concentration, Eq. (IV-2U)
z	spatial coordinate, characteristic coordinate
Z	dimensionless z, Eq. (IV-31)

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APPENDIX A
(A:5) POrlRAI? Lis*:ri.T of Simulation Program
I /	(0451.CHE,45,40),'VAN IER...FIN IS'
// EXEC FORTGCLGfPARM.FORT^MAP* ,PARM.LKED= 'MAP'
//FORT.SYSIN DO *
C MPROG
C THIS PROGRAM SIMULATES THE PERFORMANCE OF A GRANULAR ACTIVATED
C CARBON COLUMN FOR WASTE WATER TREATMENT. THE MAIN PROGRAM
C READS INPUT DATA FOR THE SIMULATION, PRINTS A RECORD OF ALL
C PARAMETERS TO RE USED, AND CALLS IJPUN VARIOUS CALCULATION
C SUBROUTINES. THE MAIN ASSUMPTIONS OF THE SIMULATION ARE AS
C FOLLOWS :
C (I) THE COLUMN CAN BY REPRESENTED AS A CONTINUOUS MEDIUM. THUS
C CONCENTRATIONS WITHIN THE COLUMN ARE SMOOTH FUNCTIONS OF
C Z AND T, AND MEANINGFUL AVERAGE VALUES EXIST FUR PARTICLE
C DIAMETER.VOID FRACT ION , FLOW RATE,ETC.
C (2) PLUG FLOW EXISTS IN THE COLUMN. THUS, THE EFFECT OF RADIAL
C CONCENTRATION GRADIENTS AND AXIAL DISPERSION IS ASSUMED TO
C BE NEGLIGIBLE.
C (3) THE ORGANIC CONTAMINANTS IN THE WASTE WATER CAN BE
C ADEQUATELY REPRESENTED BY THE TOTAL ORGANIC CARBON IN
C SOLUTION { TOC ), AND THE VOLUMETRIC SUSPENDED SOLIDS
C CONCENTRATION ( SS ).
C (4) THESE CONCENTRATIONS ARE SMALL (DILUTE SOLUTION).
C (5) ISOTHERMAL CONDITIONS PREVAIL DURING THE SIMULATION.
C (M INTRAPARTICLE CONCENTRATION GRADIENTS CAN BE REPRESENTED BY
C AN AVERAGE SOLID CONCENTRATION AND SUITABLE SOLID PHASE MASS
C TRANSFER EQUATIONS.
C (7) THE VOLUMETRIC FLOW RATE IS A FUNCTION OF TIME ONLY.
C (R) THE PHYSICAL PHENOMENA INVOLVED ARE ADSORPTIONIFFUSION,
C FILTRATION, AND BIOCHEMICAL REACTION.
C THESE 8 ASSUMPTIONS ARE INTERRELATED, AND IN SOME SENSE
C ESSENTIAL TO THE MODEL. A SECOND SET OF ASSUMPTIONS IS ADOPTED
C FOR SIMPLIFICATION, (BUT IS NOT ESSENTIAL) :
C (A) THE MEAN FLOW VELOCITY IS CONSTANT.
C (B) THE MEAN VOID SPACE IS CONSTANT.
C (C) THE BIOCHEMICAL EFFECTS ARE ABSENT.
C (D) BACKWASHING REMOVES A FIXED FRACTION OF THE PARTICLE SLIME
C LAYER.
C < E > THE SLIME LAYER (RETENTION) CAUSES THE LIQUID FILM MASS
C TRANSFER COEFFICIENT TO DECREASE LINEARLY.
C (F) A QUADRATIC FREUNDLICH ISOTHERM REPRESENTS THE ADSORPTION
C EQUILIBRIUM AT THE EXTERNAL PARIT ICLE SURFACE.
C (G) SOLID PHASE MASS TRANSFER IS GOVERNED BY A LINEAR GLUECKAUF
C TYPE DRIVING FORCE EXPRESSION.
C (H) FILTRATION AND PRESSURE DROP PARAMETERS EVALUATED FOR AN
C INITIALLY CLEAN BED CAN BE USED AFTER BACKWASHING AND SLIME
C LAYER GROWTH.
C THE FOLLOWING SUBROUTINES ARE USED :
C	MOC : SOLVES PLUG FLOW EQUATIONS BY 3RD ORDER METHOD OF
C CHARACTERISTICS AND PRINTS CONCENTRATION PROFILES.
C	BWASH : SIMULATES EFFECT OF BACKWASHING ON THE FIRST
C SECTION OF THE COLUMN.
C	REGEN : REMOVES THE FIRST SECTION OF THE COLUMN AND
C PLACES A FRESH SECTION AT THE END.
C	UZT : CHARACTERISTIC NORMAL ORDINARY DIFFERENTIAL
C EQUATIONS (5) FOR ADSORPTION AND FILTRATION.
C	ZO	J INITIALIZATION ALONG T=0 AXIS* AND AFTER EACH
C	BACKWASH. INTEGRATION BY 4TH ORDER RUNGE-.KUTTA
C	METHOD WITH RESPECT TO Z. (CALLED BY MOC )
C	TO	: INITIALIZATION AT COLUMN INLET (Z=0). INTEGRATION
C	BY 4TH ORDER RUNGE-.KUTTA METHOD WITH RESPECT TO T.
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APPENDIX A (Cont'd.)
C	(CALLED BY MOO
C	liZ : SPECIAL FORM OF CHARACTERISTIC NORMAL EQUATIONS IN
C	C AND Y NEEDED ALONG T=CONSTANT. (CALLED RY ZO)
C	UT : SPECIAL FORM OF CHARACTERISTIC NORMAL EQUATIONS IN
C	0 AND S NEEDED ALONG Z=0. (CALLED RY TO)
C 	INPUT OATA FORMAT	
C USERS SHOULD READ THE FORTRAN 4 SPECIFICATIONS FOR NAME LISTS.
C ANY NUMBER OF SIMULATION RUNS CAN RE MADE CONSECUTIVELY, EACH
C RUN CORRESPONDING TO AM INPUT DATA SET. EACH INPUT DATA SET
C CONSISTS OF DESCRIPTIVE COMMENT CARDS FOLLOWED RY 4 NAME LISTS
C AS SHOWN BELOW :
C CARD (1J	STARTING IN COLUMN 1,
C CARBON
C CARDS (2) , (3) , . .. (N) ,	ANY FORMAT,
C ANY NUMBER OF COMMENT CARDS.
C CARD ( N + l )	STARTING IN COLUMN 1,
C ENQCOMMENT
C CARDS (N+2),(N+3),(N+4),(N+5), ETC,	STARTING IN COLUMN 2
C 6FBP PR=?,UO=?,DP=?,VOIDS=?,L=?,NS=?, SEND
C OPCON CO=?,YO=?,SSCF=?, &END
C 6REQP DS=?,KLO=?,FI=?,?,FC=?,?,?,RDKL=?,PRD=?,?SRF=?, SEND
C &CONTRL TBA = ?,T PR INi=?,TMAX=?,KPX=?,DPMAX=?,CMAX=?,TREGEN=?,
C X=?,?,&END
C THE 4 NAMELISTS MUST BE IN THIS ORDER, RUT WITHIN A NAME LIST
C THE DATA MAY HAVE ANY ORDER. THE FIRST DATA SET MUST BE
C COMPLETE. THEREAFTER, ONLY THOS PARAMETERS WHICH ARE TO BE
C CHANGED NEED BE MENTIONED IN FRESH DATA SETS. HOWEVER, DUE TO
C THE VAGARIES OF FORTRAN U, EACH NAMELIST MUST CONTAIN AT LEAST
C ONE ELEMENT OF DATA. THE PROGRAM GIVES A DEFINITION OF THESE
C INPUT PARAMETERS WITH APPROPRIATE UNITS. SEE ALSO THE PH.D.
C DISSERTATION BY C.R.VANIER, S.U., 1970 . IN PARTICULAR, NOTE :
C	X(l) AND X(2) ARE THE INTEGRATION INCREMENTS IN SPACE AND
C TIME RESPECTIVELY. THESE ARE AOJUSTED RY THE PROGRAM SO THAT
C THEY ARE APPROPRIATE PROPER FRACTIONS (IF L AND TBA. IF SPECIFIC
C TIME INCREMENTS ARE DESIRED, DEFINE TBA TO BE AN INTEGER
C MULTIPLE OF X(2>. SIMILARLY, A SPECIFIC SPACE INCREMENT CAN BE
C OBTAINED BY ENSURING THAT L/NS IS AN INTEGER MULTIPLE OF X(l).
C EITHER THE FILTRATION OR THE ADSORPTION FFFECTS CAN RE
C EFFECTIVELY 'TURNED OFF' BY SETTING Y0=0.0, OR C0=0.0 RESP.
C SOME TYPICAL INCREMENTS WHICH GIVE ABOUT 4 DECIMALS ACCURACY :
C FOR ADSORPTIOM : X ( I ) = I FT, X(2)=32 HOURS (DS = 2.'5E-11 , KL0=5)
C FOR FILTRATION : X(l) = .25 FT, X(2) = .5 HOURS (FC = *ft,67,2.5 )
C THESE ARE NOT GUARANTEED OUTSIDE OF THE PARAMETER RANGE INWHICH
C THEY WERE OBTAINED. NOTE THAT THE FILTRATION EQUATIONS MAY
C DESTABILISE THE ADSORPTION EQUATIONS DUE TO COUPLING BY KLO .
C T PR I NT AND KPX CONTROL THE EXTENT OF PRINTOUT AS FOLLOWS :
C	TPRINT	IS AN INTEGER WHICH SPECIFIES HOW MANY TIME-STEPS
C	MUST ELAPSE BETWEEN PROFILE PRINTOUTS.
C	KPX	IS AN INTEGER WHICH SPECIFIES THE NUMBER OF SPACE-STEPS
C	BETWEEN EACH ELEMENT OF U(l,J) TO RE PRINTED. THIS AFFEC1S
C	COMBINED TUC PROFILES, COMPLETE PROFILE PRINTOUTS, AND
C	PROFILES AFTER BACKWASH INO AIVD REGENERATION.
C TO SIMULATE COLUMNS WITH MORE "THAN 100 SLICES* CHANGE THE
C SECOND DIMENSION FOR U ,U 1 , t)2 , DU , DU1 , Dl 12 EVERYWHERE THEY OCCUR.
C (AND ALSO THE IF STATEMENT IN THE MAIN PROGRAM NUMBERED 111).
C	 PROGRAM AUTHOR : CHRIS VANIF.R	
C	SYRACUSE UNIVERSITY, CHEM ENG DEPT	
C	FORTRAN 4 VERSION	
DIMENSION TITLE (16), C0MENT(16)
COMMON U(5 100),U1(5,100)DU1(4,100),N< 5),TIMEtNSLICE
-58-

-------
APPENDIX A (Cont'd.)
COMMON /P AR M 1/P B, IJO, DP, VOIDS, L,NS
COMMON /PARM2/CO,YO,SSCF
COMMON /PARM3/DS,KLO,FI(2),FC(3),RDKL,PRD(2),SRF
COMMON /KSS/XINC,KS,KLOK
COMMON /PARM4/TBA,TREGEN,TMAX,DPMAX,CMAX,X(2),TPRINT,KPX
REAL*4 KL0,N,L,KS,LSAVE
INTEGER TPR I NT
DATA DUMMY/'AAAA/,ENDC/ENDC'/,CARB/'CARB'/
ONAMELI ST /FBP/PB,UO,DP,VOI OS,L,NS /OPCON/CO,YO,SSCF
1	/REOP/DS,KLO,FI,FC,RDKL ,PRD, SRF
2	/CONTRL/TRA,TPRI NT,TMAX,DPMAX,CMAX,X,TREGEN,KPX
NRUN=0
C
C READ IN DATA
C
10 READ<1,20,END=999,ERR=920> RCODE
20 FOR MAT(16A4)
IF(RCODE.NE.CARB) GO TO 10
NRIJN=NRUN+1
WRITE(3,30)NRUN
30 F0RMAT('1','CARRON COLUMN SIMULATION. RUN NO. ',13/
1  37('-')//)
35 READ!1,20,ERR=920,ENP=999) COMENT
IF(COMENT(1).EO.ENDC>G0 TO 50
WRITE(3,40)COMENT
AO FORMAT( '  , 16A4)
GO TO 35
50 READ(1,FBP,ERR=920,END=999)
READ(1,0PC0N,ERR=920,END=999>
READ(1,RE0P,ERR=920,END=999)
READ(1,C0NTRL,ERR=920,END=999)
XS2=X(2)
X1 SAVE = X( 1 )
USAVE=UO
LSAVE=L
PSAVE=PRD(1)
TBSAVE=TBA
T SAVE = TREGEN
C
C CONVERT UNITS AND INITIALIZE
C
JMAX = IFI X C L/( FLOAT< NS)*X(1))) +1
IFIJMAX.EO.(2*1JMAX/2))) JMAX=JMAX+1
NSLICE^l+NSMJMAX-l)
111 IF(NSLICE.LT.100) GO TO 55
WR IT E(3 ,53) NSLICE
53 FORMAT ('ONUMBER OF SLICES = M10, / CHANGE PROGRAM DIM',
1  ENS IONS OR CHOOSE A SMALLER L/XI2) RATIO'/)
GO TO 920
55 X(l)=1.0/ FLOAT(NSLICE-1)
XINC=LSAVF.*X< 1)
XS1=L*X(1)
U0=U0*244.46
L=L*30.4fl
KL0K=0
X (2) =TBA/ FLOAT( I FI X(TBA/X(2) ))
PRD(1)=PRD(1)*L
X1CM=XS1*30.4R
X2MIN=XS2*60.0
U(l,l=1.0
-59-

-------
APPENDIX A (Cont'd.)
1)13,1 ) = 1.0
Dn 15 J=1,NSLICE
HI(5  J) = 0.0
U ( 5  J)=0.0
on 15 K=2,4,2
15 U ( K,J)=0.0
C
C LIST SIMULATION PARAMETERS
C
WRIT t ( 3 ,60)
60 FORMA T('OF IXED BED PARAMETERS',/1 ',20('-')/)
WR I TE (3*62) PRUO,USAVEDP,VOIDSL,LSAVE*NS
62 FORMAT(1 BULK DENSITY DF CARBON (PR) = ',F7.4,
1* GM/CM**3'/' MEAN SUPERFICIAL FLOW RATE (UO) = ',F7.2
2,* CM/HR =  F7.3,' GPM/FT**2'/t
3 MEAN PARTICLE DIAMETER (DP) = ',F7.4,' CM'/
4	 VOID FRACTION OF RED = ',F7.A/
5' TOTAL LENGTH OF COLUMN (L) = ,F8.2, CM = ,F7.3, FT'/
61 NUMBER OF SECTIONS (NS) = ',13//)
WRITE(3,70)
70 FORMAT( 'OOPERATING CONDI 1 I ONIS  * /  ',20(,-M/)
WR ITE(3*72) C0*Y0,SSCF
72 FORMAT( DISSOLVED TOC INPUT CONCENTRATION (CO) = ',E11.3,
1	' GM CARB0N/CM*3 WASTE WATER'/
2	 SUSPENDED SULIDS INPUT CONCENTRATION (YO) = ',E11.3,
3	' VOLUMES/VOLUME WASTE WATER'/
A ' SUSPENDED SOLIDS CONVERSION FACTOR (SSCF) = ',F.3,
5	 GM T0C/CM**3 SUSPENDED SOLIDS'//)
WRITE(3,80)
80 FORMAT ('ORATE AN[) EQUILIBRIUM PARAMETERS'./' SSll'-M/)
WRITE(3,82) DS*KL0,FI,FC,RDKL,PSAVt,PRD( 2),SRF
82 FORMAT( ' EFFECTIVE SOLID DIFFUSION COEFFICIENT (OS) = '
1	E12.4 ,  CM**2/SEC 1/
2	' LIQUID FILM MASS TRANSFER COEFFICIENT FOR CLEAN GRANULE'
3	,'S (KLO) = *E12.4,' 1/HR */
4	 FREUNDLICH ISOTHERM PARAMETERS IFI) = ',2E12.4/
5	 NOTE : 0 = FI(1 I*C**FI(2)'/
5	 FILTER PARAMETERS = ' ,3E12.4/
6	 RETENTION DEGRADATION OF LIQUID FILM MASS TRANSFER COEF'
7	,'FICIENT (RDKL) = ,E12.4/  PRESSURE DROP PARAMETERS (P'
8	,'RD) = S2E12.4,' ATM/CM, DI MENS IONL E SS ' /
9	' SLIME RESIDUE FACTOR (SRF) = *,F7.3//)
WRITE(3,90)
90 FORMAT( 'OCONTROL DATA FOR SIMULAT ION',/' r 27('-)/)
WRITE(3,92 ) TRA,TREGEN,TPRINT,KPX,TMAX,DPMAX,CMAX,XS1,
1XS2,X1CM,X2MIN
92 FORMAT(' BACKWASH INTERVAL (TBA) = ',F7.1, HRS'/
1 ' REGENERATION TIME (TREGEN) = ,F7.1,' HRS'/
1  PRINT EVERY ',13, TIME-STEPS AND EVERY M3,' SPACE-ST'
1	, 'EPS'/
2	 SIMULATION INTERVAL (TMAX) = '*F8.1, HRS'/
3	' PRESSURE OROP MAXIMUM (DPMAX) = ,E12.4,' ATMOSPHERES'/
A ' BREAKTHROUGH CONCENTRATION (CMAX) = ,E11.3*' GM TOC/CM'
5	,.##31/1 INTEGRATION INCREMENTS (X) = 21:11.3*' FT, HRS'
6	/ 1 27X , '= ~2E11.3,' CM, MIN'//)
C
C CALCULATION OF DI MENS IONLESS SCALE FACTORS
C
KS=15.0*DS*3600.0/DP#*2
X(2)=X(2)*KS
-60-

-------
APPENDIX A (Cont'd.)
TBA=TBA*KS
TD=L*VOIDS/UO
N ( 1 )=L*KL0/U0
IF(CO.NE.O.O) GO TO 94
N(1)=0.0
N ! 2)=0.0
GO TO 9ft
94 N!2)=KL0/!KS*PB*FI!l)*CO)
96	IF(Y0.NE.0.0) GO TO 97
N(3)=0,0
GO TO 98
97	N(3)= L*FC(1)/U0
9R N!4)=Y0*FC( 1 )/KS
C
C EXECUTION PHASE
C
WRITE ( 3, 100) 
-------
APPENDIX A (Cont'd.)
C 11(4,J) = S , SOLID PHASE DEPOSITED SOLIDS (RETENTION)
C U(5 ~ J) = P , PRESSURE DIFFERENCE FROM INLET
C
DIMENSION CHAR(5),Z(100)
DIMENSION 1)2(5,100) , DU ( 4, 100) ,DU2(4, 100) t TOC ( 100)
COMMON U(5,100),U1(5,100),DU1(4,100),N(5), TIME,NSLICE
COMMON /PARM2/C0,YO,SSCF
COMMON /PARM3/0S,KL0,FI(?),FC(3),RDKL,PRD(2),SRF
COMMON /PARM4/TBA,TREGEN,TMAX,DPMAX,CMAX,X(2)TPRINTKPX
COMMON /KSS/XINC ,KS,KLOK
RE AL*4 KL0,N,L,KS,LSAVE
INTEGER T PR I NT
DATA CHAR/'C'. 0,,,Y , 'S't 'P'/
IFIT1ME.NE.0.0) GO TO 10
X13=X(1)/3.0
X 23 = X(2)/3  0
X 1 2 = X( 1 )/2.0
X22=X(2)/2.0
X1M2=X(1 )*2.0
TREG = TR EGEN
XP=PRD(2)*X13
DO 5 J=1,NSLICE
5 Z(J)=XINC* FLOAT ( J 1 )
C
C INITIALIZATION ALONG T=0 AXIS
C
10 T=0.0
K = 0
CALL ZO
DO 20 J = 1,NSLICE
DO 15 1=1,4
15 U1(I,J)=U( I , J)
CALL UZT(J)
DO 20 1=1,4
20 DU(I,J )=DU1(I,J)
C
C ROUNDARY CONDITION AT INLET
C
27 K = K + 1
T = K*X(2 )
KLOK=KLOK+1
TIME= X(2)* FLOAT(KLOK) /KS
DO 120 J=2, NSLICE
CALL TO
CALL UZT(1)
C
C FIRST APPROXIMATION
C
30 DO 40 1=1,3,2
U1(I,J)=U1(I,J-ll+U(I,J)-U-
1 DU(I,J-l) )
40 U1(I+1,J)=U1(I+1,J-l)+U( 1 + 1,J)-U(1+1,J-l)+X< 2)#(DU(1 + 1,J)-
1 DU(1+1,J-l ) )
CALL UZT(J)
C
C SECOND APPROXIMATION
C
DO 60 1=1,3,2
Ul( I , J)=U1(I.J)+X12*(DU1(I,J)+DU(I,J-l)-DU1(I,J-l)-DU(I,J) )
60 U1(I+1,J)=U1(1+1*J)+X2 2*(DU1(I+1,J)+0U
-------
APPENDIX A (Cont'd.)
1 )-DU(I+l,J))
CALL U.7 T{ j)
C
C APPLICATION OF SIMPSON'S RULE
C
NJ= 1- (J-2**Y0)*1.0E6
WRITE(3, 163) (TOC(J),J=1,NSLICE,KPX)
163	FORMAT! 5E15.6)
C
C LOGICAL CHECKS TO DETERMINE DETAILED PRINTOUTS, BACKWASHES,ETC
C
IF
175 FORMAT (  I (FEET ) * / ( BF15.5))
DO 166 1=1,5
WRIT E(3,169) CHAR( I )
169 FORMAT! ,A4)
166 WRITE(3,167) IU(I,J),J = 1 ,NSLICE ,KPX)
-63-

-------
APPENDIX A (Cont'd.)
167 FOR MAT < fif: I 5 . 6 I
180 IF(TIME.LT .TMAX) GO TO 165
I STQP= 1
WRI1EI3, 190)
190 FORMAT('OS IMUL AT ION TIME LIMIT REACHED'/)
GO TO 300
1*5 IF(TUCINSLICC).LT.
-------
APPENDIX A (Cont'd.)
C CONCENTRATIONS U(2,J) AND U(A,J) IN THE BOTTOM SECTION.
C
COMMON U(5,100),U1(5,100),DU1(A,100),N< 5)TI ME,NSLICE
COMMON /PARMA/TBA,TREGEN,TMAX,OPMAX,CMAX,X(2), TPRINT,KPX
DIMENSION CHAR(5)
REALA KL 0,N,L
INTEGER TPR I NT
DATA CHAR/'C'1Q ,Y*' S', ' P  /
00 10 J=JMAX,NSLICE
K=J-JMAX+ 1
U(2,K)=U(2,J)
UI2,K)=U(2,J)
10 U(A,K)=U(A,J)
KMAX=NSLICE-JMAX+1
DO 20 J=KMAX  NSLICE
U(2,J)=0.0
20 U { A, J) =0 . 0
WR I T E(3 30)
30 FORMAT(OREGENERATED SECTION ADOEO. 1ST SECTION DROPPED.'/)
DO 50 1=2,4,2
50 WRITE(3,AO) CHAR(I),(UU,J),J=1,NSLICE,KPX)
AO FORMAT!  '.Al/C S5E15.6))
RETURN
END
SUBROUTINE UZT(J)
C
C THIS SUBROUTINE IS THE HEART OF THE SIMULATION. IT CONTAINS
C THE MASS BALANCE AND RATE EQUATIONS FOR ALL COMPONENTS.
C 0U(1,J) = OUlll.J) = DC/DZ
C DU ( 2 , J ) = DIJ1 ( ? , J ) = DY/DZ
C DU(3,J) = DU1(3,J) = DO/DT
C DU( A, J ) = DUl(JfJ) = DS/DT
C.
COMMON UI 5,100),U1(5,100),DU1(A,100),N(5)TI ME ,NSLICE
COMMON /PARM3/DS,KL0,FI(2),FC(3)~RDKL,PRO(2)SRF
RFAL*A KLO,N,L
YA=1.0-RDKL*U1(A,J)
Y1=N(2)*YA
Y2=U1( 1, J)-0.5*(-Yl+ S0RT(Yl*Yl + A*(Ul(2tJ)+Yl'ttUl II, J) ) ) )
Y3=l) 1 ( 3 , J ) / ( 1+FC(2)*Ul (A,J) )**FC(3)
DU 1 ( 1, J )=-N(1)*YA*Y2
DU1(2,J)=Y1#Y2
DU1(3,J)=-N<3)*Y3
DU1(A,J)=N(A)*Y3
RETURN
END
SUBROUTINE TO
C
C VALUES MUST BE SUPPLIED FOR C(0,T) AND Y(0,T).
C ASSUME C=Y=1 AT Z=0, AND INTEGRATE OVER ONE TIME STEP.
C IN GENERAL, THE INPUT CONCENTRATIONS C=U1(1,1),AND Y=Ul(3,l)
C MAY BE ARBITRARY FUNCTIONS OF TIME. THE FOLLOWING CODE ASSUMES
C THAT THEY DO NOT VARY FROM THEIR INITIAL VALUE OF UNITY.
C
COMMON U(5100),U1(5,100),DU1(A,100),N(5),TIME,NSLICE
COMMON /PARMA/T BA, TREGEN,TMAX,DPflAX,CMAX,X(2) , TPR I NT, KPX
REAL*A KLO,N,L
DIMENSION DF
-------
APPENDIX A (Cont'd.)
Y(1)=U< 2,1)
Y(2)=U(4,1)
CALL IJT (Y,DF, 1 )
DO 20 1=2,4
Y ( 1)= Y<1 >+DF(1-1,1 )*X(2)/D( I )
Y(2)= Y12)+DF(I-1,2)*X(2)/D(I)
20 CALL UT(Y * DF * I )
U1(2,1)=U(2,1)+X26*(DF(i,l)+2.0*(DF(2,l)+UF(3,l)) +DF(4,1) )
U1 ( 4 , 1 ) = U (4, 1 )+X26*(DF(1,2) + 2.0*(DF(2,2)+0F(3,2) )+DF(4,2) )
RETURN
END
SUBROUTINE UT(Y,DF,I)
C
C EVALUATION OF FUNCTIONS TO BE INTEGRATED ALONG Z=0 AXIS
C Y(1)=0, Y ( 2 ) = S  DF( I , 1)=DQ/DT  DF( I,2)=DS/DT
C
COMMON U(5,100),U1(5,100),DU1(4,100),N(5),TIME,NSLICE
COMMON /PARM3/DS,KL0,F1(2),FC<3),RDKL*PRD(2)~SRF
RE AL*"4 KLO, N, L
DIMENSION Y(2) ,DF(4,2)
Y1= 1-RDKL*Y(2)
Y2=N(2)*Y 1
DF(I,1)=	Y2*(1.0-0.5*(-Y 2+ SORT(Y2*Y2 + 4*(Y(1)+Y2 ) ) ) )
DF(I,2)=N(4)/(1+FC(2)*Y(2))**FC(3)
RETURN
END
SUBROUTINE ZO
C
C INTEGRATION ALONG T=0 BY 4TH ORDER RUNGE-KUTTA METHOD.
C IT IS ASSUMED THAT Q
-------
APPENDIX A (Cont'd.)
Y2 = W(I1)*U(4,J) + WU,2)*U(4J+1)
Y1 = 1.0 -RDKL#Y2
Y3=N(2)*Y1
OH ( I , 1)=-N( 1 )Yl#( Y( 1 )-0.5*( ( Y3 ) + SORT ( Y3*Y3+4MW( I tl ) *
1 IM 21 J ) + W( I 2)*li<2J+l) + Y3*Y(l) ) ) J )
DH(I 2 > = -N(3)*Y<2)/(1.O + FC(2>*Y2)**FC(3
RETURN
END
/*
//GO.SYSIN DD *
CARBON
THIS DATA SET CORRESPONDS TO NO.(6) IN TARLE (6-1)
EXTENDED CARBON COLUMN SIMULATION
REGULAR BACKWASHING NO PRESSURE BREAKTHRU ALLOWED
DATA COMES FROM JEFF'S COLUMN
DARCO CARBON AT 30 DEG.CENT I GRADE
EN DC
CF BP
UO=3.2DP=.064 8 ,V0IDS=.5L=16.0,NS=1,PB=.39,
CEND
COPCON
Y0=10.0E-6, SSCF = 1.0 C0 = 40.0E-6,
CEND
CR EOP
KLO=100.0,FI = 1.35AlEt)t2.0 ,FC=4fl.0,67.Ot2.5, DS=4.0E-10,
PRD=1.bE-3,28 3,SRF=.05,RDKL=10.0,
f. END
CCONTRL
TPRINT=12,TMAX=1200.0, DPMAX=3.00*CMAX=2.00E-6, TBA=96.0 KPX=2,
TREGEN=5000.0,
X=.50,2.0 CENO
CARBON
NNNNNNN22222
ENDC
CF BP NS= 2 SEND
COPCON SSCF = 1.0 CEND
tREOP KLO=100.0 CEND
CCONTRL KPX=2 CEND
CARBON
ENDCOMMENT
CFBP NS = 4 CEND
COPCON SSCF = L.0 CEND
CREOP KLO=100.0 CEND
CCONTRL K PX = 2 CEND
CARBON
TEST 3333333
ENDCOMMENT
CF BP NS=8 CEND
COPCON SSCF = 1.0 CEND
CR EOP K. L 0 =100.0 CEND
CCONTRL KPX=2 CEND
CARBON
TEST 2222222222
ENDCOMMENT
CF BP NS=16 CEND
COPCON SSCF=1.0 CEND
CR EOP KL0=100.0 CEND
CCONTRL KPX=2 CEND
CARBON
THIS IS DATA SET NO{3) IN T ABL E(6-1)
-67-

-------
APPENDIX A (Cont'd.)
COMPARISON WITH PAYATAKES & MEHTER
RUN NO. 16, TARLES E8,E9 REFERENCE PAYATAKES
CARBON TYPE : DARCO 20*40 MESH, DXL-0-3892
ENDCOMMENT
FBP U0=2.029,DP=.0594,VOIDS=.49,L=4.265092,NS=lPB=.39,6END
fcOPCON Y0=88.2E-6,C0=0.0 ,SSCF=1.0 END
CREOP FC=50.47,30.4fc,5.09 50,PRD=.335E-3,464.1,KL0=100.0,DS=2.5E-11,
SRF=.05,RDKl=10.0,FI=l.3451E9,2.0 GEND
tCONTRL TPRINT=2iTMAX=2.00tX=.082021?.25,TBA=48.0,KPX=2,
TREGEN=200.0 ,CMAX=1.0,
DPMAX=1.0 &END
/*
-68-

-------
APPENDIX B
Algorithm for the Numerical Integration of
Semi-Linear Hyperbolic Equations
The algorithm developed by Vanier and designated as CN553
for the numerical integration of semi-linear hyperbolic equations
is briefly described as follows. Briefly, this algorithm is aimed
at the solution of equations of the following type:
(Ul}z = fl (zl tl Ul' U2)	(A_1)
(U2) = f2 (Sl tl	U2)
(A-2)
The algorithm has a basic integration step defined on a
rectangle such as Fig. A-l
,k	6/
(n, J-l)
(n-1, j-l) X
(n, J)
3(n-l, j)
The basic grid spacing along the z direction is denoted by h
and that along t is denoted as he, the subscripts (n,j), denotes
the coordinates to be z = n*h t = (j)(h0). It is assumed all the
dependent variables and their derivatives are known as points
1, 2 and 3, and our purpose is to find U1 and Ug and their
derivatives at point 6.
-69-

-------
APPENDIX B (Cont'd.)
The basic algorithm can be described in terms of the
following steps:
(3)	(3)
(a)	Calculate values	and U2 by the following expressions:
ui3)  uinj-i+ "iT1 - ui7-i *11 (fij-i - fi j-i'  = U2 j-1 * U2 j"1 " U2 j-i + eh (f2 j"1 - f2>i (A-l4)
(3)	(3)
Note that	and	can be calculated directly, since it
is assumed that values of U^, U2 are known at points 1, 2 and 3.
(3)	(3)
(b)	If the values of	and U^ are taken as approximations
of and at point 6, one can estimate the values of
functions f^ and f^ at point 6, i.e:
fij = fi(zi + h' ti + 0h ui3) ' u23))	(A_5)
f
" = f2(zx + h, t + eh, u[3) , u^3))	(a-6)
2j
(c)	Based upon the values of f..n and f?n> a fourth order
1	1
approximation of and can be 0calculated:
+  (-f n + f n-"''  f n f n )	(a-7)
U1 ~ U1 + 2 Ulj flj-l flj U-l; K U
(U) _ (3) . 6h , n n-1 n-1 n *	, Q)
2 " U2 + 2 2 j + 2 j-1 2j " 2 j-1
(d)	Using	and	as approximations, one can evaluate
f ^ and f ^ (i.e. repeating step b). Based on these new values
In 2n
^The superscript (3) denotes that the expression is of third-order approxi-
mation of the dependent variables U and Ug.
-70-

-------
APPENDIX B (Cont'd.)
fin ^2n' ^ ^-mProve<^ va-lue of at point 6 can be obtained
according to:
Un = Un +  (f n +iifn +fn)	(A-Q)
Ulj lj-2 3 Ulj-2 4I1J-1 Xlj;	(A
This step, however, is carried only for j being the odd number.
(e) Similarly, for U0, the calculation is made according to:
U n = U n-^ +  (f n-^ + Uf n ^ + f n)	(A-10)
2j 2J 3 k 2j *2J Z2iJ	U)
for n being the odd number.
-71-

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Accession Numbci
Subject Field &, Group
05D
SELECTED WATER RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
Organization
Department of Chemical Engineering and Metallurgy
Syracuse University
Title
Carbon Column Operation in Waste Water Treatment
10
Authors)
Vanier, Christopher
and Chi Tien
16
Project Designation
17020 DZO
21 Note
22
Citation
Descriptors (Starred First)
*waste water treatment, *Activated carbon, *Adsorption, *Carbon,
*Filtration
25
27
Identifiers (Starred First)
Physical-chemical separation, wastewater purification
Abstract A mathematical model has been devised to simulate the adsorption
and filtration of waste water in an isothermal column packed with granular
activated carbon. The adsorption process is considered to be controlled
by a combination of liquid phase diffusion and interparticle diffusion
which can be approximated by a solid phase rate expression based upon
Glueckauf's Linear driving force. The filtration rate equation is assumed
to be the same as that of filtration of clay suspension in a carbon bed,
which was investigated experimentally as a companion study in this work.
The effects of column backwashing and regeneration are also included in
the model.
Abstractor Joseph F. Roesler
Institution
EPA, WQO
WR.I02 (REV JULY (969)
WR SI C
SEND TO: WATER RESOURCES SCIENTIFIC INFORMATION CENTER
U S DEPARTMENT OF THE INTERIOR
WASHINGTON, D. C 20240
 GPO: 1969~ 359-339

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