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Sorption in Flow Through Porous Media
SORPTION IN FLOW THROUGH POROUS MEDIA DAVIDW. HENDRICKS
INTRODUCTION
In one dimensional flow of a dissolved solute through a saturated homogeneous porous media, in which the solute sorbs on the granular material, the concentration of that solute species in the liquid phase C ( Z , t ) , and in the solid phase, X ( 2 , t ) , will vary with time, t , and distance, Z . It is the objective herein to outline the fundamental principles relevant to the solutions in time and space for both solid and liquid phases, C ( 2 , t ) and X ( Z , t ) . Such solutions are useful in designing and in specifying operation of ionexchange and adsorption columns, and in evaluating the movement characteristics of a ground water contaminant. MATHEMATICAL DESCRIPTION
The elements of the overall problem are readily delineated by a conventional mass balance analysis on a column slice of infinitesimal thickness, which results in the differential equation
ac az
ac
-- -O-+D
at
net rate of (change of sorbate conc.
) ( =
-a2c --paz2
net convection) rate
+
ax
1- P P at net (dispersion) rate
(sorption ) rate
where : C = concentration of sorbate species in liquid phase (gmlml) t = time from a convenient reference point, such as initial introduction of the sorbate (min.) Z = distance from beginning of flow path (cm) 6 = interstitial flow velocity (cmlmin.) D = coefficient of disperion for the porous media at velocity v (cm’jmin.) p = density of the individual granular particles comprising the porous media (gm3/cm) P = porosity of the porous media (expressed as a decimal fraction) 8 = concentration of sorbate species in the solid phase (gm sorbate/gm sorbent 384
Sorption in Flow Through Porous Media
385
Eq. 1 is intuitively meaningful, if each of the several groupings of variables is expressed in terms of its respective physical significance, as written below Eq. 1. Thus at any given location on the flow path, the rate of change of sorbate concentration depends upon the transport rate to that location by convection and dispersion, and the rate of removal by the sorbent. The conditions change, of course, in time and space, and the solution C ( Z , t) is two dimensional, being expressed as C ( Z ) ,, the concentration profile at a fixed time, or C ( t ) , , known as the “breakthrough curve,” at a given distance, Z . The variables 6 , ( D / a ) , p , and P must be evaluated by measurement, which is straightforward. ‘The term D / a is the measurable property of the porous media (see Rifai, Kaufman and Todd, 1956). The kinetic term, &?/at, is the most profound mathematically, and also is the most difficult term to evaluate. KINETICS
Three kinetic mechanisms are presented in the literature (Helferrich, 1962; Hiester and Vermeullen, 19521; these are: (1) chemical reaction kinetics, (2) liquid phase diffusion, and (3) solid phase diffusion. It is generally conceded that (1) is not rate limiting (Hiester and Vermeullen, 1952). Either or both of the other two mechanisms may be rate limiting, however, depending upon chemical and flow conditions; it is very difficult to discern the conditions controlling the respective kinetic mechanisms. Obtaining a practical rate law for either category of diffusion is also difficult. Fick’s first law describes either solid or liquid phase diffusion, but is difficult to apply in its differential form. Hiester and Vermeullen (1952) have presented approximations to Fick’s first law (flux density = diffusion coefficient x grad. concentration) for both liquid and solid phase diffusion which, when combined with the mass action equilibrium equations, results in expressions for each mechanism having a form identical to the equation for second-order reaction kinetics for the sorption reaction. Keinath and Weber (1968) have applied the methods of Hiester and Vermeullen to predict breakthrough curves for fluidized beds; their experimental results agree well with predicted breakthrough curves. As pointed out by Helferrich (1962), many rate equations are suggested in the literature, and may apply quite well under the conditions in the systems investigated, but fail under other conditions. For the work reported herein, the experimental data (X vs r ) from batch sorption tests were found to be best described by the equation :
(z)p
=bC(8*
- R)
where,
(axpt),,= kinetic term when solid phase diffusion is rate controlling
David W. Hendricks
386
b R*
=
diffusion coefficient in solid phase (gm-ml/gm-min.)
= equilibrium concentration of sorbate in solid phase, corre-
sponding to the liquid phase concentration, C (gm sorbate/gm sorbent)
Experimental data using two experimental systems (rhodamine-B dye and Dowex 50 resin and rhodamine B dye and Eau Claire sand), were fitted to several empirical kinetic equations, but Eq. 2 was the best fit. It can also be argued that Eq. 2 has some rational basis, having similarity with second order kinetics (the rate is proportional to the product of two reactants). While a suitable kinetic equation having a broad application is desirable, in the absence of such an equation, or if it is difficult to apply, it may be more pragmatic to find an empirical equation which fits the data for the range of interest. Application of Eq. 2 was somewhat more formidable than immediately apparent, however. It is necessary to evaluate the isotherm, ,P*(C),, for the particular sorbate-sorbent system in question. The Langmiur isotherm is both widely used and has a rational basis, and fits the experimental data, so was used in this work; the Langmiur isotherm is defined
R*
(3)
x,
-
aC*
l+uC*
where,
X,
= ultimate capacity of sorbent (gm sorbate/gm sorbent)
= Langmiur coefficient C* = equilibrium liquid phase concentration (gmlml) (C is used for calculating ;P*, however) This equation was experimentally defined for two temperatures 23°C and 21°C having respective Langmiur constants
a
u = .96, X , = 4000 and a = 0.62,
X,
= 2760
(the values given for a and X , are for the rhodamine-B-Dowex 50 system and should be multiplied by to obtain units in grams). This strong temperature sensitivity caused considerable instability in the C(Z), measured profiles. Another impediment in using Eq. 2 lies in evaluating the term b, which unfortunately is not a constant. This term was found to have the empirical form (4)
b
= (.00417 x 10
--I .5
x/x
)c-
.02
which also can be rationalized. The fit for this equation to the data is shown in Fig. 1, which is for the rhodamine B-Dowex 50 system. Eq. 4 was corroborated also for the rhodamine-B-Eau Claire sand system with more limited data,
Sorption in Flow Through Porous Media
387
10
. Y
u .-
.-U z 0
U
W *
a 4
W
0. I
Y
a *
a
3
.01
Concentration Rh-B solution, (pg/mL)
Fig. 1. Uptake rate coefficient, D, for Dowex 50 resin as affeced by solution and relative degree of resin saturation
concentraticn
the constants were different, of course. Alphabetic points shown are from measurements on column profiles, which provides a n independent corroboration of the equation. At this point it should be noted that Eqs. 2, 3, and 4 comprise the description of kinetics, as measured for the experimental system studied. Conditions for C and X change with time and space, which implies that Eq. 2 also changes both along the distance, Z , and with time. CONVECTIVE DISPERSION
A fourth kinetic mechanism is convective-dispersion. This mechanism is rate limiting for the portion of the C(Z), profile from the inflection point forward.
David W. Hendricks
388
When this condition prevails the sorbate is sorbed as quickly as’it is delivered to the sorbent neighborhood, and aC/at x 0. Thus Eq. 1 becomes
-ac= o = at
- -o -ac +D
az
a2c az2
p--1 - P
P
ax at
Solving for aX/at give
The subscript “CD” refers to the convective-dispersion mechanism. The task now is to find an expression for C ( Z ) which will allow solution for Eq. 6. This function is unique for the particular porous media in question and is influenced also by flow velocity. Thus C ( 2 ) must be an experimentally determined function under conditions such that (aX/dt>,, < ( a X / a t ) , . The nature of the C ( Z ) function is reasoned as follows. Consider a single sorbate particle traveling with the interstitial stream. As the stream bifurcates and undergoes velocity changes, in a random manner, a single sorbate particle will have a certain fixed probability, unique for the particular porous media, of making a collision with the solid phase. Velocity of the fluid stream is relevant t o the collision since this affects the zone of diffusion influence. Thus for a given porous medium and a given flow velocity, a single sorbate particle will have a probability of say 0.50 of making a collision with the solid phase within a certain distance of travel (call it the “half distance”). Consider now 100 sorbate particles in the fluid stream initially; 50 will remain after one half distance, 25 after the next half distance, and so on. This suggests a decay of the form (7) where,
C ’ , = C(Zl,) Z ’ , = the pivot point, where (aX/at), = (aX/af),, Z’ = z - Z ‘ , I = collision probability coefficient. Taking derivatives on Eq. 7 gives
(9) and
=
azc
-- az2
I * C(Z) A2 . C ( 2 )
Sorpiiotr in Flow Through Porous Media
389
Substitutine Eqs. 9 and 10 in Eq. 6 gives
The terms 17,D, and C ( Z ) in Eq. 1 1 are indicative of the transport rate at the position Z ; the term I is the collision probability coefficient. This coefficient is determined experimentally by Eq. 7. Fig. 2 illustrates the existence of Eq. 7 for a measured C ( Z ) , curve. Fig. 3 shows the effect of flow rate (veloctiy) upon the collision probability coefficient, A.
D i s t a n c e along column.
z
(Cm)
Fig. 2. Illustration of decay equation for measured concentration profile at t Eau Claire sand and rhodamine B dye
=
30 min. for
CALCULATIONS
Either Eq. 2 , or Eq. 11 may be rate controlling. The computer must test each of these equations at each Z increment along the column, and for each new time increment. The equation having the smaller value for ( 6 Z j d t ) governs,
390
David W. Hendricks 1.0
.9
7
.3 .2
I 0
2 00
100
300
Q (mllmin)
Fig. 3. Sorbate - sorbent collision probability, 1, as affected by flow rate in 1.5" dia. column of 16-20 mesh Dowex 50 resin
-
I00 m a n s balance
computer solution of m a s s balance and particle k i n e t i c s equations for t = 2 0 h r . F i g . 5
80
60
. 2
w
U
40
2onl O
\
particle kinetics
I
L
.
.
8
12
z (cm 1
16
20
2 i
28
Fig. 5.1 Illustration of model components showing zones af influence of governing equations
D i s t a n c e along c o l u m n , Z ( c m )
E x p e r i m e n t a l input data for computer program
.-i
I&
392
David W . Hendricks
and is used in the finite difference calculation of Eq. 1 for a given time and distance. It has been proven (Hendricks, 1965) that the inflection point in the C ( Z ) , profile is the transition point from Eq. 2 to Eq. 11. Fig. 5 summarizes the zones of applicability of Eqs. 1 , 2 , and 11. Fig. 4 is a computer output for solutions of Eq. 1 in the form C ( Z ) , , C(Z)r2,... C(Z),". These solutions are compared also with several measured profiles, shown as dashed lines in Fig. 5; the match is only nominal due to fluctuations in room temperature which had a marked effect on the isotherm. The computer solutions did behave in an identical manner to the measured solutions, however, when the same variations in room temperature and flow rate were imposed. Computer solutions to X ( Z ) , were also found ; this solution consists merely with a single measured solution was quite favorable. Further work is now underway to investigate kinetics more exhaustively under a variety of chemical and physical conditions. Work is also proceeding on bacterial adsorption which will be related to the mechanisms of transport of bacteria through granular porous media.
ACKNOWLEDGMENT
This investigation was supported, in part, by a research fellowship (number 1-Fl-WF 24125-01) from the Division of Water Supply and Pollution Control U.S. Public Health Service during the academic year 1964-65. This report is based upon a Ph.D. dissertation completed in June 1965 at the University of Iowa. Dr. H. S. Smith was major professor for the work; his criticisms are gratefully acknowledged. The assistance of Dr. G. F. Lee is also appreciated.
REFERENCES
1. HENDRICKS, D. W. (1965), Sorption in flow through porous media, unpublished Ph.D. dissertation, Univ. of Iowa, Iowa City. 2. HELFFERICH, F. (1962), Ion Exhange, McGraw-Hill, New York. 3. HIESTER,.N.K. AND T. VERMEULEN (1952), Saturation performance of Ionexchange and Adsorption columns, Chemical Engineering Progress, 48, 505-516. 4. KEINATH, T. M. AND W. J. JR. (1968), A predictive model for the design of fluid-bed adsorbers, J. WPCF, May. 5. RIFAI,M. N. E., W. J. KAUFMAN AND D. K. TODD(1956), Dispersion Phenomena in Laminarpow through porous media, Report No. 2, r.E.R. Series 90, Sanitary Engin. Research. Lab., Univ. of Calif., Berkeley.
UTAH WATERRESEARCH LABORATORY, UTAHSTATEUNIVERSITY, LOGAN, UTAH 84321
INTRODUCTION
In one dimensional flow of a dissolved solute through a saturated homogeneous porous media, in which the solute sorbs on the granular material, the concentration of that solute species in the liquid phase C ( Z , t ) , and in the solid phase, X ( 2 , t ) , will vary with time, t , and distance, Z . It is the objective herein to outline the fundamental principles relevant to the solutions in time and space for both solid and liquid phases, C ( 2 , t ) and X ( Z , t ) . Such solutions are useful in designing and in specifying operation of ionexchange and adsorption columns, and in evaluating the movement characteristics of a ground water contaminant. MATHEMATICAL DESCRIPTION
The elements of the overall problem are readily delineated by a conventional mass balance analysis on a column slice of infinitesimal thickness, which results in the differential equation
ac az
ac
-- -O-+D
at
net rate of (change of sorbate conc.
) ( =
-a2c --paz2
net convection) rate
+
ax
1- P P at net (dispersion) rate
(sorption ) rate
where : C = concentration of sorbate species in liquid phase (gmlml) t = time from a convenient reference point, such as initial introduction of the sorbate (min.) Z = distance from beginning of flow path (cm) 6 = interstitial flow velocity (cmlmin.) D = coefficient of disperion for the porous media at velocity v (cm’jmin.) p = density of the individual granular particles comprising the porous media (gm3/cm) P = porosity of the porous media (expressed as a decimal fraction) 8 = concentration of sorbate species in the solid phase (gm sorbate/gm sorbent 384
Sorption in Flow Through Porous Media
385
Eq. 1 is intuitively meaningful, if each of the several groupings of variables is expressed in terms of its respective physical significance, as written below Eq. 1. Thus at any given location on the flow path, the rate of change of sorbate concentration depends upon the transport rate to that location by convection and dispersion, and the rate of removal by the sorbent. The conditions change, of course, in time and space, and the solution C ( Z , t) is two dimensional, being expressed as C ( Z ) ,, the concentration profile at a fixed time, or C ( t ) , , known as the “breakthrough curve,” at a given distance, Z . The variables 6 , ( D / a ) , p , and P must be evaluated by measurement, which is straightforward. ‘The term D / a is the measurable property of the porous media (see Rifai, Kaufman and Todd, 1956). The kinetic term, &?/at, is the most profound mathematically, and also is the most difficult term to evaluate. KINETICS
Three kinetic mechanisms are presented in the literature (Helferrich, 1962; Hiester and Vermeullen, 19521; these are: (1) chemical reaction kinetics, (2) liquid phase diffusion, and (3) solid phase diffusion. It is generally conceded that (1) is not rate limiting (Hiester and Vermeullen, 1952). Either or both of the other two mechanisms may be rate limiting, however, depending upon chemical and flow conditions; it is very difficult to discern the conditions controlling the respective kinetic mechanisms. Obtaining a practical rate law for either category of diffusion is also difficult. Fick’s first law describes either solid or liquid phase diffusion, but is difficult to apply in its differential form. Hiester and Vermeullen (1952) have presented approximations to Fick’s first law (flux density = diffusion coefficient x grad. concentration) for both liquid and solid phase diffusion which, when combined with the mass action equilibrium equations, results in expressions for each mechanism having a form identical to the equation for second-order reaction kinetics for the sorption reaction. Keinath and Weber (1968) have applied the methods of Hiester and Vermeullen to predict breakthrough curves for fluidized beds; their experimental results agree well with predicted breakthrough curves. As pointed out by Helferrich (1962), many rate equations are suggested in the literature, and may apply quite well under the conditions in the systems investigated, but fail under other conditions. For the work reported herein, the experimental data (X vs r ) from batch sorption tests were found to be best described by the equation :
(z)p
=bC(8*
- R)
where,
(axpt),,= kinetic term when solid phase diffusion is rate controlling
David W. Hendricks
386
b R*
=
diffusion coefficient in solid phase (gm-ml/gm-min.)
= equilibrium concentration of sorbate in solid phase, corre-
sponding to the liquid phase concentration, C (gm sorbate/gm sorbent)
Experimental data using two experimental systems (rhodamine-B dye and Dowex 50 resin and rhodamine B dye and Eau Claire sand), were fitted to several empirical kinetic equations, but Eq. 2 was the best fit. It can also be argued that Eq. 2 has some rational basis, having similarity with second order kinetics (the rate is proportional to the product of two reactants). While a suitable kinetic equation having a broad application is desirable, in the absence of such an equation, or if it is difficult to apply, it may be more pragmatic to find an empirical equation which fits the data for the range of interest. Application of Eq. 2 was somewhat more formidable than immediately apparent, however. It is necessary to evaluate the isotherm, ,P*(C),, for the particular sorbate-sorbent system in question. The Langmiur isotherm is both widely used and has a rational basis, and fits the experimental data, so was used in this work; the Langmiur isotherm is defined
R*
(3)
x,
-
aC*
l+uC*
where,
X,
= ultimate capacity of sorbent (gm sorbate/gm sorbent)
= Langmiur coefficient C* = equilibrium liquid phase concentration (gmlml) (C is used for calculating ;P*, however) This equation was experimentally defined for two temperatures 23°C and 21°C having respective Langmiur constants
a
u = .96, X , = 4000 and a = 0.62,
X,
= 2760
(the values given for a and X , are for the rhodamine-B-Dowex 50 system and should be multiplied by to obtain units in grams). This strong temperature sensitivity caused considerable instability in the C(Z), measured profiles. Another impediment in using Eq. 2 lies in evaluating the term b, which unfortunately is not a constant. This term was found to have the empirical form (4)
b
= (.00417 x 10
--I .5
x/x
)c-
.02
which also can be rationalized. The fit for this equation to the data is shown in Fig. 1, which is for the rhodamine B-Dowex 50 system. Eq. 4 was corroborated also for the rhodamine-B-Eau Claire sand system with more limited data,
Sorption in Flow Through Porous Media
387
10
. Y
u .-
.-U z 0
U
W *
a 4
W
0. I
Y
a *
a
3
.01
Concentration Rh-B solution, (pg/mL)
Fig. 1. Uptake rate coefficient, D, for Dowex 50 resin as affeced by solution and relative degree of resin saturation
concentraticn
the constants were different, of course. Alphabetic points shown are from measurements on column profiles, which provides a n independent corroboration of the equation. At this point it should be noted that Eqs. 2, 3, and 4 comprise the description of kinetics, as measured for the experimental system studied. Conditions for C and X change with time and space, which implies that Eq. 2 also changes both along the distance, Z , and with time. CONVECTIVE DISPERSION
A fourth kinetic mechanism is convective-dispersion. This mechanism is rate limiting for the portion of the C(Z), profile from the inflection point forward.
David W. Hendricks
388
When this condition prevails the sorbate is sorbed as quickly as’it is delivered to the sorbent neighborhood, and aC/at x 0. Thus Eq. 1 becomes
-ac= o = at
- -o -ac +D
az
a2c az2
p--1 - P
P
ax at
Solving for aX/at give
The subscript “CD” refers to the convective-dispersion mechanism. The task now is to find an expression for C ( Z ) which will allow solution for Eq. 6. This function is unique for the particular porous media in question and is influenced also by flow velocity. Thus C ( 2 ) must be an experimentally determined function under conditions such that (aX/dt>,, < ( a X / a t ) , . The nature of the C ( Z ) function is reasoned as follows. Consider a single sorbate particle traveling with the interstitial stream. As the stream bifurcates and undergoes velocity changes, in a random manner, a single sorbate particle will have a certain fixed probability, unique for the particular porous media, of making a collision with the solid phase. Velocity of the fluid stream is relevant t o the collision since this affects the zone of diffusion influence. Thus for a given porous medium and a given flow velocity, a single sorbate particle will have a probability of say 0.50 of making a collision with the solid phase within a certain distance of travel (call it the “half distance”). Consider now 100 sorbate particles in the fluid stream initially; 50 will remain after one half distance, 25 after the next half distance, and so on. This suggests a decay of the form (7) where,
C ’ , = C(Zl,) Z ’ , = the pivot point, where (aX/at), = (aX/af),, Z’ = z - Z ‘ , I = collision probability coefficient. Taking derivatives on Eq. 7 gives
(9) and
=
azc
-- az2
I * C(Z) A2 . C ( 2 )
Sorpiiotr in Flow Through Porous Media
389
Substitutine Eqs. 9 and 10 in Eq. 6 gives
The terms 17,D, and C ( Z ) in Eq. 1 1 are indicative of the transport rate at the position Z ; the term I is the collision probability coefficient. This coefficient is determined experimentally by Eq. 7. Fig. 2 illustrates the existence of Eq. 7 for a measured C ( Z ) , curve. Fig. 3 shows the effect of flow rate (veloctiy) upon the collision probability coefficient, A.
D i s t a n c e along column.
z
(Cm)
Fig. 2. Illustration of decay equation for measured concentration profile at t Eau Claire sand and rhodamine B dye
=
30 min. for
CALCULATIONS
Either Eq. 2 , or Eq. 11 may be rate controlling. The computer must test each of these equations at each Z increment along the column, and for each new time increment. The equation having the smaller value for ( 6 Z j d t ) governs,
390
David W. Hendricks 1.0
.9
7
.3 .2
I 0
2 00
100
300
Q (mllmin)
Fig. 3. Sorbate - sorbent collision probability, 1, as affected by flow rate in 1.5" dia. column of 16-20 mesh Dowex 50 resin
-
I00 m a n s balance
computer solution of m a s s balance and particle k i n e t i c s equations for t = 2 0 h r . F i g . 5
80
60
. 2
w
U
40
2onl O
\
particle kinetics
I
L
.
.
8
12
z (cm 1
16
20
2 i
28
Fig. 5.1 Illustration of model components showing zones af influence of governing equations
D i s t a n c e along c o l u m n , Z ( c m )
E x p e r i m e n t a l input data for computer program
.-i
I&
392
David W . Hendricks
and is used in the finite difference calculation of Eq. 1 for a given time and distance. It has been proven (Hendricks, 1965) that the inflection point in the C ( Z ) , profile is the transition point from Eq. 2 to Eq. 11. Fig. 5 summarizes the zones of applicability of Eqs. 1 , 2 , and 11. Fig. 4 is a computer output for solutions of Eq. 1 in the form C ( Z ) , , C(Z)r2,... C(Z),". These solutions are compared also with several measured profiles, shown as dashed lines in Fig. 5; the match is only nominal due to fluctuations in room temperature which had a marked effect on the isotherm. The computer solutions did behave in an identical manner to the measured solutions, however, when the same variations in room temperature and flow rate were imposed. Computer solutions to X ( Z ) , were also found ; this solution consists merely with a single measured solution was quite favorable. Further work is now underway to investigate kinetics more exhaustively under a variety of chemical and physical conditions. Work is also proceeding on bacterial adsorption which will be related to the mechanisms of transport of bacteria through granular porous media.
ACKNOWLEDGMENT
This investigation was supported, in part, by a research fellowship (number 1-Fl-WF 24125-01) from the Division of Water Supply and Pollution Control U.S. Public Health Service during the academic year 1964-65. This report is based upon a Ph.D. dissertation completed in June 1965 at the University of Iowa. Dr. H. S. Smith was major professor for the work; his criticisms are gratefully acknowledged. The assistance of Dr. G. F. Lee is also appreciated.
REFERENCES
1. HENDRICKS, D. W. (1965), Sorption in flow through porous media, unpublished Ph.D. dissertation, Univ. of Iowa, Iowa City. 2. HELFFERICH, F. (1962), Ion Exhange, McGraw-Hill, New York. 3. HIESTER,.N.K. AND T. VERMEULEN (1952), Saturation performance of Ionexchange and Adsorption columns, Chemical Engineering Progress, 48, 505-516. 4. KEINATH, T. M. AND W. J. JR. (1968), A predictive model for the design of fluid-bed adsorbers, J. WPCF, May. 5. RIFAI,M. N. E., W. J. KAUFMAN AND D. K. TODD(1956), Dispersion Phenomena in Laminarpow through porous media, Report No. 2, r.E.R. Series 90, Sanitary Engin. Research. Lab., Univ. of Calif., Berkeley.
UTAH WATERRESEARCH LABORATORY, UTAHSTATEUNIVERSITY, LOGAN, UTAH 84321