- Email: [email protected]
A simplified solution technique for carbon adsorption model
War. Res. Vol. 27, No. 6, pp. 1033--1040,1993 Printed in Great Britain.All rightsreterved
0043-1354/93 $6.00+ 0.00 Copyright© 1993Pergamon~ Ltd
A SIMPLIFIED SOLUTION TECHNIQUE FOR CARBON ADSORPTION MODEL DIPAK ROY*, GUANG-TE WANG and DONALD DEAN ADRIAN@ Department of Civil Engineering, Louisiana State University, Baton Rouge, LA 70803, U.S.A.
(First received December 1991; accepted in revisedform December 1992) Almmet--A new method of solving the homogeneous surface diffusion model for activated carbon absorption wakes use of Laplace transforms on the equations developed by applying orthogonal collocation. The simultaneous equations which are developed are non-linear so an iterafive method is introduced to make them solvable when one wishes to calculate the surface diffusion coeff~ent, D,, and the film transfer ~ t , Kf, from batch adsorption data. The proposed model has the advantage of being continuous in time in contrast to earlier models which relied on finite difference numerical methods to solve the system of equations comprisin8 the homogeneous surface diffusion model. The model is applied to one data set from the literature. The proposed method gave values of/~ and D. nearly identical to values found by other investigators. The model was applied to three new data sets obtained in our laboratories: an agricultural waste, a dye waste and an aqueous waste stream containing a herbicide. The industrial wastes were composed of a mixture of species, whereas the herbicide waste contained a single organic compouod. The model produced a good match between experiment and pgediction for all waste streams. The final form of the equations is presented in a form readily useable by the person interested in applications to carbon adsorption bed practice. Key words--carbon adsorption, orthogonal collocation, Laplace transforms, modeling, surface diffus/on, fiquid film transfer, Freundlich isotherm, agricultural waste, dye waste, herbicide waste
q.(t)fadsorbed phase adsorbate concentration at solid-liquid interface (MM-') pp : apparent particle density OdL -~) q0 = initial adsorbed phase adsorbate concentration (MM -i) r ffi radial coordlnage (L) R ffi dimen~onless radial coordinate t--time(r) T = dimca~ionle, time, also used to show the dimemion time V ffi fiquid volume in closed batch test (L3)
NOMgNCLATUItE A~- collocation matrix to replace gradient B~,= collocation matrix to replace Lapladan operator Bi = Blot number based on surface diffusion (dimen,~onleu) C~,= matrix for collocation points Cb0),-bulk liquid phase adsorbate concentration (ML -]) C~(t)-non-dimeus/onal bulk liquid concentration C.(t) ,= liquid phase adsorbate concentration at mild-liquid interface (ML -3) C*.(t)-(Ynnensionless fiquid phase concentration at interrace Co = initial liquid phase concentration (ML -3) -matrices for collocation points = ,dsorbate particle diameter 0~) = surface distribution parameter (dimensionless) s tanface diff~on coefficient (LT -~)
~D~
f (D., IQ) -
re~Juai
function
K = Fmmdlich isotherm capacity constant, dimensions vary with n r ~ - liquid film mau tramfer coeflkient (LT -t) M-total mau of carbon in closed batch teat (M) n - Freundlich isotherm comtant (dimensionless) q(r, t) - adsorbed phase adsorbate concentration (MM -l) q~(t) -. adsorbed phase admrbate concentration at ith collocation point (MM -t) q.~ =.averaF adsorbed phase adsorbate concentration (MM-') q~n" non-dimem/onal average surface concentration (MM-') *Author to whom all correspondence should be addressed. wx ~ - o
INTRODUCTION The homogeneous surface diffusion model (HSDM) has been developed and analyzed by several researchers (Weber and Chakravorty, 1974; Mathews and Weber, 1975; Crittenden and Weber, 1978; Traegner and Suldan, 1989) to describe the process of adsorption on granular activated carbon (GAC). The mathematical formulation involves non-linear equations which include physical and kinetic paramcters. These equations are solved numerieaUy. Numerical solutions used in the past are powerful but have the drawback that each time the equations are solved the solution must start from the initial conditions, then proceed forward in time, The model can also be applied for parameter identification when experimental concentration distribution data are available, Since the H S D M has been developed, a number of investigators have applied it to estimate parameters such as the transfer coefficient, Kr, and
1033
1034
Dw~
R O Y et ai.
the surface diffusion coefficient, D,, by matching experimental results and the HSDM prediction. In brief, closed batch tests are performed and the HSDM is applied while varying Kf and D, until a satisfactory agreement between the model and experimental observations is obtained, In some cases carefully planned experiments reduce the number of parameters to be estimated. Kim et al. 0978) selected the method of qumilineafization in conjunction with the orthogonal collocation method to estimate the model parameters from the batch experimental data. Hand et al. (1983) were able to experimentally eliminate Kf in batch adsorption experiments, then they were able to deterifiine D, by applying the HSDM. A method to simultaneously estimate both K t and D, without the above experimental restrictions was developed by Traegner and Suidan (1989). The proposed collecting data from batch tests, then using the Levenburg--Marquardt algorithm while varying Kf and D, to minimize the sum of the squared residuals between experimental data and the model prediction. It would be desirable to have a solution technique which was continuous in time rather than one which only solved for the dependent variables at the node points of a numerical grid. The primary purpose of this investigation is to introduce new techniques to reduce the complexity of the solution methodology for the HSDM, to develop solutions which are continuous in time, and to make it useful to practicing engineers. A secondary objective of the investigation is to apply the HSDM to mixed organic wastes whose characteristics are measured by a gross parameter such as COD. The improved solution methodology will be used to estimate /Q and /3,. The steps which will be followed include: (l) presenting the several partial differential equations which govern the dynamics of adsorption; (2) using the orthogonal collocation method to discretize the partial differential equations to obtain a system of ordinary differential equations; (3) utilizing the Laplace transform to transfer ordinary differential equations into a set of algebraic equations which include the Laplace transform complex variable, then solving the algebraic equations; and (4) minimizing the difference between the model calculations and actual batch data to determine the kinetic parameters /~ and D,. Note that steps (1)-(3) result in developing the solution to the prediction model. The addition of step (4) enhances the procedure to include parameter estimation. HOMOGENEOUS SURFACE DIFFUSION MODEL
The equations describing the homogeneous surface diffusion model are presented in Table I following the form and notation used by Traegner and Suidan (1989). When the exponent n ffi I in equation (7), the linear adsorption case, the HSDM is a linear system of seven equations, for n ffi 0 the HSDM becomes a
Table 1. Equations for the homoseneons surface di~esion model
Equation No.
Rok
Equation
I
~Vf-M--~t 3
Mare bniance for dined batch test
t~pa
ae
z), a / 2aq~
3
~ ffi-~"Or ~r ~r}
4
q(r, 0) ffi 0
5
&/ffi 0 for r = 0 Or
Diffusion equation for a
sphericalparticle Initial condition
Boundary condition for the center of the splm.kni
parade
6
ppDl~-=~K~Cb-C,)" or
Boundary eon~tion for continuity of flux
at, = dd2 7
q, - KC~
Freuadlich isotherm for equilibrium at solid-liquid interface
simple case of diffusion in a sphere whose analytical solution is constant, is presented by Carslaw and Jaeger (1959). For n ~ 1 or n ~0, equation (7) is non-linear making the HSDM a non-linear problem requiring numerical methods for its solution. The non-dimensional form of the HSDM equations are shown in Table 2, where C f = C/Co, q*---q/qo, T ffi 4D, t / ~ , and R ffi 2r/dp are the dimensionless liquid phase concentration, adsorbed phase concentration, time and radial distance, respectively. The non-dimensional solute distribution is Mqo D, = Y Co
05)
The boundary condition at the solid-liquid interface, contains the non-dimensional Biot number, Biffi
KfdpC0 = 2D, ppqo
(16)
Kf
D,
where ~ffi(dpCo)/(2ppqo). The eiot number represents the ratio of the transport across the liquid layer to the rate of diffusion within the particle. Table 2. Non-dimensional form of the homolleneom surfaordiffusion model
Equation No.
8 9 10
Equation
C~ + Dsq',q - ! q.*,,s- 3
aq*(R,T)
q*~dR
I d f 2dq*\
11
q*(& 0) - 0
12
aq,(O,aR ______~_~- o
13
14
0"(I, 7 3 . ai(C~ OR
q?-Cr
Ct)
Homogeneous surface diffusion model
dq~ 41).
DISCiflgTIZING THE PAgrlAL DWFEiIENTIAL EQUATION USING THg OilTHOGONAL COLIJ)CATION METHOD
dt
The orthogonal collocation method developed by Villadsen and Stewart (1967) is appropriate for discretizing the partial differential equation to obtain an approximate solution. The orthogonal polynomial expansion is a trial-function such that the residual which results from the substitution of the trial function into the differential equation can be forced to zero at N specified collocation points (Finlayson, 1980). As the orthogonal collocation method has been presented in similar applications (Kim et al., 1978), we present here the essential equations. A trial function in terms of R 2 iS l-N
q*(R,T)=q*(I,T)+(I-R 2) ~ a,P,_t(R 2) (17) Since P~_, (R 2) is a polynomial of degree i - 1 in R 2, the trial function is a polynomial of degree i in R 2. Equation 07) can be rewritten as I-N
q*(R, 73 ffi Z d~R''-~
1035
(iS)
~ ' [ B 3 , q ~ + B32q* + B33q~+ B~q [] (28) p
A4,ql* + A42q2* + A43q3* + A~q,* --
Bi(C[
-
C*)
(29) q* ffi C,*"
(30)
TRANSFERRING ORDINARY DIIq~itENTIAL EQUATIONS INTO ALGEIli~IC EQUATIONS USING LAPLACE TRANSlFOitM AND SOLVING THE PREDICTION PROBLEM
Equations (25)-(30) can be solved using the Runge-Kutta method of integration (Sewell, 1988). In this paper another way to solve equations (25)-(30) has been demonstrated which results in a simple solution that is continuous in time by applying the Laplace transform, solving the resulting transformed equations, and taking the inverse Laplace transform. The Laplace transforms of equations (26)--(28) are ,
4D, [-
,
.
The spatial derivatives are replaced with the following matrices: dq,
~ t-N+ I
E Aoq?
dR
I d
R2 ~
R 2 dR
(19)
i-I
= ~ Boq?
4D, [sq2, (s) ffi -~p LBztq,, . .t$) _+ B22qe(s)
(20)
+.~q,(s)+~:
I-I
where Ao and Bo for N = 1 and 2 are available (Finlayson, 1980). The A0 and B0 for N ffi 3 were computed in the same manner as Finlayson (1980) and are provided in the Appendix. Then equations (9), (10) and (13) are reduced by substitution to
*] (32)
4Ds r ~ ' (.,) ffi--~[S3~q~, (s) + S~q['(s)
N+I
q*,~= 3 ~" W~qt
(21)
o r = ~" B~q*
(22)
where the transform is defined as
.~{q*} ffi ~ : e-Stq*(t) dt ~ q*(s)
(34)
.~{~t } ffi s.~{q*} - q*(O)
(35)
N+I
Z A.+,gq?ffiBi(C[
C*)
(23)
l~l
At the solid--liquid interface
Also, in anticipation of our solution methodology to
q~,+ , ffi c ? "
(24) be discussed later, we treat q[ as a constant.
Equations (31)--(35) can be rearranged as follows:
In the above equations, B, A and W are collocation matrices to replace the Laplacian, gradient and integral operators, respectively. If N ffi 3 collocation points, the HSDM becomes: "q[~ = 3[W,q~ + W~q r + w, qr + IV,q []
dq~ 4D, dt = ~
~*
4.0,
,
[.B,,q, + .e,~q* + B,~q~' + B,,q[]
-~p" Bt , -- s)aq,* (s) +--~'pB,~q2 4sD, , (s) [4D. 4sDs
(26)
dt =-~v [B,,q* + B..q~ + Bnq~ + Buq*] (27)
,
4D,
,
4D,
..
+-~p B,,q, (s) = --~p B,,C,
(25) 4sD,
,
[4D, 4sD,
(36)
DIPAK ROYet
exf
+($%-sp:(s)=
-y+cy
(38)
Equations (36)-(38) can be solved by using a tedious elimination method or they can be rewritten in matrices: EQ=-F
(39)
where
ai. - 142.634t
($)I
- 0.22628C:” exp[- 39.9961 $ (
)I
-0.219297C:” exp[-9.86861 i)]
(45)
Also, substituting q& from equation (25) in equation (0 C: = 1.0 - D,q,:,
E=
= LO-$yW,B:(t)+
w,qZ(t)
0
+
W,q:(t)
+
W,C:ll
=I(). -vc[0.0457g:(‘)+0.1259gf(r) 3Mqo 0
+
1
4:(s) Q = q$(s) [ q?(s)
(41)
- l.O727q:(r) + 5.3256q:(t) - 20.753qf(t) + 16.5Cy = Bi(C,+- C:)
40. R B,,Cf J UP After substituting B,, values to equation (40), the
L
eigenval~ and eigenvector of equation (40) can be obtained by using the IMSL subroutine (IMSL, 1989). Then one takes the inverse Laplace transform to obtain the solution which is continuous in time: q:(t) = 0.9!NCj” - 0.10727C:
(47)
One now has a set of five non-linear algebraic equations and five unknowns. Fortunately, after a trial value of C: is assumed, the equations reduce to a set of linear algebraic equations which are easy to solve. When time t is given, equations (43), (44) and (45) are solved. After q:(r),q:(t),q:(t) arc calcula@ qtw and Ct will be obtained from equation (46). C$ can also be obtained from equation (47). If the values of C2 obtained from equations (46) and (47) are different, then a new trial value of C: is calculated from the two C,+ values and the iteration is continued until the convergence criterion is sati&d. A number of iteration equations could be applied, but a simple one is
142.6341@)]
+ 0.702755C:” ex[-
39996t ($1
- 1.59C:“ex{-9.86861@)]
(43)
q:(t) = o.%WCy + 0.239778C: xex{-142.6341($)] -044308C:“exp[
-39996r ($1
- 0.797097C3:" exp[ -9.86861@)] q:(t) -
(46)
Furthermore, substituting the matrix elements from the Appendix in equation (29), one obtains:
(42)
x ex[-
O.l34Og;(t) + 0.0278C:“)
l.OOC,u- 0.554656C:
(44)
where Cz, and Cc are obtained from equations (46) and (47), respectively, and Z is a factor which can be held constant or increased as the number of iterations kdeases. If the initial guess of C: is far from its true value, the dilkrence between C,, and C, will be high. In such cams, a high Z value needs to be used resulting in a slower convergence and longer computation time for the iteration scheme. On the otherhand,iftheinitialgueasofC:isclosetoits truevalue,asmallerZvalutcanbeusalleading to fast convergence of the iteration procedure. In practice, 5 G Z G 20 works well. With proper selection of Z, this linear iteration method was found suitable to solve the set of nonlinear algebraic
Homogeneous surfacediffusionmodel equations and was observed to converge quickly to predict Cb. For collocation points N ffi 6, N : 7 . . . . . the same procedure can be used to get a more precise solution. However, it should be noted that the solution obtained for N ffi 3 is easier to apply and its accuracy is acceptable for engineering practice. The prediction problem from the HSDM is now solved. One now has a procedure for calculating q*(t), q*(t) and q*(t) as continuous functions of time, not just at discrete values of time specified in a finite difference or a Runge-Kutta algorithm. Of course, one would transform the results from their non-dimensional form to their dimensional form as in Table 1, IDF-~qTIFICATION OF THE MODEL P ~ M E A N S OF O F I I M A L TECHNIQUE
BY
Now that a conventional method is available to solve the prediction problem, one can address the inverse problem of using the HSDM to identify the model parameters. The kinetic parameters D, and /~ cannot be measured directly. Traegner and Suidan (1989) recommended a technique to find D. and Y..fby minimizing the sum of the squares of the difference between the corresponding model solutions, which strongly depend on the parameters, and experimental data collected from batch adsorption experiments. Starting from some initial guess for the values of the unknown parameters, the program should vary these values until the sum of the squared differences between the model solution and the experimental data is a minimum. The objective function is Minimize If(D,, Kf)] = ~ r2(D,, Kf)
(49)
I--I
rt(D,,/~) = C [ ( t ) - C * ( t ) i ~ D ~ , and m is t h e number of data points. For Bi < 1, external mass transport resistance, /~, controls the mass transfer rate, while for Bi ~, 100, surface diffusion, D,, controls the mass transfer rate (Traegner and Suidan, 1989). In either of these extreme cases, the objective function mainly depends only on a single parameter so that a very simple one-dimensional search method can be used to optimize the controlling parameter while keeping the other parameter a constant. A Biot number between 1 and 100 indicates that both mass transfer mechanisms are important for the particular process. In this case, Traegner and Suidan (1989) use the Levenberg-Marquardt algorithm to minimize the residual function. In this paper, a quasi.Newton method and a finite-difference gradient are used to find the minimum of the residual function, equation (49). Given a starting point X~, the search direction is computed according to the formula where
d ffi - - B - I g ,
(50)
1037
where B is a positive definite approximation of the Hessian, and gj is the gradient evaluated at X,. A line search is then used to find a new point
x,+, =x,+za
(51)
where z is a constant ranging between 0 and 1. When optimality is not achieved, B is updated according to the following formula
B, + B, SSrB, B~+i :
SrB, S
yyr + Y--~S
(52)
where SffiX~+t-X~ and Y = g l + l - - g i . For more details, see Dennis and Schnable (1983). The quasi-Newton method and a finite difference gradient used to minimize the residual function were those provided in the International Mathematical and Statistical Library (IMSL, 1989) which is available in most computer libraries. Rv-SULTS
The proposed method was tested usin8 the data presented by Traegner and Suidan (1989). Their results and the results generated by the proposed technique are presented in Table 3 and Fig. I. The results are nearly identical. The simplified solution technique was then applied to two industrial wastes containing mixed organics and to one synthetic waste containing a commonly used herbicide, 2,4dicJalorophenoxy acetic acid (2,4.D), as the only organic compound. The two industrial waste streams were obtained from a multinational chemical cornparty manufacturing agricultural chemicals and commercial grade dyes. The agricultural waste stream known as chIoroacetate waste contained sodium chloroacetate and acetone as the major constituents with methanol, isopropanol and toluene as the minor constituents in a mixture with chemical oxygen demand (COD) of 2657 mg/! (Boudreau, 1991). The dye waste contained a commercial grade organic sulfonate dye with 1770mg/l COD. The absorption spectra of the dye waste indicated multiple peaks between 500 and 540 nm wavelength. Detailed procedure of the batch adsorption experiments using these industrial wastes has been reported by Boudreau (1991). Experimental results of these studies are summarized in Table 4. Applying the Laplace transform-prediction method and the identification methods to the experimental data, the mass transfer coefficients for the three waste streams (Table 5) were obtained by
Table 3. The coml~rison of Immramule~ D,, Kr <~culaled from di1~nt Ull~lOdl Based on T r a q p ~ ' s data
Pm.im¢,lm
Tmqlner method
In this m d y
Kr D,
1.022 x I0 -) 1.282 x I0 -I°
1.02 x I0) 1.2"15 x I0 -i°
1038
Dw~J~ RoY et al. 1.05
F
1.00 ~
+ Simplified solution
0.95
x Traegner's method
0.90 o 0.85
o
0.80 --
" " "
0.75
0.70 0.65
-
0.60
0
I
I
I
I
I
200
400
600 T i m e (rain)
800
1000
1200
Fig. 1. Comparison of the proposed method with Traegner's method (data from Traegner and Suidan, 1989).
two methods. The agreement in results supports the concentration Cb can only be calculated at discrete validity of the new approach for pure as well as mixed time steps. These time steps may not coincide with the organic wastes. The kinetic parameters D, and K¢ time at which experimental data were collected. In for agricultural waste were determined to be addition, each numerical integration starts from zero 2.376 × 10-7 and 6.56 × 10-5, respectively. For dye so that a lot of computer time may be used to waste, the kinetic parameters D, and /Q were calculate the concentration for long times. In this 5.996 x 10-' and 3.004 x 10-~, respectively. Figure 2 paper, the Laplace transform method b used to shows the results of the curved fitted to the exper- convert the linear ordinary differential equations imental data on three waste streams using the opti- to algebraic equations. These linear algebraic equations with an additional three non-linear boundmum parameter values. ary condition equations are applied to solve for the bulk liquid phase adsorbate concentration C*. A total of N + 3 non-linear algebraic equations include SUMMARY N + 3 unknown values. Fortunately, after C* is In this paper, the orthogonal collocation method assumed, the equations reduce to linear algebraic was used to discretize the partial differential equations which are. easy to solve. An iterative proequations of the HSDM to ordinary differential cedure improves the accuracy of the solution. The equations with initial conditions. Similar work was equations developed in this paper are continuous carried out by Kim et al. (1978) who used the in time so that solution can be applied only at orthogoual collocation method to replace spatial times of interest to the engineer. An engineer does operators with matrices to convert the partial dilfer- not have to be concerned with the derivation of enthtl equations to a system of ordinary differential the system of equations, but only has to solve a equations which can be integrated numerically by system of simultaneous linear algebraic equations. using techniques such as the Runge--Kutta method, Therefore, the new method has the following advanthe Enler method or the predictor--corrector method. tages: In their study, a subroutine, DIFSUB, developed by (I) The Laplace transform is used to convert the Gear (1971a, b) was employed. A disadvantage of ordinary differential equations to algebraic using the numerical integrating techniques lies in the equations, thus avoiding the errors from requirement that the bulk liquid phase edsorhate
Table 4. EZl~rimeatal results and input data for three wastes
Parameters d~ ( m )
~
( l ~ m 3)
n
Co (COD)(ml/I) *as m ~ l of 2A-D.
~unfl 0.26
waste
Dye waste 0.059 0.457
0.43 8.85 x I0 -s 0.6399
2.1066 x 10 -4 0,973
2657
two
2,4-D 0.059 0.417 0.043988 0.2?8
254.78*
Table 5. Thecomparisonof D, and Kfusingtl~eeindustrialwasles Waste Parsmeter Tra~n~ This ~.dy Alpicultural Dye
2,4-D
Kr D, Kr D, Kf /)1
6.48 x 2.32 x 6.46 x 1.26 x 3.776 x 3.29 x
10 "s 10 -7 10 -6 I0-* 10 -3 I0 -s
6.56 x 2.376 x 6.26 x 1.258 x 3.8 x 3.329 x
10 -s 10-7 10-* I0 -s t0 -s I0 -a
Homogeneous surface diffusion model
1039
1.1 1.0 x Dye waste
0.9 o
0.8
~
0.~
~ o
0,6
~
0,5
N
0.4
+ 2,4-D
0.3 0.2 0.1
(2)
(3)
(4)
(5)
0
t
I
I
0.2
0.4
I I l 0.6 0,8 1.0 Time (T/To) Fig. 2. Prediction of batch carbon experiments by the proposed method.
1.2
finite differences. Therefore, it is more accurate t h¢n the numerical integration method. In addition, the solution is continuous in time. All the unknown values included in the non-linear algebraic equations are explicit functions of time. When any time is given, such as the time at which experimental data were collected, the bulk liquid phase adsorbate concentration C~ can be obtained by using an iterative method. This ability to calculate at any specified time avoids tedious step by step calculations common to numerical integration methods. When C* is assumed, the equations are reduced to finear equations~ Linear equations are easier to solve than non-linear equations and by an iterative procedure converge to the real C,* quickly. The proposed procedures are easy for engineers to use, when the kinetic parameters ( ~ , D,), the geometric and volumetric quantities (pp, M, V, a~) and the Freundlich parameters (K, n) are given. The practicing energies only substitute these parameters into N + 3 equations and obtain the bulk fiquid concentration C*. It is also convenient for researchers or engineers to estimate the kinetic parameters (Kf, D,) by using some optimization method to fit the experimental data, since the C* is now easy to calculate.
,,lcknowbd£ement:--Theauthors acknowledge the support obtained from the Louisiana Educational Quality Support Fund and the Louisiana Transport Research Center. In addition, the authors thank Richard Boudreae, Staff Environmental Chemist, CIBA-GEIGY Corporation, for furnishing two data sets used in this study.
REgERlgNCF~ Boudreau R. B. (1991) Laboratory and pilot evaluation of aerobic fixed film biotreatment as a pt~reatment method for environmental management of chemical plant wastewater streams. M.Sc. thesis, Louisiana State University. Carslaw H. S. and Jaeger J. C. (1959) Conduction of Heat in Solids, pp. 230-237. Oxford University Press, New York. Crittenden J. C. and Weber W. J. (1978) A predictive model for design of fixed-bed abeorbers: tingle component model verification. J. enviv. Engng Div., Am. Soc. Cir. Engrs 104, 433-443. Dennis J. E. and Schnabel R. B. (1983) Nun~rical Methods for Unconstrained Opttm~ation and Nonlinear Equatknts. Prentice-Hail, Englewood Cliffs, N. J. Finiayson B. A. (1980) Nanflnear Analysis in Chemical ~rinf. McGraw-Hill, N e w York. Hand D. W., Crittenden J. C. and Tbacker W. E. (1983) User oriented batch reactor solutions to the homogeneous surface diffusion model. J. envir. En£ng Div., Am. Soc. Cir. En£rs 109, 82. International Mathematics and Statistical Library (IMSL) (1989) Fortran Subroutine for Mathematical Applkations. Kim B. R., Schmitz R. A., Snoeyink V. L. and Tauxe G. W. (1978) Analysis of models for dichloramine removal by activated carbon in batch and packed-bed reactor using quasilinearization and orthogonal collocation methods. War. Res. 12, 317. Mathews A. and Weber W. J. (1975) Mathematical modeling of muiticomponent adsorption kinetics. Prefeated at the 68th Annual M e e t s , American Institute of Chemical Eng~ers, November 1975, Los Angeles, Calif. Sewell G. (1988) The Numerkal Solution of Ordinary and Partial Differential Equations. Academic Press, New York. Traegner U. K. and Suidan M. T. (1989) Evaluation of surface and film diffxmon coefficients for carbon adsorption. War. Res. 23, 267-273. Vi~ J. V. and Stewart W. E. (1967) Solution of boundary-value problem by orthognnal collocation. Chem. Engn£ S¢i. 22, 1483-1501. Weber W. J. and Chakravorty R. K. (1974) Pore and sofid diffusion modeb for fixed bed adsorben. A.LCh.E. JI. 20, 228-238.
1040
DIPAK ROY e t a / . APPENDIX
N~3 i = 1 i -- 2 i -- 3 i --4 A4~ Wj
The Values of Matrices Bq, A¢j, W/ B~j jffil jffi2 j~3 jffi4 --23.853 11.100 --3.323 -33.676 - 1.073 0.0457
30.594 -43.238 38.357 152.375 5.326 0.1259
--9.746 40.819 -- 125.409 -311.199 - 20.753 0.1340
3.005 --8.681 90.375 192.500 16.500 0.0278
0043-1354/93 $6.00+ 0.00 Copyright© 1993Pergamon~ Ltd
A SIMPLIFIED SOLUTION TECHNIQUE FOR CARBON ADSORPTION MODEL DIPAK ROY*, GUANG-TE WANG and DONALD DEAN ADRIAN@ Department of Civil Engineering, Louisiana State University, Baton Rouge, LA 70803, U.S.A.
(First received December 1991; accepted in revisedform December 1992) Almmet--A new method of solving the homogeneous surface diffusion model for activated carbon absorption wakes use of Laplace transforms on the equations developed by applying orthogonal collocation. The simultaneous equations which are developed are non-linear so an iterafive method is introduced to make them solvable when one wishes to calculate the surface diffusion coeff~ent, D,, and the film transfer ~ t , Kf, from batch adsorption data. The proposed model has the advantage of being continuous in time in contrast to earlier models which relied on finite difference numerical methods to solve the system of equations comprisin8 the homogeneous surface diffusion model. The model is applied to one data set from the literature. The proposed method gave values of/~ and D. nearly identical to values found by other investigators. The model was applied to three new data sets obtained in our laboratories: an agricultural waste, a dye waste and an aqueous waste stream containing a herbicide. The industrial wastes were composed of a mixture of species, whereas the herbicide waste contained a single organic compouod. The model produced a good match between experiment and pgediction for all waste streams. The final form of the equations is presented in a form readily useable by the person interested in applications to carbon adsorption bed practice. Key words--carbon adsorption, orthogonal collocation, Laplace transforms, modeling, surface diffus/on, fiquid film transfer, Freundlich isotherm, agricultural waste, dye waste, herbicide waste
q.(t)fadsorbed phase adsorbate concentration at solid-liquid interface (MM-') pp : apparent particle density OdL -~) q0 = initial adsorbed phase adsorbate concentration (MM -i) r ffi radial coordlnage (L) R ffi dimen~onless radial coordinate t--time(r) T = dimca~ionle, time, also used to show the dimemion time V ffi fiquid volume in closed batch test (L3)
NOMgNCLATUItE A~- collocation matrix to replace gradient B~,= collocation matrix to replace Lapladan operator Bi = Blot number based on surface diffusion (dimen,~onleu) C~,= matrix for collocation points Cb0),-bulk liquid phase adsorbate concentration (ML -]) C~(t)-non-dimeus/onal bulk liquid concentration C.(t) ,= liquid phase adsorbate concentration at mild-liquid interface (ML -3) C*.(t)-(Ynnensionless fiquid phase concentration at interrace Co = initial liquid phase concentration (ML -3) -matrices for collocation points = ,dsorbate particle diameter 0~) = surface distribution parameter (dimensionless) s tanface diff~on coefficient (LT -~)
~D~
f (D., IQ) -
re~Juai
function
K = Fmmdlich isotherm capacity constant, dimensions vary with n r ~ - liquid film mau tramfer coeflkient (LT -t) M-total mau of carbon in closed batch teat (M) n - Freundlich isotherm comtant (dimensionless) q(r, t) - adsorbed phase adsorbate concentration (MM -l) q~(t) -. adsorbed phase admrbate concentration at ith collocation point (MM -t) q.~ =.averaF adsorbed phase adsorbate concentration (MM-') q~n" non-dimem/onal average surface concentration (MM-') *Author to whom all correspondence should be addressed. wx ~ - o
INTRODUCTION The homogeneous surface diffusion model (HSDM) has been developed and analyzed by several researchers (Weber and Chakravorty, 1974; Mathews and Weber, 1975; Crittenden and Weber, 1978; Traegner and Suldan, 1989) to describe the process of adsorption on granular activated carbon (GAC). The mathematical formulation involves non-linear equations which include physical and kinetic paramcters. These equations are solved numerieaUy. Numerical solutions used in the past are powerful but have the drawback that each time the equations are solved the solution must start from the initial conditions, then proceed forward in time, The model can also be applied for parameter identification when experimental concentration distribution data are available, Since the H S D M has been developed, a number of investigators have applied it to estimate parameters such as the transfer coefficient, Kr, and
1033
1034
Dw~
R O Y et ai.
the surface diffusion coefficient, D,, by matching experimental results and the HSDM prediction. In brief, closed batch tests are performed and the HSDM is applied while varying Kf and D, until a satisfactory agreement between the model and experimental observations is obtained, In some cases carefully planned experiments reduce the number of parameters to be estimated. Kim et al. 0978) selected the method of qumilineafization in conjunction with the orthogonal collocation method to estimate the model parameters from the batch experimental data. Hand et al. (1983) were able to experimentally eliminate Kf in batch adsorption experiments, then they were able to deterifiine D, by applying the HSDM. A method to simultaneously estimate both K t and D, without the above experimental restrictions was developed by Traegner and Suidan (1989). The proposed collecting data from batch tests, then using the Levenburg--Marquardt algorithm while varying Kf and D, to minimize the sum of the squared residuals between experimental data and the model prediction. It would be desirable to have a solution technique which was continuous in time rather than one which only solved for the dependent variables at the node points of a numerical grid. The primary purpose of this investigation is to introduce new techniques to reduce the complexity of the solution methodology for the HSDM, to develop solutions which are continuous in time, and to make it useful to practicing engineers. A secondary objective of the investigation is to apply the HSDM to mixed organic wastes whose characteristics are measured by a gross parameter such as COD. The improved solution methodology will be used to estimate /Q and /3,. The steps which will be followed include: (l) presenting the several partial differential equations which govern the dynamics of adsorption; (2) using the orthogonal collocation method to discretize the partial differential equations to obtain a system of ordinary differential equations; (3) utilizing the Laplace transform to transfer ordinary differential equations into a set of algebraic equations which include the Laplace transform complex variable, then solving the algebraic equations; and (4) minimizing the difference between the model calculations and actual batch data to determine the kinetic parameters /~ and D,. Note that steps (1)-(3) result in developing the solution to the prediction model. The addition of step (4) enhances the procedure to include parameter estimation. HOMOGENEOUS SURFACE DIFFUSION MODEL
The equations describing the homogeneous surface diffusion model are presented in Table I following the form and notation used by Traegner and Suidan (1989). When the exponent n ffi I in equation (7), the linear adsorption case, the HSDM is a linear system of seven equations, for n ffi 0 the HSDM becomes a
Table 1. Equations for the homoseneons surface di~esion model
Equation No.
Rok
Equation
I
~Vf-M--~t 3
Mare bniance for dined batch test
t~pa
ae
z), a / 2aq~
3
~ ffi-~"Or ~r ~r}
4
q(r, 0) ffi 0
5
&/ffi 0 for r = 0 Or
Diffusion equation for a
sphericalparticle Initial condition
Boundary condition for the center of the splm.kni
parade
6
ppDl~-=~K~Cb-C,)" or
Boundary eon~tion for continuity of flux
at, = dd2 7
q, - KC~
Freuadlich isotherm for equilibrium at solid-liquid interface
simple case of diffusion in a sphere whose analytical solution is constant, is presented by Carslaw and Jaeger (1959). For n ~ 1 or n ~0, equation (7) is non-linear making the HSDM a non-linear problem requiring numerical methods for its solution. The non-dimensional form of the HSDM equations are shown in Table 2, where C f = C/Co, q*---q/qo, T ffi 4D, t / ~ , and R ffi 2r/dp are the dimensionless liquid phase concentration, adsorbed phase concentration, time and radial distance, respectively. The non-dimensional solute distribution is Mqo D, = Y Co
05)
The boundary condition at the solid-liquid interface, contains the non-dimensional Biot number, Biffi
KfdpC0 = 2D, ppqo
(16)
Kf
D,
where ~ffi(dpCo)/(2ppqo). The eiot number represents the ratio of the transport across the liquid layer to the rate of diffusion within the particle. Table 2. Non-dimensional form of the homolleneom surfaordiffusion model
Equation No.
8 9 10
Equation
C~ + Dsq',q - ! q.*,,s- 3
aq*(R,T)
q*~dR
I d f 2dq*\
11
q*(& 0) - 0
12
aq,(O,aR ______~_~- o
13
14
0"(I, 7 3 . ai(C~ OR
q?-Cr
Ct)
Homogeneous surface diffusion model
dq~ 41).
DISCiflgTIZING THE PAgrlAL DWFEiIENTIAL EQUATION USING THg OilTHOGONAL COLIJ)CATION METHOD
dt
The orthogonal collocation method developed by Villadsen and Stewart (1967) is appropriate for discretizing the partial differential equation to obtain an approximate solution. The orthogonal polynomial expansion is a trial-function such that the residual which results from the substitution of the trial function into the differential equation can be forced to zero at N specified collocation points (Finlayson, 1980). As the orthogonal collocation method has been presented in similar applications (Kim et al., 1978), we present here the essential equations. A trial function in terms of R 2 iS l-N
q*(R,T)=q*(I,T)+(I-R 2) ~ a,P,_t(R 2) (17) Since P~_, (R 2) is a polynomial of degree i - 1 in R 2, the trial function is a polynomial of degree i in R 2. Equation 07) can be rewritten as I-N
q*(R, 73 ffi Z d~R''-~
1035
(iS)
~ ' [ B 3 , q ~ + B32q* + B33q~+ B~q [] (28) p
A4,ql* + A42q2* + A43q3* + A~q,* --
Bi(C[
-
C*)
(29) q* ffi C,*"
(30)
TRANSFERRING ORDINARY DIIq~itENTIAL EQUATIONS INTO ALGEIli~IC EQUATIONS USING LAPLACE TRANSlFOitM AND SOLVING THE PREDICTION PROBLEM
Equations (25)-(30) can be solved using the Runge-Kutta method of integration (Sewell, 1988). In this paper another way to solve equations (25)-(30) has been demonstrated which results in a simple solution that is continuous in time by applying the Laplace transform, solving the resulting transformed equations, and taking the inverse Laplace transform. The Laplace transforms of equations (26)--(28) are ,
4D, [-
,
.
The spatial derivatives are replaced with the following matrices: dq,
~ t-N+ I
E Aoq?
dR
I d
R2 ~
R 2 dR
(19)
i-I
= ~ Boq?
4D, [sq2, (s) ffi -~p LBztq,, . .t$) _+ B22qe(s)
(20)
+.~q,(s)+~:
I-I
where Ao and Bo for N = 1 and 2 are available (Finlayson, 1980). The A0 and B0 for N ffi 3 were computed in the same manner as Finlayson (1980) and are provided in the Appendix. Then equations (9), (10) and (13) are reduced by substitution to
*] (32)
4Ds r ~ ' (.,) ffi--~[S3~q~, (s) + S~q['(s)
N+I
q*,~= 3 ~" W~qt
(21)
o r = ~" B~q*
(22)
where the transform is defined as
.~{q*} ffi ~ : e-Stq*(t) dt ~ q*(s)
(34)
.~{~t } ffi s.~{q*} - q*(O)
(35)
N+I
Z A.+,gq?ffiBi(C[
C*)
(23)
l~l
At the solid--liquid interface
Also, in anticipation of our solution methodology to
q~,+ , ffi c ? "
(24) be discussed later, we treat q[ as a constant.
Equations (31)--(35) can be rearranged as follows:
In the above equations, B, A and W are collocation matrices to replace the Laplacian, gradient and integral operators, respectively. If N ffi 3 collocation points, the HSDM becomes: "q[~ = 3[W,q~ + W~q r + w, qr + IV,q []
dq~ 4D, dt = ~
~*
4.0,
,
[.B,,q, + .e,~q* + B,~q~' + B,,q[]
-~p" Bt , -- s)aq,* (s) +--~'pB,~q2 4sD, , (s) [4D. 4sDs
(26)
dt =-~v [B,,q* + B..q~ + Bnq~ + Buq*] (27)
,
4D,
,
4D,
..
+-~p B,,q, (s) = --~p B,,C,
(25) 4sD,
,
[4D, 4sD,
(36)
DIPAK ROYet
exf
+($%-sp:(s)=
-y+cy
(38)
Equations (36)-(38) can be solved by using a tedious elimination method or they can be rewritten in matrices: EQ=-F
(39)
where
ai. - 142.634t
($)I
- 0.22628C:” exp[- 39.9961 $ (
)I
-0.219297C:” exp[-9.86861 i)]
(45)
Also, substituting q& from equation (25) in equation (0 C: = 1.0 - D,q,:,
E=
= LO-$yW,B:(t)+
w,qZ(t)
0
+
W,q:(t)
+
W,C:ll
=I(). -vc[0.0457g:(‘)+0.1259gf(r) 3Mqo 0
+
1
4:(s) Q = q$(s) [ q?(s)
(41)
- l.O727q:(r) + 5.3256q:(t) - 20.753qf(t) + 16.5Cy = Bi(C,+- C:)
40. R B,,Cf J UP After substituting B,, values to equation (40), the
L
eigenval~ and eigenvector of equation (40) can be obtained by using the IMSL subroutine (IMSL, 1989). Then one takes the inverse Laplace transform to obtain the solution which is continuous in time: q:(t) = 0.9!NCj” - 0.10727C:
(47)
One now has a set of five non-linear algebraic equations and five unknowns. Fortunately, after a trial value of C: is assumed, the equations reduce to a set of linear algebraic equations which are easy to solve. When time t is given, equations (43), (44) and (45) are solved. After q:(r),q:(t),q:(t) arc calcula@ qtw and Ct will be obtained from equation (46). C$ can also be obtained from equation (47). If the values of C2 obtained from equations (46) and (47) are different, then a new trial value of C: is calculated from the two C,+ values and the iteration is continued until the convergence criterion is sati&d. A number of iteration equations could be applied, but a simple one is
142.6341@)]
+ 0.702755C:” ex[-
39996t ($1
- 1.59C:“ex{-9.86861@)]
(43)
q:(t) = o.%WCy + 0.239778C: xex{-142.6341($)] -044308C:“exp[
-39996r ($1
- 0.797097C3:" exp[ -9.86861@)] q:(t) -
(46)
Furthermore, substituting the matrix elements from the Appendix in equation (29), one obtains:
(42)
x ex[-
O.l34Og;(t) + 0.0278C:“)
l.OOC,u- 0.554656C:
(44)
where Cz, and Cc are obtained from equations (46) and (47), respectively, and Z is a factor which can be held constant or increased as the number of iterations kdeases. If the initial guess of C: is far from its true value, the dilkrence between C,, and C, will be high. In such cams, a high Z value needs to be used resulting in a slower convergence and longer computation time for the iteration scheme. On the otherhand,iftheinitialgueasofC:isclosetoits truevalue,asmallerZvalutcanbeusalleading to fast convergence of the iteration procedure. In practice, 5 G Z G 20 works well. With proper selection of Z, this linear iteration method was found suitable to solve the set of nonlinear algebraic
Homogeneous surfacediffusionmodel equations and was observed to converge quickly to predict Cb. For collocation points N ffi 6, N : 7 . . . . . the same procedure can be used to get a more precise solution. However, it should be noted that the solution obtained for N ffi 3 is easier to apply and its accuracy is acceptable for engineering practice. The prediction problem from the HSDM is now solved. One now has a procedure for calculating q*(t), q*(t) and q*(t) as continuous functions of time, not just at discrete values of time specified in a finite difference or a Runge-Kutta algorithm. Of course, one would transform the results from their non-dimensional form to their dimensional form as in Table 1, IDF-~qTIFICATION OF THE MODEL P ~ M E A N S OF O F I I M A L TECHNIQUE
BY
Now that a conventional method is available to solve the prediction problem, one can address the inverse problem of using the HSDM to identify the model parameters. The kinetic parameters D, and /~ cannot be measured directly. Traegner and Suidan (1989) recommended a technique to find D. and Y..fby minimizing the sum of the squares of the difference between the corresponding model solutions, which strongly depend on the parameters, and experimental data collected from batch adsorption experiments. Starting from some initial guess for the values of the unknown parameters, the program should vary these values until the sum of the squared differences between the model solution and the experimental data is a minimum. The objective function is Minimize If(D,, Kf)] = ~ r2(D,, Kf)
(49)
I--I
rt(D,,/~) = C [ ( t ) - C * ( t ) i ~ D ~ , and m is t h e number of data points. For Bi < 1, external mass transport resistance, /~, controls the mass transfer rate, while for Bi ~, 100, surface diffusion, D,, controls the mass transfer rate (Traegner and Suidan, 1989). In either of these extreme cases, the objective function mainly depends only on a single parameter so that a very simple one-dimensional search method can be used to optimize the controlling parameter while keeping the other parameter a constant. A Biot number between 1 and 100 indicates that both mass transfer mechanisms are important for the particular process. In this case, Traegner and Suidan (1989) use the Levenberg-Marquardt algorithm to minimize the residual function. In this paper, a quasi.Newton method and a finite-difference gradient are used to find the minimum of the residual function, equation (49). Given a starting point X~, the search direction is computed according to the formula where
d ffi - - B - I g ,
(50)
1037
where B is a positive definite approximation of the Hessian, and gj is the gradient evaluated at X,. A line search is then used to find a new point
x,+, =x,+za
(51)
where z is a constant ranging between 0 and 1. When optimality is not achieved, B is updated according to the following formula
B, + B, SSrB, B~+i :
SrB, S
yyr + Y--~S
(52)
where SffiX~+t-X~ and Y = g l + l - - g i . For more details, see Dennis and Schnable (1983). The quasi-Newton method and a finite difference gradient used to minimize the residual function were those provided in the International Mathematical and Statistical Library (IMSL, 1989) which is available in most computer libraries. Rv-SULTS
The proposed method was tested usin8 the data presented by Traegner and Suidan (1989). Their results and the results generated by the proposed technique are presented in Table 3 and Fig. I. The results are nearly identical. The simplified solution technique was then applied to two industrial wastes containing mixed organics and to one synthetic waste containing a commonly used herbicide, 2,4dicJalorophenoxy acetic acid (2,4.D), as the only organic compound. The two industrial waste streams were obtained from a multinational chemical cornparty manufacturing agricultural chemicals and commercial grade dyes. The agricultural waste stream known as chIoroacetate waste contained sodium chloroacetate and acetone as the major constituents with methanol, isopropanol and toluene as the minor constituents in a mixture with chemical oxygen demand (COD) of 2657 mg/! (Boudreau, 1991). The dye waste contained a commercial grade organic sulfonate dye with 1770mg/l COD. The absorption spectra of the dye waste indicated multiple peaks between 500 and 540 nm wavelength. Detailed procedure of the batch adsorption experiments using these industrial wastes has been reported by Boudreau (1991). Experimental results of these studies are summarized in Table 4. Applying the Laplace transform-prediction method and the identification methods to the experimental data, the mass transfer coefficients for the three waste streams (Table 5) were obtained by
Table 3. The coml~rison of Immramule~ D,, Kr <~culaled from di1~nt Ull~lOdl Based on T r a q p ~ ' s data
Pm.im¢,lm
Tmqlner method
In this m d y
Kr D,
1.022 x I0 -) 1.282 x I0 -I°
1.02 x I0) 1.2"15 x I0 -i°
1038
Dw~J~ RoY et al. 1.05
F
1.00 ~
+ Simplified solution
0.95
x Traegner's method
0.90 o 0.85
o
0.80 --
" " "
0.75
0.70 0.65
-
0.60
0
I
I
I
I
I
200
400
600 T i m e (rain)
800
1000
1200
Fig. 1. Comparison of the proposed method with Traegner's method (data from Traegner and Suidan, 1989).
two methods. The agreement in results supports the concentration Cb can only be calculated at discrete validity of the new approach for pure as well as mixed time steps. These time steps may not coincide with the organic wastes. The kinetic parameters D, and K¢ time at which experimental data were collected. In for agricultural waste were determined to be addition, each numerical integration starts from zero 2.376 × 10-7 and 6.56 × 10-5, respectively. For dye so that a lot of computer time may be used to waste, the kinetic parameters D, and /Q were calculate the concentration for long times. In this 5.996 x 10-' and 3.004 x 10-~, respectively. Figure 2 paper, the Laplace transform method b used to shows the results of the curved fitted to the exper- convert the linear ordinary differential equations imental data on three waste streams using the opti- to algebraic equations. These linear algebraic equations with an additional three non-linear boundmum parameter values. ary condition equations are applied to solve for the bulk liquid phase adsorbate concentration C*. A total of N + 3 non-linear algebraic equations include SUMMARY N + 3 unknown values. Fortunately, after C* is In this paper, the orthogonal collocation method assumed, the equations reduce to linear algebraic was used to discretize the partial differential equations which are. easy to solve. An iterative proequations of the HSDM to ordinary differential cedure improves the accuracy of the solution. The equations with initial conditions. Similar work was equations developed in this paper are continuous carried out by Kim et al. (1978) who used the in time so that solution can be applied only at orthogoual collocation method to replace spatial times of interest to the engineer. An engineer does operators with matrices to convert the partial dilfer- not have to be concerned with the derivation of enthtl equations to a system of ordinary differential the system of equations, but only has to solve a equations which can be integrated numerically by system of simultaneous linear algebraic equations. using techniques such as the Runge--Kutta method, Therefore, the new method has the following advanthe Enler method or the predictor--corrector method. tages: In their study, a subroutine, DIFSUB, developed by (I) The Laplace transform is used to convert the Gear (1971a, b) was employed. A disadvantage of ordinary differential equations to algebraic using the numerical integrating techniques lies in the equations, thus avoiding the errors from requirement that the bulk liquid phase edsorhate
Table 4. EZl~rimeatal results and input data for three wastes
Parameters d~ ( m )
~
( l ~ m 3)
n
Co (COD)(ml/I) *as m ~ l of 2A-D.
~unfl 0.26
waste
Dye waste 0.059 0.457
0.43 8.85 x I0 -s 0.6399
2.1066 x 10 -4 0,973
2657
two
2,4-D 0.059 0.417 0.043988 0.2?8
254.78*
Table 5. Thecomparisonof D, and Kfusingtl~eeindustrialwasles Waste Parsmeter Tra~n~ This ~.dy Alpicultural Dye
2,4-D
Kr D, Kr D, Kf /)1
6.48 x 2.32 x 6.46 x 1.26 x 3.776 x 3.29 x
10 "s 10 -7 10 -6 I0-* 10 -3 I0 -s
6.56 x 2.376 x 6.26 x 1.258 x 3.8 x 3.329 x
10 -s 10-7 10-* I0 -s t0 -s I0 -a
Homogeneous surface diffusion model
1039
1.1 1.0 x Dye waste
0.9 o
0.8
~
0.~
~ o
0,6
~
0,5
N
0.4
+ 2,4-D
0.3 0.2 0.1
(2)
(3)
(4)
(5)
0
t
I
I
0.2
0.4
I I l 0.6 0,8 1.0 Time (T/To) Fig. 2. Prediction of batch carbon experiments by the proposed method.
1.2
finite differences. Therefore, it is more accurate t h¢n the numerical integration method. In addition, the solution is continuous in time. All the unknown values included in the non-linear algebraic equations are explicit functions of time. When any time is given, such as the time at which experimental data were collected, the bulk liquid phase adsorbate concentration C~ can be obtained by using an iterative method. This ability to calculate at any specified time avoids tedious step by step calculations common to numerical integration methods. When C* is assumed, the equations are reduced to finear equations~ Linear equations are easier to solve than non-linear equations and by an iterative procedure converge to the real C,* quickly. The proposed procedures are easy for engineers to use, when the kinetic parameters ( ~ , D,), the geometric and volumetric quantities (pp, M, V, a~) and the Freundlich parameters (K, n) are given. The practicing energies only substitute these parameters into N + 3 equations and obtain the bulk fiquid concentration C*. It is also convenient for researchers or engineers to estimate the kinetic parameters (Kf, D,) by using some optimization method to fit the experimental data, since the C* is now easy to calculate.
,,lcknowbd£ement:--Theauthors acknowledge the support obtained from the Louisiana Educational Quality Support Fund and the Louisiana Transport Research Center. In addition, the authors thank Richard Boudreae, Staff Environmental Chemist, CIBA-GEIGY Corporation, for furnishing two data sets used in this study.
REgERlgNCF~ Boudreau R. B. (1991) Laboratory and pilot evaluation of aerobic fixed film biotreatment as a pt~reatment method for environmental management of chemical plant wastewater streams. M.Sc. thesis, Louisiana State University. Carslaw H. S. and Jaeger J. C. (1959) Conduction of Heat in Solids, pp. 230-237. Oxford University Press, New York. Crittenden J. C. and Weber W. J. (1978) A predictive model for design of fixed-bed abeorbers: tingle component model verification. J. enviv. Engng Div., Am. Soc. Cir. Engrs 104, 433-443. Dennis J. E. and Schnabel R. B. (1983) Nun~rical Methods for Unconstrained Opttm~ation and Nonlinear Equatknts. Prentice-Hail, Englewood Cliffs, N. J. Finiayson B. A. (1980) Nanflnear Analysis in Chemical ~rinf. McGraw-Hill, N e w York. Hand D. W., Crittenden J. C. and Tbacker W. E. (1983) User oriented batch reactor solutions to the homogeneous surface diffusion model. J. envir. En£ng Div., Am. Soc. Cir. En£rs 109, 82. International Mathematics and Statistical Library (IMSL) (1989) Fortran Subroutine for Mathematical Applkations. Kim B. R., Schmitz R. A., Snoeyink V. L. and Tauxe G. W. (1978) Analysis of models for dichloramine removal by activated carbon in batch and packed-bed reactor using quasilinearization and orthogonal collocation methods. War. Res. 12, 317. Mathews A. and Weber W. J. (1975) Mathematical modeling of muiticomponent adsorption kinetics. Prefeated at the 68th Annual M e e t s , American Institute of Chemical Eng~ers, November 1975, Los Angeles, Calif. Sewell G. (1988) The Numerkal Solution of Ordinary and Partial Differential Equations. Academic Press, New York. Traegner U. K. and Suidan M. T. (1989) Evaluation of surface and film diffxmon coefficients for carbon adsorption. War. Res. 23, 267-273. Vi~ J. V. and Stewart W. E. (1967) Solution of boundary-value problem by orthognnal collocation. Chem. Engn£ S¢i. 22, 1483-1501. Weber W. J. and Chakravorty R. K. (1974) Pore and sofid diffusion modeb for fixed bed adsorben. A.LCh.E. JI. 20, 228-238.
1040
DIPAK ROY e t a / . APPENDIX
N~3 i = 1 i -- 2 i -- 3 i --4 A4~ Wj
The Values of Matrices Bq, A¢j, W/ B~j jffil jffi2 j~3 jffi4 --23.853 11.100 --3.323 -33.676 - 1.073 0.0457
30.594 -43.238 38.357 152.375 5.326 0.1259
--9.746 40.819 -- 125.409 -311.199 - 20.753 0.1340
3.005 --8.681 90.375 192.500 16.500 0.0278