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Development of Solutions to Two-Resistance Mass Transport Models Based on External and Pore Diffusion. Part I Theoretical Development

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Development of Solutions to
Two-Resistance Mass Transport
Models Based on External and Pore
Diffusion. Part I: Theoretical
Gordon McKay, Bushra Al-Duri and Stephen McKee
Department of Chemical Engineering, The Queen's University
of Belfast, Belfast BT9 5AG, Northern Ireland
A two-resistance mass transfer model has been developed to describe the adsorption
of various pollutants onto adsorbents in batch adsorbers. The mechanism is based on
external liquid-phase mass transfer and pore-diffusion controlling internal mass
transport. A previous analytical moder' was developed for a pseudo-irreversible
isotherm, but it was only applicable to systems in which operating lines and tie lines
terminated on the isotherm monolayer. The application of the original model has
been extended by incorporating several curve fitting equations into the analysis and
differentiating them, thus enabling mass transfer coeflcients to be predicted for
operating lines over the whole range of the equilibrium isotherm. A comparison of
the curve fitting equations was made to test which, i f any, gave the 'best-fit' to
experimental data.
Adsorption operations exploit the ability of certain solids to preferentially
concentrate specific substances from solution onto their surface. The technique
has found widespread use in the treatment of wastewaters containing dyestuffs or
colourless organics.2" Adsorption on granular activated carbon has found
widespread use, because it is an effective and economic means of removing low
concentrations of adsorbable pollutants from liquids. Recently, water pollution
control has become more stringent worldwide and carbon adsorption is
particularly useful for removing organics which are chemically too stable to treat
by conventional biological treatment.7-9 In the adsorption of a substance from
solution by an adsorbent, three rate controlling steps are usually considered,10s"
these are:
* To whom correspondence should be addressed.
Developments in Chemical Engineering and Mineral Processing, Vol. 1, No. 2/3,Page 129
G. McKay, B. Al-Duri and S. McKee
(1) Resistance to mass transfer in the liquid film surrounding each particle;
(2) The diffusional mechanism by which adsorbate is carried to the internal
surfaces of the particles;
(3) The rate of uptake of adsorbate at the sorption sites.
The number of controlling resistances operating in any system depends upon
the specific system, the physical properties of adsorbent, and the chemical
properties of adsorbates. However, the number of resistances operating in any
adsorption system affects the kinetics of the process and therefore the
mathematical description of the latter varies accordingly. Consequently, several
mathematical mass-transfer models classified by the controlling resistances have
been proposed to describe the system kinetics, based on experimental inputs of
system conditions. Mathematical modelling and computer simulation are
invaluable techniques for this area of research and development.
The purpose of this paper is to present a mass transport model based on
external film mass transfer and pore-diffusion controlled internal mass transfer.
The basic model was developed by Spahn and Schlunder12 and based on the
unreacted core theory.13-15 The model utilised graphical differentiation to develop
the solution equations, an aspect which is susceptible to errors, and hence iving
inaccurate results. An analytical solution to this model has been developed% but
it has limited application to adsorption systems which exhibit a rectangular
isotherm, some examples of its application are presented in this paper. The present
paper tests a number of techniques for incorporating curve-fitting equations into
the computer program for this model, thus providing a pseudo-analytical solution
of the mass transport equations. The techniques are based on: (i) finite difference;
(ii) polynomial equations; (iii) exponential equations.
The models were tested using batch adsorption data for phenol on carbon,
Astrazone Blue dye on silica, and dyestuffs on chitin.
Experimental Details
Kinetics studies were carried out in an agitated batch adsorption system which
consisted of a 2 dm3 glass vessel (diameter 0.13 m) filled with 1.7 dm3 of
solution, giving a solution height of 0.13 m. The solution was agitated by a
six-flat-blade impeller driven by a Heidolph electric motor, with a speed range
from 100-600 revolutions per minute. The width of the impeller blades was
0.013 m. Complete mixing was facilitated by eight baffles, each of width 0.01 m,
distributed at 45' around the circumference of the beaker and held in position by
a polystyrene baffle holder.
Kinetic and equilibrium data were analysed using a Perkin Elmer SP550
spectrophotometer. The optical densities of the sorbates were measured at their
maximum absorbance wavelength. The sorbate concentration is related to the
optical density by a predetermined calibration graph.
Further details of the experimental system and some results have been reported
Theoretical Development
The film-pore diffusion mechanism is based on the unreacted shrinking core mass
transfer model proposed by Le~enspiel,'~
where reaction starts at the particle
Two-Resistance Mass Transport Models
surface forming a reacted zone which moves inwards with a certain velocity.
Hence, during the whole reaction time there is an unreacted core, shrinking in size
as the reaction proceeds.
Theoretical basis
Spahn and Schlunder'* have developed the film-pore diffusion mathematical
model based on the unreacted shrinking core mass transfer model of Leven~piel,'~
and including the porous structure of carbon. These authors proposed the
following assumptions:
(1) Equilibrium exists between the pore liquid and particle interior, that is, the flow
of solution to the pores is much faster than its uptake at sorption sites.
(2) Mass transfer in the pores is solely by molecular diffusion, i.e. it follows Fick's
law and is measured by the pore diffusion coefficient, Deff(cm2s-').
(3) Solute concentration in the adsorbed phase is independent of that in the pore
liquid, therefore adsorption is irreversible, i.e. producing a rectangular
isotherm, and Y* can be substituted by Yh*, which is constant for all Cs
(4)Solute concentration in the pore liquid is very small, relative to that in the
adsorbed phase, and can therefore be neglected.
In Figure 1, the concentration decreases from Ct at the liquid bulk to C, at the
particle outer surface. This is due to the external resistance controlled by the
external mass transfer coefficient, PL (cm s-'). The value of C, drops further to
zero at a point rF in the particle interior due to the internal resistance measured by
the effective pore diffusion coefficient, Deff (cm s-'). The unreacted zone moves
inwards in a well defined concentration front with a variable velocity, and at all
times there is an unreacted core shrinking in size. Therefore, PL and Defi are the
main parameters that describe the profile of a concentration decay curve.
Concn. in liquid, C
Figure 1 Film-pore diffusion model.
G. McKay, B. Al-Duriand S. McKee
Fundamental equations
Equations of the internal and external mass transfer fluxes, together with the
material balance necessary to describe the adsorption system, are sufficient to
describe the system kinetics.
Barch analysis: The generalised definitions of the external, PL, and internal, Ps,
are given by equations (1) and (2).
(1) Mass transfer in the external boundary layer:
(2) Mass transfer across the boundary layer is then followed by transfer into the
(3) Material balance:
Solving equations (1) and (3) for C, yields:
Two-Resistance Mass Transport Models
At t = 0, all resistance is in the boundary layer since diffusion is entirely
external; therefore, by drawing the initial straight line slope to the concentration
decay curve at t = 0, the gradient (dCt/dt)+o is obtained and the external mass
transfer coefficient (PL) can be determined. l 2
Equilibrium is described by the Langmuir isotherm equation, as given by:
K CJ(1
+ bC,)
Hence, the above equations yield values of Ct, Y, C, and Y* which are used to
determine the internal mass transfer coefficient, Ps,t (cm s-'), shown directly from
equations (1) and (2) to be:
Values of Ps can then be obtained, which are expressed as a function of
experimental and theoretical Sherwood numbers.
Calculation of adsorption rate: The rate of adsorption onto a spherical particle
can be determined as follows:
(1) Mass transfer from the external solution, from equation (1):
4%R2/[B~ (Ct
- C,)]
(2) Diffusion in the pore solution according to Fick's first law:
N -(ltrD,ff)C,,
t/[ (l/rf)
(1/R) I
G. McKay, B. Al-Duri and S. McKee
where D,ff is the effective diffusion coefficient in the pore solution.
(3) The velocity of the concentration-front is obtained from the mass balance on a
spherical element:
(Amount of solid adsorbed, mg) = (weight of ‘reacted’ adsorbent, g)
x (Equilibrium solid phase concentrtation, mg g-’)
From a mass balance on a partially loaded particle:
The rate of adsorption is then obtained by differentiation:
-4s If2P, yh* (drf/dt)
(4) The average concentration of adsorbate in the particle is determined as follows:
The adsorbate concentration is given by equation (14). If instead there was an
average concentration (Y)throughout the particle:
- (4x/3)R3p,Y
then equating (14) and (16) and rearranging:
Equating equations (1 1) and (15):
- -4*rf2Yh*pS(drf/dt)
Substituting equation (20) into (22):
G. McKay, B. Al-Duri and S. McKee
Dimensionless time is given by:
Equation (25) may now be rewritten:
The dimensionless liquid phase concentration is given by:
Therefore, .substituting equations (27) and (29) into (28):
Two-Resistance Mass Transport Models
The dimensionless solid phase concentration is given by:
Relating equations (31) and (17):
Substituting for rf and drf into equation (30):
G.McKay, B. Al-Duri and S. McKee
The Biot number is given by:
Substituting equation (38) into (37) and rearranging:
This reduces to the adsorption rate equation:
Therefore, the adsorption rate to the individual particle is a function of the
adsorbate concentration in the liquid phase, in the solid phase, and of the Biot
Two-Resistance Mass Transport Models
For a mass balance on the system:
- (Y - Y,)W
Assuming Yo= 0, and rearranging:
- F w/v
The capacity factor (Ch) is given by:
Rearranging equation (42) and substituting in (43), (29) and (31):
Substituting equation (44) into (40):
Development of solution methods
Equations (1) and (8) cannot be easily solved because of the term (dCt/dt), in
equation (8). It has been solved by graphical differentiation, enabling a constant
effective diffusivity to be determined which describes the concentration decay
curves over the whole range of the isotherm applied to the system.I2 This method
is tedious and subject to a significant error, therefore other solutions have been
Pseudo-analytical batch model (PAM): For this model, an analytical solution
using the fundamental equations (1) and (8) has been d e ~ e l o p e d ~ but
' - ~ ~this
solution involves applying the saturated isotherm constraint. It is a practical,
easy-to-use model and with its pseudo-irreversible assumption it is applicable to
G. McKay, B. Al-Duri and S. McKee
system conditions where the operating line terminates on the isotherm plateau.
Therefore, a ‘pseudo-irreversible’ isotherm approximation is valid and so:
Y* - Y
- y-
This allows equation (45) to be integrated, giving:
The integration limits for equation (47)are:
z=O,q=Oandx= 1
T = T, q = q and x = x
By converting dimensionless time (T) into real time (t) it is possible to
qompare experimental and theoretical concentration decay curves.
,Pseudo analytical model using a polynomial curve fitting equation: This model is
used to determine the best combination of the two mass transfer resistances to
Two-Resistance Mass Transport Models
describe experimental resistances. The computer program incorporates a
subroutine which fits a polynomial by least squares to the experimental data
relating to the concentration decay curve. The success of the fit is assessed by
comparing actual and calculated point concentrations and determining the
correlation coefficient for various sets of data. The subroutine ‘Polyfit’ fits a
number of polynomial equations of different order to the experimental
concentration data using the NAG 24 subroutine E02ACF.24 The general form of
the polynomial equation is given for the concentration (C,) as a function of time
.A + Alt + A2t2
.......... + A#
The best-fit polynomial can then be differentiated within the program, in
another subroutine, giving the appropriate value of (dC/dt), for substitution and
solution of equation (4). The concentration gradient has the form:
+ ......... + NANtN-l
Program arrays are used to present experimental and calculated data in tabular
form including time, liquid concentration, calculated liquid concentration,
concentration gradient, internal mass transfer coefficient, and theoretical and
experimental Sherwood numbers. The success of the model can be determined by
comparing the experimental and theoretical Sherwood numbers.
Exponential curve-fit model: The solution equation (47) has given accurate
r e ~ u l t s ~but
~ *is~ ’only applicable to a limited range of system conditions. Another
method has been introduced where a computer program originally used by
Provencher26and Provencher and V ~ g e has
l ~ been
~ adopted. The program applies
an exponential series function to the set of concentration-time points. This
subroutine uses Fourier transforms in the fit and yields the following general
solution function:
G. McKay, B. Al-Duri and S. McKee
i. e.
a1 a+'
. .... + ane - A nt
A wide range of values of
and Defi is supplied to the program for various
system conditions. The program responds with the five best-fit curves based on
the best combinations of PL and Deff depending on the system conditions.
A subroutine has been utilised to determine the concentration gradients at the
experimental data points. The form is as follows:
C - A a exp(-X t)
1 J
i. e.
- .... -
The values of (dc/dt) are then used to solve equation (4) and a comparison can
be made between experimental and theoretical Sherwood numbers, their accuracy
being weighted by the residual.
The modified Sherwood number (Shtheo) is proportional to the adsorption
rate1* and is given by:
The experimental Sherwood number is calculated using the experimental
adsorption rate, and is evaluated (R in cm) from:
The theoretical Sherwood number is obtained by substituting equation (45)
into ( 5 3 , thus yielding:
Two-Resistance Mass Transport Models
The extent of agreement between She,, and Shth,, is measured by the residual,
R'. This is calculated by the program through a subroutine:
The models developed in this paper are applied to generate theoretical
concentration versus time decay curves which are compared with experimental
data in Part I1 of this paper.
simplifying term in analytical solution [(I - ch)/Ch]"
adsorbent particle surface area (cm2)
Langmuir constant (dm3 mg-')
simplifying term in analytical solution [ 1 - (pi-')]
Biot number (PLR/Defi)
liquid phase concentration of solute (mg dm-3)
bulk equilibrium liquid phase concentration of solute (mg dm-3)
initial liquid phase concentration of solute (mg dm-3)
equilibrium liquid phase concentration of solute at surface of particle
(mg dm-3)
liquid phase concentration of solute at time t (mg dm-3)
capacity factor (WYh*NCo)
diameter of particle (cm or pm)
effective diffusion coefficient (cm2 s-')
Langmuir equilibrium constant (dm3 g-')
amount of solute adsorbed (mg)
rate of adsorption (mg s-')
radial distance from centre of particle (cm)
radius of concentration front (cm)
radius of adsorbent particle (cm or pm)
specific surface of adsorbent particle (cm-')
modified Sherwood number (theoretical)
G. McKay, B. Al-Duri and S. McKee
experimental Sherwood number = (psR/De.f)(psY */106Co)
time (s)
volume of solution (dm3 or cm3)
mass of adsorbent (mg or g)
simplifying term in analytical solution = (1 -1l1''~
mean concentration of solute on adsorbent (mg g-'
initial concentration of solute on adsorbent (mg g- l))
concentration of solute on adsorbent at equilibrium (mg g-')
hypothetical equilibrium concentration of solute on adsorbent (mg g-'1
Greek Symbols
mass transfer coefficient in liquid, solid (cm s-')
dimensionless liquid-phase concentration (CJC,)
dimensionless solid-phase concentration (Y/Yh*)
density of solid (adsorbent) (g ~ m - ~ )
dimensionless time = (CoDefft)/(psYh*R2)
1 McKay, G. 1984. Analytical solution using a pore diffusion model for a
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3 Hassler, J.W. 1974. Purification with Activated Carbon. Chemical Publishing Company,
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4 McKay, G., Aga, J.A., Allen, S.J., Bino, J.J., McConvey, I.F.. Otterburn, M.S. and
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6 Usinowicz, P.L. and Weber, W.J. 1973. Mathematical Simulation and Prediction of
Adsorbent Performance for Complex Waste Mixtures. Summary Report, EPA
Research Project No. 17020, University of Michigan, Ann Arbor, June.
7 Onano, E.A.R., Sudo, K. and Suziki, T.W. 1978. Water pollution control in developing
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8 Sudo, Y. and Suzuki, M. 1978. Concentration dependence of effective surface diffusion
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10 Furusawa, T. and Smith, J.M. 1973. Fluid-particle and intra-particle mass transport rates
in slurries. Ind. Eng. Chem. Fundam., 12(2), 197-204.
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19 McKay, G., Blair, H.S. and Gardner, J.G. 198%. The adsorption of dyes on chitin; Part I:
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20 McKay, G., Blair, H.S. and Gardner, J.G. 1982b. The adsorption of dyes on chitin; Part
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21 Neretnieks, I. 1974. Adsorption of components having a saturation isotherm. Swedish
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22 Weber, T.W. 1978. Batch adsorption for pore diffusion with film resistance and an
irreversible isotherm. Can. J. Chem. Eng., 56, 187-195.
23 McKay, G. 1983. The adsorption of dyestuffs from aqueous solution using activated
carbon: Analytical solution for batch adsorption based on external mass transfer and
pore diffusion. Chem. Eng. J., 27, 187-194.
24 NAG Numerical Algorithms Group (UK) Ltd. 1983. Fortram Library, Volume 5, Central
Office: 7 Banbury Road, Oxford OX2 6NN, England.
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26 Provencher, S.W. 1976. A Fourier method for the analysis of exponential decay curves.
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27 Provencher, S.W. and Vogel, R.H. 1980. Analysis and formulation of series for
exponential decay curves. Math. Biosci., 50.25 1-257.
Received: 27 February 1991; accepted: 10 April 1992.
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