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

код для вставкиСкачатьDevelopment of Solutions to Two-Resistance Mass Transport Models Based on External and Pore Diffusion. Part I: Theoretical DeveIopment 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. Introduction 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 129 130 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 previously.17-21 Theoretical Development Description 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 131 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 values. (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. 0 Concn. in liquid, C Figure 1 Film-pore diffusion model. G. McKay, B. Al-Duriand S. McKee 132 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 particles: (3) Material balance: N - -V(dCt/dt) - W(d?/dt) Solving equations (1) and (3) for C, yields: Initially, Ct - co and cs,t - (3) Two-Resistance Mass Transport Models I33 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: Y* - K CJ(1 + bC,) (9) 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): N 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 I34 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-’) (13) From a mass balance on a partially loaded particle: The rate of adsorption is then obtained by differentiation: N - -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: N - (4x/3)R3p,Y then equating (14) and (16) and rearranging: Y - Yh* - (rfnl31 Equating equations (1 1) and (15): N - kR2P~(Ct - C,) - -4*rf2Yh*pS(drf/dt) I35 Rearranging: Substituting equation (20) into (22): Expanding: G. McKay, B. Al-Duri and S. McKee 136 Rearranging: Dimensionless time is given by: and Equation (25) may now be rewritten: The dimensionless liquid phase concentration is given by: c - (C,/C,) Therefore, .substituting equations (27) and (29) into (28): Two-Resistance Mass Transport Models The dimensionless solid phase concentration is given by: (I - %Yh* Relating equations (31) and (17): Differentiating: drf - (-R/3)(1 - Substituting for rf and drf into equation (30): I37 I38 G.McKay, B. Al-Duri and S. McKee Rearranging: 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 number. Two-Resistance Mass Transport Models 139 For a mass balance on the system: - V(C, C) - (Y - Y,)W Assuming Yo= 0, and rearranging: (C0 - C) - 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 developed. 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 140 system conditions where the operating line terminates on the isotherm plateau. Therefore, a ‘pseudo-irreversible’ isotherm approximation is valid and so: - Y* - Y - Yh* - y- (46) This allows equation (45) to be integrated, giving: where, B - (1 - Bi-’) (48) (49) 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 141 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 (t): Ct - .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: (dC/dt) - A1 + 2A2t + 3A3t2 + ......... + NANtN-l (52) 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 142 i. e. C, - + a1 a+' . + a2 . .... + ane - A nt PL 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: (dCt/dt) - (dCt/dt) -- n C - A a exp(-X t) 1 J j j-0 (544 i. e. Xlale-Xlt - X2a2c-X2t - .... - Xnan (54b) 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 143 The extent of agreement between She,, and Shth,, is measured by the residual, R'. This is calculated by the program through a subroutine: 1 R'-- 2 n C i-1 - Shtheo) Shexp + i Shexp - Shtheo Shtheo (58) i 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. Nomenclature a A b B Bi C ce. CO cs c, ch d 4 f K N N r 'f R S Shh 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 144 She,, t V W x Y Yo Y* Yh* 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 pL,s 5 mass transfer coefficient in liquid, solid (cm s-') dimensionless liquid-phase concentration (CJC,) q dimensionless solid-phase concentration (Y/Yh*) density of solid (adsorbent) (g ~ m - ~ ) dimensionless time = (CoDefft)/(psYh*R2) ps z References 1 McKay, G. 1984. Analytical solution using a pore diffusion model for a pseudo-irreversible isotherm for the adsorption of basic dye on silica. AIChEJ, 30(4), 692-697. 2 Bruke, T., Hyde, R.A. and Zabel, T.F. 1980. Performance and Costs of Activated Carbon for the Control of Organics, 73-81. IWES Seminar - Trace Organics, 21 October, London Institute of Marine Engineers. 3 Hassler, J.W. 1974. Purification with Activated Carbon. Chemical Publishing Company, New York. 4 McKay, G., Aga, J.A., Allen, S.J., Bino, J.J., McConvey, I.F.. Otterburn, M.S. and Sweeney, A.G. 1984. The design of fixed bed adsorption systems for the treatment of coloured and colourless organics in effluents. Chem. Eng. Symp. Series, No.77,217. 5 Perrich, J.R. (ed.) 1981. Activated Carbon Adsorption for Wastewater Treatment. CRC Press Inc., Florida, USA. 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 countries. Proc. Int. Con$, Bangkok, Thailand, 21-25 February, 210-217. 8 Sudo, Y. and Suzuki, M. 1978. Concentration dependence of effective surface diffusion coefficients in aqueous phase adsorption on activated carbon. Chem. Eng. Sci., 33, 1287-1293. 9 Mathews, A.P. and Weber, W.J. 1980. Mathematical modelling of adsorption in multicomponent systems. J. Am. Chem. SOC., 27-36. 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. 11 Furusawa, T. and Smith, J.M. 1974. Intra-particle mass transport in slurries by dynamic adsorption studies. AIChEJ, 20( I), 88-95. 12 Spahn, H. and Schlunder, E.U. 1975. The scale-up of activated carbon columns for water purification, based on results from batch tests (Part 1). Chem. Eng. Sci., 30, 529-537. 13 Levenspiel, 0. 1972. Chemical Reaction Engineering. J. Wiley and Sons, New York. 14 Yagi, S. and Kunii, D. Performance of fluidised-solids reactors. J. Chem. SOC. Jap., 56, 131-138. Two-Resistance Mass Transport Models 145 15 Yagi, S. and Kunii, D. 1961. Fluidised-solidsreactors with continuous solids feed (Part 11). Chem. Eng. Sci., 16, 372-379. 16 Al-Duri, B. and McKay, G. 1990. Comparison in theory and application of several mathematical models to predict kinetics of single component batch adsorption systems. Proc. Safety Environ. Prom (Trans. I. ChemE.), 68B,254-268. 17 McKay, G.and Allen, S.J. 1980. Surface mass transfer processes using peat as an adsorbent for dyestuffs. Can. J. Chem. Eng., 58,521-529. 18 McKay, G., Poots, V.J.P. and Alexander, F. 1978. Adsorption kinetics and diffusional mass transfer processes during colour removal from effluent using silica. Ind. Eng. Chem. Fundam., 17,20-24. 19 McKay, G., Blair, H.S. and Gardner, J.G. 198%. The adsorption of dyes on chitin; Part I: Equilibrium studies. J. Appl. Poly. Sci., 27, 3043-3055. 20 McKay, G., Blair, H.S. and Gardner, J.G. 1982b. The adsorption of dyes on chitin; Part 11: External mass transfer processes. J. Appl. Poly. Sci., 27,42514259, 21 Neretnieks, I. 1974. Adsorption of components having a saturation isotherm. Swedish Paper J., 77,407413. 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. 25 McConvey, I.F. and McKay, G. 1985. Mass transfer model for the adsorption of basic dyes on woodmeal in agitated batch adsorbers. Chem. Eng. Proc., 19,267-277. 26 Provencher, S.W. 1976. A Fourier method for the analysis of exponential decay curves. Biophys. J., 16, 27-33. 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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