The Chemical and Biological Defense Program (CBDP) Dr. Gerald Parker Sept 19, 2012 Advance Planning Briefing for Industry (APBI)
Report CopyRight/DMCA Form For : Adsorption And Desorption Kinetics For Diffusion
where qo qs and Q q qo The diffusivity ratio over the concentration range 0. Q 1 is given by r 1 1 If the equilibrium isotherm is highly favorable b r. can be very large,2 Sorption Kinetics, For an isothermal parallel sided slab thickness 2l subjected at time zero to a step. change in the surface concentration of sorbate the adsorption desorption kinetics. assuming internal diffusion control are governed by the partial differential equation. where f Q 1 1 Q D Do X x and Dot 2 with the initial and boundary. conditions,0 Q 0 ads or Q 1 des for all x,0 Q 0 1 ads or Q 0 0 des for all 4. Since Eq 3 is non linear and the boundary conditions are finite an analytic solution is. not practical Numerical solutions for the corresponding problem in spherical coordinates. were presented many years ago by Garg and Ruthven 2. In the initial region the concentration front has not penetrated to the centre of the slab. so the system behaves as a semi infinite medium In this situation the last boundary. condition of Eq 4 may be replaced by,x 0 Q x 0 ads or 1 0 des 5. This allows the Boltzmann transformation y x 2 Do t to be used to reduce Eq 3. to the ordinary differential equation, Note that distance x is measured from the external surface at which x 0. with the initial boundary conditions,t 0 Q y 0 ads or 1 0 des. t 0 Q 0 1 0 ads or 0 des for all t 7,Q y 0 ads or 1 0 des. Integration of Eq 6 yields,ydQ f Q f Q 8,2 dy y 0 2 dy y. Since the gradient of concentration is zero for large y this simplifies to. A ydQ y 0 ads y 0 des 9, where A is the area under the Q vs y profile corresponding to the amount of material. adsorbed or desorbed Eq 9 provides a convenient check on the validity of numerical or. analytic approximations to the solution for the concentration profile. 3 Adsorption Profiles, A formal analytic solution to this problem for adsorption with f Q 1 Q 1 has. been given in parametric form by Fujita 3 summarized by Crank 4. Q 1 e 1 e for r large y,I f Q 2 2 ln r 11,The parameter is defined by the relationship. I 1 lnr 12,When is small f,and I I 1 ln ln r 13, With these approximations the limiting expression for the concentration profile for r. large may be obtained in explicit form,Q 1 exp 2 y 14. The limiting slope is given by,r y 0 2 15, This approximation is valid only for the region where Q 1 0 but the range of. validity increases when r is large,Variation of r with. r 1 00E 04, 1 00E 08 1 00E 06 1 00E 04 1 00E 02 1 00E 00 1 00E 02. Fig 1 Variation of diffusivity ratio r with parameter calculated from Eq 12. Another limiting explicit approximation which is valid as Q 0 may be obtained by. noting that for 0,I 2 erfc u 16,where u 2 ln We thus obtain. y 2 erfc u Q 2 erfc y 17,as the asymptotic expressions for the limit 0 u. The variation of the parameter with r calculated by numerical integration of. Eq 12 is shown in figure 1 Concentration profiles for various values of r calculated. from Eq 10 are shown in figure 2 Also shown are the limiting profiles calculated from. the asymptotic expressions Eqs 14 and 17 for selected values of r When r is large. Eq 14 provides a good approximation over most of the range but Eq 17 is valid only at. very low values of Q Since y x 2 Do t X 2 one may plot directly the profiles. of Q vs X for a finite slab for times less than the time at which the penetrating wave. reaches the centre of the slab Such a plot is shown in figure 3 from which the form of. the penetrating wave is apparent Note that in contrast to the shockwave type of. behavior observed for a macropore diffusion controlled system with irreversible. adsorption 5 the concentration wave in the present system disperses as it penetrates. Concentration Profiles Adsorption,Q r 1 r 10 100 640 3 50E 04 2 50E 07. 0 0 5 1 1 5 2 2 5 3 3 5 4, Fig 2 Concentration profiles for adsorption showing how the form of the. concentration wave changes with r Note that when r is large the asymptotic. expression Eq 14 provides a good approximation except when Q is small. Adsorption Profiles Finite Slab,0 8 r 3 5 E4,0 001 0 003 0 01 0 02. 0 0 2 0 4 0 6 0 8 1, Fig 3 Profiles of figure 2 replotted in the coordinates Q vs X for a finite slab for. various values of Note the increasing spread of the wave form as increases. 4 Desorption Profiles, Fujita presented only the solution for adsorption but by following his procedure the. corresponding desorption problem may be solved to yield for the transient profile. 1 exp 2 I I 1,y f exp I 1 I, where the symbols and integrals have the same meanings as for the adsorption case. It is shown in the appendix that when r 1 Eqs 10 and 18 both reduce to the well. known error function forms for a constant diffusivity system Following a procedure. similar to that used to derive Eq 14 the following explicit approximation valid only for. small C and large r can be obtained, Concentration profiles calculated from Eqs 18 are shown in figure 4 As is to be. expected and in contrast to a linear system the profiles for desorption and adsorption are. quite different The desorption profiles show no inflexion The initial slope increases. only slightly with increasing r but the curvature increases so that when r is large the. gradient at some distance from the surface is small Physically this form is to be. expected since when r is large the desorption rate is controlled by diffusion through the. surface region in which D Do,Profiles for Desorption. r 1 r 2 r 20,0 1 2 3 4 5 6, Fig 4 Profiles for desorption from a semi infinite medium for selected values of r. 5 Uptake Rates,The adsorption rate is given by,dm dq Dq0dQ. rD0 x 0 y 0 20,dt dx 2 D0 t dy,and on integration,mt r D0 t q0 y 0 21. When the diffusivity ratio is large 1 0 r the concentration profile. assumes the form of a penetrating wave see figure 2 so the semi infinite medium. approximation will be valid throughout most of the uptake In this regime the fractional. approach to equilibrium m qo will be given by,mt D0 t dQ dQ. r y 0 r y 0 22,m A 2 dy dy, This expression shows that the t law is still valid even when the diffusivity is. strongly concentration dependent, Prediction of the uptake rate depends on estimating the concentration gradient at the. surface dQ dy y 0 When r is large we may use Eq 15 which yields. By comparison with a linear system for which the approach to equilibrium is given. we see that the effective diffusivity De for adsorption is given by. The product r decreases with increasing r so the effective diffusivity will increase. continuously rather than approaching an asymptotic limit as the isotherm approaches. the rectangular form, For desorption the semi infinite medium approximation is valid only at short times. since as may be seen from figure 4 when r is large the concentration far from the surface. decreases quite rapidly The expression corresponding to Eq 22 is. but this expression is of limited utility since it applies only in the limit of very short. times A more useful approximation for a finite slab may be obtained by recognizing. that when r is large most of the desorbing material comes from the interior region where. the concentration profile is quite flat As a simple model we therefore approximate the. system as a semi infinite medium in which the concentration qo is essentially uniform. except in the surface region but decreases with time. dq0 dq q0 D0 dq,A D0 x 0 y 0 27,dt dx 2 t dy, We assume that y 0 remains approximately constant true for large r With. this approximation Eq 27 can be easily integrated to yield. Since q q0 1 mt m this yields for the desorption curve. This expression explains why for strongly adsorbed species desorption is very slow. and plots of ln 1 mt m vs t which are linear for a constant diffusivity system. commonly show a decreasing slope 2,Variation of Sorption Rates with r. Adsorption,des or rk ads,Desorption,1 10 100 1000, Fig 5 Variation of initial adsorption and desorption rate semi infinite medium. with r calculated from the limiting slopes of the Q y profiles according. to Eq 22 ads or Eq 26 des, At short times Eq 29 reduces to Eq 26 which is of the same form as Eq 22 for. adsorption so the initial rates of adsorption and desorption for a semi infinite medium. may be compared simply on the basis of the limiting slopes and the diffusivity ratio. Such a comparison is shown in figure 5 For both adsorption and desorption the rate. increases with r but the increase is greater for adsorption leading to an increasing. difference between initial adsorption and desorption rates with increasing non linearity 6. 6 Conclusion, Although the equations describing transient adsorption and desorption with a. concentration dependent diffusivity can be solved numerically this task is not trivial when. r is large and the equations are strongly non linear The analytic solutions originally. obtained by Fujita and extended here to desorption yield greater understanding and. provide a basis for simple asymptotic approximations which are practically useful for. modelling adsorption desorption kinetics,A defined by Eq 8. b Langmuir constant Eq 1,D Fickian diffusivity,Do Limiting or corrected diffusivity Eq 2. De effective diffusivity,I I 1 Integrals defined by Eqs 11 and 12. half thickness of parallel sided slab, mt mass of sorbate adsorbed or desorbed during time t. m value of mt as t i e at equilibrium,p sorbate partial pressure. q adsorbed phase concentration, qo value of q at equilibrium with external gas phase. qs saturation limit Eq 1,Q dimensionless concentration q qs. r diffusivity ratio D Q D0,x distance measured from external surface. X dimensionless distance x,y dimensionless distance x 2 D0 t. limiting slope dQ dy y 0 for desorption Eq 28,constant defined by Eq 12. dimensionless time Dot 2,Q parameter defined by Eqs 11 and 12. References, 1 D M Ruthven Principles of Adsorption and Adsorption Processes pp 125. 170 John Wiley New York 1984, 2 D R Garg and D M Ruthven Chem Eng Sci 27 417 424 1972. 3 H Fujita Textile Res J 22 757 and 823 1952, 4 J Crank Mathematics of Diffusion p 167 Oxford University Press London 1956. 5 N K B r B Balcom and D M Ruthven I and EC Research 41 2320 2329. Appendix Reduction to Linear System for r 1 0, The linear limit r 1 0 corresponds to as may be seen from figure 1. When is large we have,f 2 lnu 2 u where u2 ln,e du erfc u. I 1 lnr r exp,Both I and I 1 are small so exp I 1 I 1 erfc u. Hence we have,1 e erfc u,Adsorption Q,Desorption Q 1 exp 2 I I 1 erf u. y f exp I 1 I u exp, These are the well known solutions for a constant diffusivity system. Acknowledgement, Financial support from the National Science Foundation Grant CTS 0553861 is.
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