Lecture 4. Kinetics and Transport in Sorption (1)

Transcription

1 Leture 4. Kinetis and Transport in Sorption () Mass-Transport Proesses in Adsorption Convetion-Dispersion Model Modes of Time-Dependent Sorption Operation Chromatogram Solute Conentration Distriution Separation Effiieny Chromatography esolution

2 Conseutive Mass-Transport Proesses in Adsorption (). Solute transport y ulk flow (onvetion) and dispersion through intersties of the ed. External (interphase) solute transport from the ulk flow to the outer perimeter of the adsorent through a thin film or oundary layer 3. Internal (intraphase) solute transport y diffusion in quiesent fluid-filled pores 4. Surfae diffusion along the internal porous surfae of the adsorent

3 Conseutive Mass-Transport Proesses in Adsorption () External transport - Convetive dispersion of the solute within the ulk fluid - Diffusion though a oundary layer surrounding adsorent partiles Axial dispersion of individual solute moleules in the ulk fluid ours primarily y mirosale, fluid-phase phenomena suh as mixing via solid ostrutions to flow, eddies, and reirulation from regional pressure gradients Boundary-layer transport, internal transport, and surfae diffusion: random Brownian motion Kineti interation depends on solute orientation and frequeny of surfae ollisions - Instantaneous for physisorption - Can eome a ontrolling step for hemisorption (ond formation)

4 Solute Conentration and Temperature Profiles egeneration of the adsorent: the reverse of the four steps ours, following desorption Conentration gradient: usually steepest within the partile The major resistane to mass transfer is in the adsorent partile Temperature gradient: usually steepest in the oundary layer surrounding the partile esistane to heat transfer is mainly external to adsorent partile

5 Convetion-Dispersion Model () Fluid-phase onentration f (z,t) for solute onveted y dispersed, plug flow of fluid at average interstitial veloity u in the axial diretion z e + u - E - - t z z e t f f f Change in solute onentration in the ulk fluid with time Change in f with axial loation arising from ulk onvetion Axial dispersion Change in solute loading on the stationary phase ε : frational interstitial void volume (ed porosity) E : Fikian, onvetive, axial-dispersion oeffiient : volume-averaged stationary-phase onentration per unit mass For a spherial partile of radius p ò p ( 4p r ) 0 dr 3 p ( 4 3) ò p p p r dr * Superfiial veloity of the fluid, u s ε u

6 Convetion-Dispersion Model () q (equilirium loading): mass of solute adsored on surfaes of the stationary phase per unit mass of adsorent : oth adsored solute and unadsored solute diffusing in the pore volume of the stationary phase At equilirium in nonadsoring gel-filtration hromatography, q 0, Linear driving fore (LDF) approximation 4 p 3, 4p p -k tot p( a - f ) 3 t e * p f k,tot : overall mass-transfer oeffiient At equilirium, solute is partitioned etween the ulk fluid and average stationary-phase loading a f e * ( + K ) p d K d : equilirium distriution oeffiient a d kd ε p* : inlusion porosity (effetive stationary-phase volume fration aessile to a speifi solute) ε p* inludes only voids penetrale y a partiular solute due to size or steri hindrane K k s p

7 Modes of Time-Dependent Sorption Operation Frontal: The moile phase (gas or liquid) is fed ontinuously to the stationary phase until the ed approahes saturation with the solute Sorption aility B < C < D Displaement: Following near-saturation of the ed with solute(s), desorption (elution) in onseutive zones of pure sustanes is effeted y sustitution with a more strongly adsored solute (displaer) Sorption aility B < C < D < E (displaer) Differential (elution): A small pulse of solute dissolved in the moile phase is loaded onto the ed the solute pulse is arried through the ed (eluted) at a rate lower than pure solvent Sorption aility A (eluent) < B < C < D

8 Chromatogram () etention time: the time elapsed from the moment the sample is fed to the olumn up to the moment the hromatographi zone of the sustane output reahes its maximum t L u s u s : average veloity of solute in olumn For pulses whih are fairly wide t L + u s tf t F : period of the feed pulse W 4s (s is the standard deviation of the peak) The non-retained peak whih exits at t 0 shows when a small, non-retained moleule will exit. t 0 is a measure of the void volume etween partiles and inside the partiles.

9 Chromatogram () t t - t, 0 - t, 0 k a k k : equilirium onstant a ij : seletivity Peak sharpness depends on olumn effiieny, and the distane etween the maxima is determined y seletivity Effiieny an e expressed quantitatively y N (the numer of theoretial hromatographi plates) t N æ ö ç è s ø s : standard deviation in time unit Height of the equivalent theoretial plate H L N L : olumn length H µ s H is proportional to the variane for hromatographi system

10 Solute Conentration Distriution Sharp-pulse input { z, 0} m0d{ z} Ae m 0 : solute mass A : ed ross-setional area d{z} : Dira delta funtion (zero everywhere exept at z0, where its magnitude is infinity) Gaussian solution for eluted peak æ - x è s ö exp max ç x : deviation from the peak maximum ø m w é 0 -( z - z0) ù { z, t} exp ê ú Ae p Hz Hz0 ë 0 û z 0 : mean solute position (wut) w : fration of solute in the moving fluid phase at equilirium w - e + e a

11 Separation Effiieny () (Separation effiieny in linear hromatography) µ H - Van Deemter equation B H A + + Cu u - A: eddy-diffusion, related to hanneling through a non-ideal paking - B: moleular diffusion in the longitudinal diretion - C: mass transfer resistane etween moile and stationary phases (C C m + C sm + C s ) C m : extrapartile mass transfer (film diffusion) C sm : diffusion in the stagnant moile phase (pore diffusion) C s : diffusion on the solid HETP Mass transfer

12 Separation Effiieny () elation etween H and overall mass-transfer oeffiient, k,tot H æ E w( - w) pu ö + ç u 3a k è, tot ø Transport-rate resistane due to onvetive dispersion p 3 æ K ö d + + * * ç k, tot k 5e pde pkae p è + Kd ø esistane in external oundary-layer transport esistane in internal intrapartile diffusion esistane in kineti sorption rate k : fluid-phase mass-transfer oeffiient D e : effetive solute diffusivity in the pore liquid

13 Chromatography esolution esolution, : a measure of how ompletely two neighoring peaks are separated from eah other ( t ), - t, w t ì ïs í N ï îw 4s + w ( ), -, ( t ), + t, N t t ( ), -V, ( V ), + V, N V esolution in terms of retention volume