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Plate models in chromatography: analysis and implications for scale-up

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Chromatography

Part of the book series: Advances in Biochemical Engineering/Biotechnology ((ABE,volume 49))

Abstract

Detailed chromatographic rate theories from the literature can be used to determine the appropriate plate count for a plate model of linear chromatography so that the bandspreading generated by the detailed rate model is reproduced by the plate model. This process provides a link between the plate count and the physical parameters that cause bandspreading. Each sample component can be assigned an appropriate plate count, thus allowing the accurate simulation of multicomponent separations even for widely differing adsorbates. Analytical solutions are presented for the Craig distribution and the continuous plate model for both finite-pulse elution and frontal chromatography. The Craig model is widely considered unsuitable because it assumes discontinuous flow; it is shown that, for a suitably corrected plate count, the Craig model is as accurate as the continuous-flow plate theory (except for the case of an unretained solute). Direct calculation of effluent histories from these plate models show excellent agreement between themselves and with results from complex rate models available in the literature. Reasonable agreement is also found when the plate models are used a priori to predict experimental scale-up results.

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Abbreviations

c:

mobile phase concentration, M

Dx :

effective micropore diffusivity, m2s−1

Dy :

effective macropore diffusivity, m2s−1

Dz :

effective axial dispersion coefficient, m2s−1

Ix(a, b):

incomplete beta function, = \(\frac{{\smallint _0^x t^{a - 1} (1 - t)^{b - 1} dt}}{{\smallint _0^1 t^{a - 1} (1 - t)^{b - 1} dt}}\)

J:

plate count in continuous-flow plate model

kf :

external film mass transfer coefficient, m s−1

K di :

dimensionless distribution coefficient; subscript i indicates ith component

K *i :

overall distribution coefficient, =εy+(1−εy)K id

k′:

retention factor, dimensionless

L:

column length, m

M:

input pulse width (discretized)

N:

number of spatial segments in the Craig simulation

Nplate :

plate count

p′i :

probability of ith species being in the mobile phase

P(a, x):

incomplete gamma function, = \(\frac{{\smallint _0^x t^{a - 1} e^{ - t} dt}}{{\smallint _0^\infty t^{a - 1} e^{ - t} dt}}\)

Pex :

microparticle Peclet number, u(2rx)/Dx, dimensionless

Pey :

pellet Peclet number, u(2ry)/Dy, dimensionless

Pez :

bed Peclet number, uL/Dz, dimensionless

\(\overline {Pe}\) :

equivalent Peclet number in the axial-dispersion model

q:

stationary phase concentration, M

q′i :

probability of ith species being bound to the stationary phase

rx :

microparticle radius, m

ry :

pellet radius, m

Sh:

Sherwood number, kf(2ry)/Dy, dimensionless

t:

time, s

tR :

retention time (first moment), s

Δt:

time increment in the Craig simulation, s

u:

superficial velocity, m s−1

v:

mobile phase velocity, m s−1

Vint :

interstitial velocity, m s−1

x:

axial coordinate, m

Δx:

axial space increment, m

F:

frontal chromatography

i:

ith component

j:

jth plate

k:

kth time interval

αi :

coefficient in the continuous-flow plate solution, =Ni/tR,i, s−1

δx :

scaled microparticle radius, rx/L, dimensionless

ψy :

scale pellet radius, ry/L, dimensionless

εx :

micropore porosity, m3 micropores per m3 microparticles

εy :

macropore porosity, m3 macropores per m3 pellet

εz :

bed porosity, m3 bed voids per m3 bed

φ:

volumetric phase ratio, dimensionless

μ0 :

zeroth temporal moment, M s

μ1 :

first temporal moment, s

μ2 :

second temporal moment, s2

\(\bar \mu _2\) :

second central temporal moment, s2

Ï„:

input pulse width, s

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Author information

Authors and Affiliations

  1. Laboratory of Renewable Resources Engineering, Purdue University, 47907, West Lafayette, IN, USA

    A. Velayudhan & M. R. Ladisch

  2. Department of Agricultural Engineering, Purdue University, 47907, West Lafayette, IN, USA

    M. R. Ladisch

Authors
  1. A. Velayudhan
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  2. M. R. Ladisch
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G. T. Tsao

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© 1993 Springer-Verlag

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Velayudhan, A., Ladisch, M.R. (1993). Plate models in chromatography: analysis and implications for scale-up. In: Tsao, G.T. (eds) Chromatography. Advances in Biochemical Engineering/Biotechnology, vol 49. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0046575

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  • DOI: https://doi.org/10.1007/BFb0046575

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56440-9

  • Online ISBN: 978-3-540-47513-2

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