## Methodology for Design

Estimating Intraparticle Peclet Number A

The importance of convective flow in permeable particles can be assessed by intraparticle Peclet number A. The convective velocity v0 inside pores can be calculated from the equality of pressure drops across the particle Ap/dp and across the bed AP/L; in laminar flow both for the bulk fluid phase and pore fluid as in HPLC we obtain:

Vo = ^ U Bb where Bb and Bp are bed and particle permeability, respectively. The fraction of flow rate entering the column which goes through the macropores by convection is:

150(1-Sb)2

Bed permeabilities Bb are obtained from the slope of the plot AP/L versus u0 and then the bed porosity sb can be calculated. As a typical example for a POROS Q/M (PE Biosystems, USA) column 4.6 mm i.d. x 100 mm long with a bed volume of 1.7 mL filled with 20 | m particles, the bed permeability Bb = 2.35 x 10"9 cm2 s"1 and sb = 0.34.

Elution Chromatography with Nonretained Proteins

Elution chromatography experiments under non-retained conditions (b = 1) allow the understanding of mass transport inside particles. Flow rates up to 10mLmin_1 corresponding to superficial velocities of 1cms_1 were used. The diffusivities of proteins myoglobin, ovalbumin and bovine serum albumin (BSA) in aqueous solution at 25°C are 16.1 x 10"7, 6.4 x 10"7 and 1 x 10"7 cm2 s"1, respectively.

The HETP is calculated from the experimental chromatographic peaks by:

where o2 is the peak variance, is the first moment of the peak, p2 is the second moment and L is the column length. Figure 8 shows the experimentally measured reduced HETP h = HETP/dp as a function of the bed superficial velocity.

The efficiency of chromatographic columns can be characterized by the HETP; for columns packed with permeable packings, eqn [3] applies. The A term accounts for eddy dispersion effects and becomes a constant at high superficial velocities, A = 2dp; B = 2Dm and so the term B/u0 can be neglected in the case of proteins. The simplified equation for HETP with permeable packings is HETP = A + Cf(A)u0. In the low velocity region

Figure 8 HETP versus superficial velocity u0 for elution chromatography of proteins under unretained conditions in a POROS Q/Mcolumn. Solvents were TRIS-HCI 50 mmol L"1, pH 8.6, mixed with NaCl 0.5 mol L"1 at 22°C. Circles, myoglobin; squares, ovalbumin; triangles, bovine serum albumin; continuous line, Rodrigues' equation. (Reprinted from Rodrigues AE et al. (1996) Protein separation by liquid chromatography using POROS Q/M particles. Chemical Engineering Journal 61: 191-201, with permission from Elsevier Science.)

Figure 8 HETP versus superficial velocity u0 for elution chromatography of proteins under unretained conditions in a POROS Q/Mcolumn. Solvents were TRIS-HCI 50 mmol L"1, pH 8.6, mixed with NaCl 0.5 mol L"1 at 22°C. Circles, myoglobin; squares, ovalbumin; triangles, bovine serum albumin; continuous line, Rodrigues' equation. (Reprinted from Rodrigues AE et al. (1996) Protein separation by liquid chromatography using POROS Q/M particles. Chemical Engineering Journal 61: 191-201, with permission from Elsevier Science.)

where pore diffusion is the controlling mechanism of mass transfer, the slope of HETP versus u0 is

30(1 +v)2 sb De At high flow rates a plateau is reached with:

plateau

Figure 9 Adsorption equilibrium isotherm for bovine serum albumin on POROS Q/M packing. (Reprinted from Rodrigues AE et al. (1996) Protein separation by liquid chromatography using POROS Q/M particles. Chemical Engineering Journal 61: 191-201, with permission from Elsevier Science.)

Frontal Chromatography Experiments

In frontal chromatography experiments a solution of protein with concentration c0 is continuously passed through the column under retained conditions. The breakthrough curve is measured from which the amount of protein retained in the adsorbent, q0 is calculated by mass balance leading to a point on the adsorption equilibrium isotherm. The adsorption equilibrium isotherm of BSA in POROS Q/M is shown in Figure 9 and follows the Langmuir equation. Breakthrough curves with BSA at feed concentration of 2mgmL"1 and various flow rates merge together when the outlet concentration is normalized by the feed concentration and the time is reduced by the stoichiometric time, as shown in Figure 10. Moreover, breakthrough curves are very sharp, indicating that the useful dynamic capacity approaches the total column capacity.

When the particle structure characterized by the intraparticle porosity, Sp, is known, the initial slope and the plateau values provide measured values of De and

The straight line at low flow rates crosses the plateau at critical point where

18De Bb dp Bp

For POROS Q/M, sp = 0.5 and Bp = 1.5 x 10"11 cm2 for an experimental h plateau of 36 (reduced HETP) and De = 7 x 10"8 cm2 s"1.

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