Unformatted text preview: e interface rc
and time t, differentiate Eq. (2) with respect to r and evaluate the differential at r = rc. This
gives:
− c A0 − c Ab
dc A
=
(3)
dr r = rc
rc
rc 1 −
rs
The rate of diffusion at r = rc is given by Fick's first law, e.g. Eq. (366):
dN A
dc A
−
= 4πrc2 De
(4)
dt
dr r = rc
1− Exercise 11.26 (continued)
Analysis: (continued)
Where,
N A is the amount of A diffused. Combining Eqs. (3) and (4): ( d N A 4πrc De c A0 − cAb
=
dt
r
1− c
rs
By material balance for the solute in the core: ) (5) dN A
ρA d 4 3
4πrc2ρ A drc
=−
πrc = −
dt
M A dt 3
M A dt
where: (6) ρ A = initial mass of solute per unit volume of solid particle
M A = molecular weight of solute Combining Eqs. (5) and (6): −ρ A 1 1 2
− rc drc = De c A0 − c Ab dt
M A rc rs
Integrating Eq. (7) and applying the boundary condition: ρ A rs2
t=
6 De M A c A0 − c Ab (7) rc = rs at t = 0 gives: r
1− 3 c
rs 2 r
+2 c
rs 3 (8) For complete extraction, rs = 0, and Eq. (8) becomes: t= ρ A rs2
6 De M A c A0 − c Ab (9) Equations (5) and (9) indicate that both the rate of extraction and time for extraction depend on
the effective diffusivity of the solute in the particle, which must be measured experimentally. Exercise 11.27
Subject: Supercritical extraction of a slurry of 0.002 kmol/h of solid βcarotene in 0.20 kmol/h
of water with a solvent of recycle carbon dioxide in a singlestage contactor operating in the
supercritical region to extract 99% of the carotene.
Given: Design of a supercritical extractor using the GroupContribution EOS of SkjoldJorgensen, at 353 K and 1013 bar, as shown in Fig. 1 of the cited article by Cygnarowicz and
Seider.
Find: Design of a similar extractor using the PengRobinson EOS with the WangSandler
mixing rules.
Analysis: In the Cygnarowicz and Seider (CS) article, the solubility of the carotene in water is
neglected and a fit of experiment data for the solubility of carotene in CO2 is given in Fig. 2 of
the CS article. Thus, it was only necessary here to model a singlestage flash of CO2 and water.
An initial run was made with the Aspen Plus program with the FLASH2 model under the
following conditions, based on the PR EOS with the WangSandler mixing rules (PRWS):
Feed of 0.2 kmol/h of water.
Solvent of 0.8004 kmol of CO2 and 0.0081 kmol/h of water.
Singlestage extraction at 353 K and 1013 bar.
The result of the flash was only a single phase containing both the feed and the solvent. Thus,
use of the PR EOS with WangSandler mixing rule did not give the required two equilibrium
phases shown in the CS article. When the extraction pressure was reduced to 101.3 bar (100
atm), the following results, which are close to the CS results, were obtained:
kmol/h:
Component Feed Solvent
C O2
Water
Total: 0.0
0.2
0.2 0.8004
0.0081
0.8085 Raffinat
e
0.0275
0.1950
0.2225 Extract
0.7729
0.0131
0.7860 However, at this much lower pressure, Fig. 2 of the CS article shows that the mole fraction of
carotene in the extract is an unacceptable 4 x 108 instead of t...
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 Spring '11
 Levicky
 The Land

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