126
H. Rehage and M. Kind
−
˙
M out + ˙
M circ,1
ξ
L
BF x
L
j,BF
(15)
Nucleation or growth are not relevant for the BF, as both take place only in the PFR.
Consequently, the particle mass fractions for the BF can be calculated by Eq. (16).
dw
S
i,BF
dt
+ w
S
i,BF ·
dln
x
S
BF
dt
+ w
S
i,BF ·
dln(M BF )
dt
=
1
M BF x
S
BF
·
˙
M sec ξ
S
sec w
S
i,sec + ˙
M circ,2 ξ
S
circ,2 w
S
i,circ,2
−
˙
M out + ˙
M circ,1
ξ
S
BF w
S
i,BF
(16)
The unknown variables of this model are the circulation mass flow rate ˙
M circ,1
and the vector
S circ,2 representing the fluid after precipitation in the PFR. A closure
for ˙
M circ,1 based on the stirrer type and size was developed by using the conceptual
idea of the similarity between the local reaction zone in a stirred tank with Rushton
Turbine and a JICF precipitation. Calculation of
S circ,2 is explained in Sect. 2.3.2.3.
Explicit Euler’s method is used to solve Eqs. (13–16) with a static time discretization
= 0.5. This value for is appropriate for the Simulation Setups C and E.
Mixing Models
The semi-batch model uses the steady-state model presented in Sect. 2.3.1 to calculate
the PFR. The mixing model for CIJMs was replaced by a mixing model for the jet in
cross flow (JICF) mixing of the feed volume flow and the circulation flow. As shown
in [29], a JICF mixer can be used to imitate the local flow environment around
the feed pipe in STRs. The reaction zone is, therefore, defined as Q prim (P), which
engulfs Q circ,1 (C) over the mixer length coordinate z. A possible influence of meso
mixing must be additionally considered, as the fluid must be meso mixed first to start
micro mixing. Consequently, zone C meso is introduced for the fluid of Q circ,1 which
is already meso mixed and therefore, can act as an engulfment environment for P
(Fig. 11).
P
C
C
P
C
meso
P
C
C
meso
P
z
Fig. 11 Fluid prim (P) engulfing fluid circ, 1 (C) in the E-model by [24]. C meso designates the fluid
which is already meso mixed and, thus, provides the environment for the engulfment process of P
H. Rehage and M. Kind
−
˙
M out + ˙
M circ,1
ξ
L
BF x
L
j,BF
(15)
Nucleation or growth are not relevant for the BF, as both take place only in the PFR.
Consequently, the particle mass fractions for the BF can be calculated by Eq. (16).
dw
S
i,BF
dt
+ w
S
i,BF ·
dln
x
S
BF
dt
+ w
S
i,BF ·
dln(M BF )
dt
=
1
M BF x
S
BF
·
˙
M sec ξ
S
sec w
S
i,sec + ˙
M circ,2 ξ
S
circ,2 w
S
i,circ,2
−
˙
M out + ˙
M circ,1
ξ
S
BF w
S
i,BF
(16)
The unknown variables of this model are the circulation mass flow rate ˙
M circ,1
and the vector
S circ,2 representing the fluid after precipitation in the PFR. A closure
for ˙
M circ,1 based on the stirrer type and size was developed by using the conceptual
idea of the similarity between the local reaction zone in a stirred tank with Rushton
Turbine and a JICF precipitation. Calculation of
S circ,2 is explained in Sect. 2.3.2.3.
Explicit Euler’s method is used to solve Eqs. (13–16) with a static time discretization
= 0.5. This value for is appropriate for the Simulation Setups C and E.
Mixing Models
The semi-batch model uses the steady-state model presented in Sect. 2.3.1 to calculate
the PFR. The mixing model for CIJMs was replaced by a mixing model for the jet in
cross flow (JICF) mixing of the feed volume flow and the circulation flow. As shown
in [29], a JICF mixer can be used to imitate the local flow environment around
the feed pipe in STRs. The reaction zone is, therefore, defined as Q prim (P), which
engulfs Q circ,1 (C) over the mixer length coordinate z. A possible influence of meso
mixing must be additionally considered, as the fluid must be meso mixed first to start
micro mixing. Consequently, zone C meso is introduced for the fluid of Q circ,1 which
is already meso mixed and therefore, can act as an engulfment environment for P
(Fig. 11).
P
C
C
P
C
meso
P
C
C
meso
P
z
Fig. 11 Fluid prim (P) engulfing fluid circ, 1 (C) in the E-model by [24]. C meso designates the fluid
which is already meso mixed and, thus, provides the environment for the engulfment process of P