Retentate Stream : T
out
V ; P
out
V ; V
out
; Y
out
ð3:5aÞ
Permeate Stream : T
out
L ; P
out
L ; L
out
; X
out
ð3:5bÞ
3.1.3 Distributed Model Algorithm: MPd-UOE
The algorithm for MPd-UOE model comprises six steps,
which are described below: [S1] Input data; [S2] Adjusting
input parameters for first permeation element; [S3] Parameters for energy balance; [S4] Initial values for first permeation element NRM; [S5] Distributed permeation and energy
balance calculations loop; [S6] Returning product data to
simulation.
[S1] Input data: Feed temperature, pressure, molar flow,
composition, and enthalpy are rescued from feed stream in
HYSYS PFD in Eq. (3.6a). Total permeation area, product
pressures, and contact type are defined by the user in the
UOE property window—Eq. (3.6b). Note that differently
from MPx-UOE, the contact type is not a specification, since
the MPd-UOE model is valid only for parallel MP. Default
values for species k permeances are depicted in Table 4, yet
the user can specify otherwise in the UOE property window,
as depicted in Eq. (3.6c).
T
in
V ; P
in
V ; V
in
; Y
in
; H
in
V from simulation environment ð3:6aÞ
A MP ; P
out
V ; P
out
L ; n elements defined by user
ð3:6bÞ
P k default from Table 4 or defined by user
ð3:6cÞ
[S2] Adjusting input parameters for first permeation
element: Feed conditions are set as main inlet parameters for
the first permeation element in Eq. (3.7a). Since there is no
second inlet stream in the MP module, the molar flow and
composition for the first element are set to zero in Eq. (3.7b).
Head-loss in retentate stream is linearly distributed through
the permeation elements, so for the first element, the outlet
retentate pressure is set by Eq. (3.7c). For the permeate
stream, the final outlet pressure was specified in Eq. (3.6b),
yet there is no inlet permeate stream, so the head-loss is
selected and fixed as 0.1 bar, and linearly distributed
through the permeation elements; thus, the outlet permeate
pressure for the first element is given by Eq. (3.7d). Permeation elements are equally distributed in the total permeation area also defined in Eq. (3.6b), so for each element,
the permeation area is a fraction of the total specification, as
shown in Eq. (3.7e).
T
in
ð0Þ
V
¼ T
in
V ; P
in
ð0Þ
V
¼ P
in
V ;
V
in
ð0Þ ¼ V
in
; Y
in
ð0Þ ¼ Y
in
;
H
in
ð0Þ
V
¼
H
in
V
ð3:7aÞ
L
in
ð0Þ ¼ 0; X
in
ð0Þ ¼ 0
ð3:7bÞ
P
out
ð0Þ
V
¼ P
in
V À ðP
in
V À P
out
V Þ=n elements
ð3:7cÞ
P
out
ð0Þ
L
¼ P
out
L þ ð0:1=n elementsÞ Ã ðn elements À 1Þ
ð3:7dÞ
A MP ¼ A MP =n elements
ð3:7eÞ
[S3] Parameters for energy balance: DT
ð0Þ
F and external
temperature (T E ) both have default values specified in
MPd-UOE—Eqs. (3.8a) and (3.8b), respectively—however,
the user can set other values in UOE property window. Note
that differently from MPx-UOE algorithm, in MPd-UOE, the
DT F specification is set as the value for the first permeation
element only (DT
ð0Þ
F ), since it is calculated for the next elements in step [S5]. Internal and external overall heat transfer
coefficients (U I and U E ) are defined in Eqs. (3.8c) and (3.8d),
respectively. The internal area for heat transfer is equal to the
permeation area of each element, via Eq. (3.8e). Equation (3.8f) shows the relation between the external and
internal areas for heat transfer.
DT
ð0Þ
F ¼ 3
C ðdefaultÞ or defined by user
ð3:8aÞ
T E ¼ 25
C ðdefaultÞ or defined by user
ð3:8bÞ
U I ¼ 5 W=m
2
K
ð3:8cÞ
U E ¼ 2 W=m
2
K
ð3:8dÞ
A I ¼ A MP
ð3:8eÞ
A E ¼ A I =276
ð3:8fÞ
[S4] Initial values for first permeation element NRM:
Eqs. (3.9a) to (3.9j) set initial values of species transmembrane molar fluxes for the first permeation area, which
depend on the number of membrane elements chosen by the
user; the more distributed, the lower the permeation area for
each element, yet the higher the number of elements, which
is quadratic, so the lower is the permeation flux. Equations (3.9k) and (3.9m), respectively, define initial values of
retentate and permeate temperatures for the first permeation
area.
N
ð0Þ
CO 2
¼ 0:5 Ã V
in
ð0Þ
CO 2
=ðA MP Ã n elements
2
Þ
ð3:9aÞ
N
ð0Þ
CH 4
¼ 0:075 Ã V
in
ð0Þ
CH 4
=ðA MP Ã n elements
2
Þ
ð3:9bÞ
N
ð0Þ
C 2 H 6
¼ 0:01 Ã V
in
ð0Þ
C 2 H 6
=ðA MP Ã n elements
2
Þ
ð3:9cÞ
N
ð0Þ
C 3 H 8
¼ 0:005 Ã V
in
ð0Þ
C 3 H 8
=ðA MP Ã n elements
2
Þ
ð3:9dÞ
Membrane-Permeation Modeling for Carbon Capture …
167
out
V ; P
out
V ; V
out
; Y
out
ð3:5aÞ
Permeate Stream : T
out
L ; P
out
L ; L
out
; X
out
ð3:5bÞ
3.1.3 Distributed Model Algorithm: MPd-UOE
The algorithm for MPd-UOE model comprises six steps,
which are described below: [S1] Input data; [S2] Adjusting
input parameters for first permeation element; [S3] Parameters for energy balance; [S4] Initial values for first permeation element NRM; [S5] Distributed permeation and energy
balance calculations loop; [S6] Returning product data to
simulation.
[S1] Input data: Feed temperature, pressure, molar flow,
composition, and enthalpy are rescued from feed stream in
HYSYS PFD in Eq. (3.6a). Total permeation area, product
pressures, and contact type are defined by the user in the
UOE property window—Eq. (3.6b). Note that differently
from MPx-UOE, the contact type is not a specification, since
the MPd-UOE model is valid only for parallel MP. Default
values for species k permeances are depicted in Table 4, yet
the user can specify otherwise in the UOE property window,
as depicted in Eq. (3.6c).
T
in
V ; P
in
V ; V
in
; Y
in
; H
in
V from simulation environment ð3:6aÞ
A MP ; P
out
V ; P
out
L ; n elements defined by user
ð3:6bÞ
P k default from Table 4 or defined by user
ð3:6cÞ
[S2] Adjusting input parameters for first permeation
element: Feed conditions are set as main inlet parameters for
the first permeation element in Eq. (3.7a). Since there is no
second inlet stream in the MP module, the molar flow and
composition for the first element are set to zero in Eq. (3.7b).
Head-loss in retentate stream is linearly distributed through
the permeation elements, so for the first element, the outlet
retentate pressure is set by Eq. (3.7c). For the permeate
stream, the final outlet pressure was specified in Eq. (3.6b),
yet there is no inlet permeate stream, so the head-loss is
selected and fixed as 0.1 bar, and linearly distributed
through the permeation elements; thus, the outlet permeate
pressure for the first element is given by Eq. (3.7d). Permeation elements are equally distributed in the total permeation area also defined in Eq. (3.6b), so for each element,
the permeation area is a fraction of the total specification, as
shown in Eq. (3.7e).
T
in
ð0Þ
V
¼ T
in
V ; P
in
ð0Þ
V
¼ P
in
V ;
V
in
ð0Þ ¼ V
in
; Y
in
ð0Þ ¼ Y
in
;
H
in
ð0Þ
V
¼
H
in
V
ð3:7aÞ
L
in
ð0Þ ¼ 0; X
in
ð0Þ ¼ 0
ð3:7bÞ
P
out
ð0Þ
V
¼ P
in
V À ðP
in
V À P
out
V Þ=n elements
ð3:7cÞ
P
out
ð0Þ
L
¼ P
out
L þ ð0:1=n elementsÞ Ã ðn elements À 1Þ
ð3:7dÞ
A MP ¼ A MP =n elements
ð3:7eÞ
[S3] Parameters for energy balance: DT
ð0Þ
F and external
temperature (T E ) both have default values specified in
MPd-UOE—Eqs. (3.8a) and (3.8b), respectively—however,
the user can set other values in UOE property window. Note
that differently from MPx-UOE algorithm, in MPd-UOE, the
DT F specification is set as the value for the first permeation
element only (DT
ð0Þ
F ), since it is calculated for the next elements in step [S5]. Internal and external overall heat transfer
coefficients (U I and U E ) are defined in Eqs. (3.8c) and (3.8d),
respectively. The internal area for heat transfer is equal to the
permeation area of each element, via Eq. (3.8e). Equation (3.8f) shows the relation between the external and
internal areas for heat transfer.
DT
ð0Þ
F ¼ 3
C ðdefaultÞ or defined by user
ð3:8aÞ
T E ¼ 25
C ðdefaultÞ or defined by user
ð3:8bÞ
U I ¼ 5 W=m
2
K
ð3:8cÞ
U E ¼ 2 W=m
2
K
ð3:8dÞ
A I ¼ A MP
ð3:8eÞ
A E ¼ A I =276
ð3:8fÞ
[S4] Initial values for first permeation element NRM:
Eqs. (3.9a) to (3.9j) set initial values of species transmembrane molar fluxes for the first permeation area, which
depend on the number of membrane elements chosen by the
user; the more distributed, the lower the permeation area for
each element, yet the higher the number of elements, which
is quadratic, so the lower is the permeation flux. Equations (3.9k) and (3.9m), respectively, define initial values of
retentate and permeate temperatures for the first permeation
area.
N
ð0Þ
CO 2
¼ 0:5 Ã V
in
ð0Þ
CO 2
=ðA MP Ã n elements
2
Þ
ð3:9aÞ
N
ð0Þ
CH 4
¼ 0:075 Ã V
in
ð0Þ
CH 4
=ðA MP Ã n elements
2
Þ
ð3:9bÞ
N
ð0Þ
C 2 H 6
¼ 0:01 Ã V
in
ð0Þ
C 2 H 6
=ðA MP Ã n elements
2
Þ
ð3:9cÞ
N
ð0Þ
C 3 H 8
¼ 0:005 Ã V
in
ð0Þ
C 3 H 8
=ðA MP Ã n elements
2
Þ
ð3:9dÞ
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167