N k ¼ P k ^ f
V
k À ^ f
L
k
À
Á
ð2:1Þ
@V k
@z
¼ ÀSaN k k ¼ 1. . .nc
ð
Þ
ð 2:2Þ
@L k
@z
¼ SaN k k ¼ 1. . .nc
ð
Þ
ð 2:3Þ
1 À C P V
q V
q V S V
2
(
)
@P V
@z
þ ÀC T V
q V
q V S V
2
(
)
@T V
@z
¼
Saq
S V
q V
q V S V
À q V gsinðhÞ À
W V pD þ Sa
ð
Þ
S V
ð2:4Þ
1 À C P L
q L
q L S L
2
(
)
@P L
@z
þ ÀC T L
q L
q L S L
2
(
)
@T L
@z
¼ À
Saq
S L
q L
q L S L
À q L g sinðhÞ À
W L Sa
S L
ð2:5Þ
1 À C PV
q V
q V S V
2
þ
T V C TV
q V
(
)
@P V
@z
þ
q
C
V
P
M V
À C TV
q V
q V S V
2
(
)
@T V
@z
¼ Àq V gsinðhÞ þ q V
Saq
q V
q V
q V S V
2
þ
X E pDq V
q V
T E À T V
ð
Þ À
XSaq V
q V
T V À T L
ð
Þ
ð2:6Þ
1 À C P L
q L
q L S L
2
þ
T L C T L
q L
(
)
@P L
@z
þ
q
C
L
P
M L
À C T L
q L
q L S L
2
(
)
@T L
@z
¼ Àq L g sinðhÞ À q L
Saq
q L
q L
q L S L
þ
XSaq L
q L
T V À T L
ð
Þ
þ
Saq L
q L
X
k
N k E
V
k À E
L
k
À
Á
ð2:7Þ
E
V
k ¼ H
V
k þ
M k
2
q V
q V S V
2
þ M k :g:hðzÞ
ð2:8Þ
E
L
k ¼ H
L
k þ
M k
2
q L
q L S L
2
þ M k :g:hðzÞ
ð2:9Þ
W V ¼
1
8
f V
q
2
V
q V S 2
V
; f V ¼ f V ðRe V ;
e V
D V
Þ; Re V ¼
4q V
} V l V
ð2:10Þ
W L ¼
1
8
f L
q
2
L
q L S 2
L
; f L ¼ f L ðRe L ;
e L
D L
Þ; Re L ¼
4q L
} L l L
ð2:11Þ
D V ¼ 4S V =} V ; D L ¼ 4S L =} L
ð2:12Þ
} V ¼ pD þ N HF pd o ; } L ¼ N HF pd i
ð2:13Þ
l V ðT V ; P V ; VÞ; l L ðT L ; P L ; LÞ
ð 2:14Þ
q V ¼
X nc
k¼1
M k :V k ; q L ¼
X nc
k¼1
M k :L k
ð2:15Þ
q TM ¼
X nc
k¼1
M k :N k
ð2:16Þ
C T V ¼
@q V
@T V
P V ;V
; C T L ¼
@q L
@T L
P L ;L
;
C P V ¼
@q V
@P V
T V ;V
; C P L ¼
@q L
@P L
T L ;L
ð2:17Þ
H
V
k ðT V ; P V ; VÞ; H
L
k ðT L ; P L ; LÞ ðk ¼ 1. . .ncÞ
ð2:18Þ
C
V
P ðT V ; P V ; VÞ; C
L
P ðT L ; P L ; LÞ
ð 2:19Þ
q V ðT V ; P V ; VÞ; q L ðT L ; P L ; LÞ
ð 2:20Þ
^ f
V
k ðT V ; P V ; VÞ; ^ f
L
k ðT L ; P L ; LÞ ðk ¼ 1. . .ncÞ
ð2:21Þ
2.1.2 Parameters of Parallel-Flow HFM Module
Three classes of geometrical and/or physical parameters
have to be specified for substitution in the HFM model in
Eqs. (2.1) to (2.21); namely: (i) HFM OD (d o ), HFM ID (d i ),
and retentate/permeate flow roughnesses (e V ; e L ); (ii) module
ID (D) and other module parameters h, Z M , N HF , T E , X, X E ;
and (iii) kth species equivalent HFM permeance
P k ðk ¼ 1. . .ncÞ.
Species equivalent HFM permeances are critical parameters defining both the selectivity of the MP module and its
gas processing capacity in terms of the trans-membrane flux
(mol s
−1 m
−2 ). Equivalent HFM permeances allow the HFM
parallel-flow model to approximately reproduce the
responses of a real MP battery processing CO 2 -rich NG
using, for example, cellulose-acetate SWM elements
(Table 1). Equivalent HFM premeances are supposed to be
constant parameters. In the present case, Table 2 shows
values of equivalent HFM permeances and heat transfer
coefficients for CO 2 removal from CO 2 -rich NG in the MP
applications of Sect. 2. They were estimated with data of
SWM MP units processing NG with %20 mol% CO 2 , so that
a parallel-flow horizontal (h = 0°) HFM module (10 m
0.2 m) with Z M = 10 m, N HF = 110,000 and approximate
150
J. L. de Medeiros et al.
V
k À ^ f
L
k
À
Á
ð2:1Þ
@V k
@z
¼ ÀSaN k k ¼ 1. . .nc
ð
Þ
ð 2:2Þ
@L k
@z
¼ SaN k k ¼ 1. . .nc
ð
Þ
ð 2:3Þ
1 À C P V
q V
q V S V
2
(
)
@P V
@z
þ ÀC T V
q V
q V S V
2
(
)
@T V
@z
¼
Saq
S V
q V
q V S V
À q V gsinðhÞ À
W V pD þ Sa
ð
Þ
S V
ð2:4Þ
1 À C P L
q L
q L S L
2
(
)
@P L
@z
þ ÀC T L
q L
q L S L
2
(
)
@T L
@z
¼ À
Saq
S L
q L
q L S L
À q L g sinðhÞ À
W L Sa
S L
ð2:5Þ
1 À C PV
q V
q V S V
2
þ
T V C TV
q V
(
)
@P V
@z
þ
q
C
V
P
M V
À C TV
q V
q V S V
2
(
)
@T V
@z
¼ Àq V gsinðhÞ þ q V
Saq
q V
q V
q V S V
2
þ
X E pDq V
q V
T E À T V
ð
Þ À
XSaq V
q V
T V À T L
ð
Þ
ð2:6Þ
1 À C P L
q L
q L S L
2
þ
T L C T L
q L
(
)
@P L
@z
þ
q
C
L
P
M L
À C T L
q L
q L S L
2
(
)
@T L
@z
¼ Àq L g sinðhÞ À q L
Saq
q L
q L
q L S L
þ
XSaq L
q L
T V À T L
ð
Þ
þ
Saq L
q L
X
k
N k E
V
k À E
L
k
À
Á
ð2:7Þ
E
V
k ¼ H
V
k þ
M k
2
q V
q V S V
2
þ M k :g:hðzÞ
ð2:8Þ
E
L
k ¼ H
L
k þ
M k
2
q L
q L S L
2
þ M k :g:hðzÞ
ð2:9Þ
W V ¼
1
8
f V
q
2
V
q V S 2
V
; f V ¼ f V ðRe V ;
e V
D V
Þ; Re V ¼
4q V
} V l V
ð2:10Þ
W L ¼
1
8
f L
q
2
L
q L S 2
L
; f L ¼ f L ðRe L ;
e L
D L
Þ; Re L ¼
4q L
} L l L
ð2:11Þ
D V ¼ 4S V =} V ; D L ¼ 4S L =} L
ð2:12Þ
} V ¼ pD þ N HF pd o ; } L ¼ N HF pd i
ð2:13Þ
l V ðT V ; P V ; VÞ; l L ðT L ; P L ; LÞ
ð 2:14Þ
q V ¼
X nc
k¼1
M k :V k ; q L ¼
X nc
k¼1
M k :L k
ð2:15Þ
q TM ¼
X nc
k¼1
M k :N k
ð2:16Þ
C T V ¼
@q V
@T V
P V ;V
; C T L ¼
@q L
@T L
P L ;L
;
C P V ¼
@q V
@P V
T V ;V
; C P L ¼
@q L
@P L
T L ;L
ð2:17Þ
H
V
k ðT V ; P V ; VÞ; H
L
k ðT L ; P L ; LÞ ðk ¼ 1. . .ncÞ
ð2:18Þ
C
V
P ðT V ; P V ; VÞ; C
L
P ðT L ; P L ; LÞ
ð 2:19Þ
q V ðT V ; P V ; VÞ; q L ðT L ; P L ; LÞ
ð 2:20Þ
^ f
V
k ðT V ; P V ; VÞ; ^ f
L
k ðT L ; P L ; LÞ ðk ¼ 1. . .ncÞ
ð2:21Þ
2.1.2 Parameters of Parallel-Flow HFM Module
Three classes of geometrical and/or physical parameters
have to be specified for substitution in the HFM model in
Eqs. (2.1) to (2.21); namely: (i) HFM OD (d o ), HFM ID (d i ),
and retentate/permeate flow roughnesses (e V ; e L ); (ii) module
ID (D) and other module parameters h, Z M , N HF , T E , X, X E ;
and (iii) kth species equivalent HFM permeance
P k ðk ¼ 1. . .ncÞ.
Species equivalent HFM permeances are critical parameters defining both the selectivity of the MP module and its
gas processing capacity in terms of the trans-membrane flux
(mol s
−1 m
−2 ). Equivalent HFM permeances allow the HFM
parallel-flow model to approximately reproduce the
responses of a real MP battery processing CO 2 -rich NG
using, for example, cellulose-acetate SWM elements
(Table 1). Equivalent HFM premeances are supposed to be
constant parameters. In the present case, Table 2 shows
values of equivalent HFM permeances and heat transfer
coefficients for CO 2 removal from CO 2 -rich NG in the MP
applications of Sect. 2. They were estimated with data of
SWM MP units processing NG with %20 mol% CO 2 , so that
a parallel-flow horizontal (h = 0°) HFM module (10 m
0.2 m) with Z M = 10 m, N HF = 110,000 and approximate
150
J. L. de Medeiros et al.