If Contact=Counter-Current Then
T
out
V ¼ T
in
V À 10
ð3:3kÞ
T
out
L ¼ T
in
V À 5
ð3:3mÞ
Elseif Contact=Parallel Then
T
out
ð0Þ
V
¼ T
in
ð0Þ
V
À 5
ð3:3nÞ
T
out
ð0Þ
L
¼ T
in
ð0Þ
V
À 15
ð3:3oÞ
End if
[S4] Lumped permeation and energy balance calculations: The Newton–Raphson method (NRM) is applied for
the target equations described in Eqs. (3.4a) to (3.4c), which
represent the transmembrane molar fluxes of species k, and
the energy balance equations for retentate and permeate
streams, respectively. The driving force in Eq. (3.4a) is the
log mean of partial pressure differences of species k (DP
LN
k ),
which varies accordingly to the contact type: if CC, it is
defined by Eq. (3.4d); if PC, by Eq. (3.4f); where P
in
V Y
in
k ,
P
out
V Y
out
k , P
out
L X
out
k
represent the partial pressures of species
k in feed, retentate, and permeate, respectively. The same
methodology applies to the log mean of temperature differences for internal heat transfer (DT
LN
I ): if contact is
counter-current, Eq. (3.4e) is applied; if it is parallel,
Eq. (3.4g) is selected. Equations (3.4h) and (3.4i) represent
the log mean of temperature differences between the retentate and the external area, and the log mean of partial molar
enthalpies of species k in retentate stream, respectively.
Equations (3.4j) to (3.4p) are applied to calculate the molar
flow rates and molar compositions of retentate and permeate
streams (V
out
; L
out
; Y
out
; X
out , respectively). The molar
enthalpies of retentate and permeate streams (H
out
V and H
out
L )
are obtained via flashðT
out
V ; P
out
V ; Y
out
Þ and flashðT
out
L ;
P
out
L ; X
out
Þ, respectively. In summary, the MPx-UOE model
comprises a system of 7nc + 6 nonlinear equations Eqs.
(3.4a) to (3.4p), to be numerically solved by NRM for
7nc + 6 variables N k ; DP
LN
k ; T
out
V ; T
out
L ; DT
LN
I ; DT
LN
E , H k
;
L
out
k ; L
out
; X
out
k ; V
out
k ; V
out
; Y
out
k .
Target Equations:
N k À P k DP
LN
k ¼ 0 ðk ¼ 1:::ncÞ
ð3:4aÞ
V
out
H
out
V À V
in
H
in
V
À U E A E DT
LN
E þ U I A I DT
LN
I
þ
X nc
k¼1
N k A I H k
¼ 0
ð3:4bÞ
L
out
H
out
L À U I A I DT
LN
I
À
X nc
k¼1
N k A I H k
¼ 0
ð3:4cÞ
Auxiliary Equations:
If Contact=Counter-Current Then
DP
LN
k ¼
P
in
V Y
in
k À P
out
L X
out
k
À
Á À P
out
V Y
out
k
À
Á
ln
P in
V Y in
k
ÀP out
L X out
k
P out
V Y out
k
0
@
1
A ðk ¼ 1:::ncÞ
ð3:4dÞ
DT
LN
I
¼
DT F À T
in
V þ T
out
L
ln
DT F
T in
V ÀT out
L
2
4
3
5
ð3:4eÞ
Elseif Contact=Parallel Then
DP
LN
k ¼
P
in
V Y
in
k
À
Á À P
out
V Y
out
k À P
out
L X
out
k
À
Á
ln
P in
V Y in
k
P out
V
Y out
k
ÀP out
L
X out
k
0
@
1
A ðk ¼ 1:::ncÞ
ð3:4fÞ
DT
LN
I
¼
DT F À T
out
V þ T
out
L
ln
DT F
T out
V ÀT out
L
2
4
3
5
ð3:4gÞ
End if
DT
LN
E ¼
T
in
V À T
out
V
ln
T E ÀT out
V
T E ÀT in
V
2
4
3
5
ð3:4hÞ
H k
¼
H
out
V k
À H
in
V k
ln
H
out
V k
H
in
V k
ðk ¼ 1. . .ncÞ
ð3:4iÞ
L
out
k ¼ N k A MP ðk ¼ 1. . .ncÞ
ð 3:4jÞ
L
out
¼
X nc
k
L
out
k
ð3:4kÞ
X
out
k ¼
L
out
k
L out ðk ¼ 1. . .ncÞ
ð 3:4mÞ
V
out
k ¼ V
in
Y
in
k À N k A MP ðk ¼ 1. . .ncÞ
ð 3:4nÞ
V
out
¼
X nc
k
V
out
k
ð3:4oÞ
Y
out
k ¼
V
out
k
V out ðk ¼ 1. . .ncÞ
ð 3:4pÞ
[S5] Returning product data to simulation: Data of
retentate and permeate streams are pasted onto the product
streams of MPx-UOE in HYSYS PFD via Eqs. (3.5a) and
(3.5b).
166
J. L. de Medeiros et al.