]>
tutorial_che_8
Model of a gas separation on a porous membrane with a metal support. The model employs
the Generalised Maxwell-Stefan (GMS) equations to predict fluxes and selectivities.
The membrane unit model represents a generic two-dimensonal model of a porous membrane
and consists of four models:
- Retentate compartment (isothermal axially dispersed plug flow)
- Micro-porous membrane
- Macro-porous support layer
- Permeate compartment (the same transport phenomena as in the retentate compartment)
The retentate compartment, the porous membrane, the support layer and the permeate
compartment are coupled via molar flux, temperature, pressure and gas composition at
the interfaces.
The model is described in the section 2.2 Membrane modelling of the following article:
- Nikolic D.D., Kikkinides E.S. (2015) Modelling and optimization of PSA/Membrane
separation processes. Adsorption 21(4):283-305.
`doi:10.1007/s10450-015-9670-z <http://doi.org/10.1007/s10450-015-9670-z>`_
and in the original Krishna article:
- Krishna R. (1993) A unified approach to the modeling of intraparticle
diffusion in adsorption processes. Gas Sep. Purif. 7(2):91-104.
`doi:10.1016/0950-4214(93)85006-H <http://doi.org/10.1016/0950-4214(93)85006-H>`_
This version is somewhat simplified for it only offers an extended Langmuir isotherm.
The Ideal Adsorption Solution theory (IAS) and the Real Adsorption Solution theory (RAS)
described in the articles are not implemented here.
The problem modelled is separation of hydrocarbons (CH4+C2H6) mixture on a zeolite
(silicalite-1) membrane with a metal support from the section 'Binary mixture permeation'
of the following article:
- van de Graaf J.M., Kapteijn F., Moulijn J.A. (1999) Modeling Permeation of Binary
Mixtures Through Zeolite Membranes. AIChE J. 45:497–511.
`doi:10.1002/aic.690450307 <http://doi.org/10.1002/aic.690450307>`_
The CH4 and C2H6 fluxes, and CH4/C2H6 selectivity plots for two cases: GMS and GMS(Dij=∞),
1:1 mixture, and T = 303 K:
.. image:: _static/tutorial_che_8-results.png
:width: 800px
${\mathit{tutorial}}_{{\mathit{che}}_{\mathit{8}}}$
Purity_feed
${\mathit{Purity}}_{\mathit{feed}}$
eAlgebraic
$${ { Purity_feed \left( { i } \right) } - { Feed.X_out \left( { i } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
Purity_permeate
${\mathit{Purity}}_{\mathit{permeate}}$
eAlgebraic
$${ { Purity_permeate \left( { i } \right) } - { Permeate.X_out \left( { i } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
Selectivity
$\mathit{Selectivity}$
eAlgebraic
$${ \left( { { Selectivity \left( { i, j, z } \right) } \cdot \left( { { Feed.X \left( { i, z } \right) } \cdot { Permeate.X \left( { j, z } \right) } } \right) } \right) - \left( { { Permeate.X \left( { i, z } \right) } \cdot { Feed.X \left( { j, z } \right) } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
j
$\mathit{j}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
Feed_Flux
${\mathit{Feed}}_{\mathit{Flux}}$
eAlgebraic
$${ { Feed.Flux \left( { i, z } \right) } - { Membrane.Flux \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
Support_Flux
${\mathit{Support}}_{\mathit{Flux}}$
eAlgebraic
$${ { Support.Flux \left( { i, z } \right) } - { Membrane.Flux \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
Permeate_Flux
${\mathit{Permeate}}_{\mathit{Flux}}$
eAlgebraic
$${ { Permeate.Flux \left( { i, z } \right) } + { Support.Flux \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
Membrane_Xinlet
${\mathit{Membrane}}_{\mathit{Xinlet}}$
eAlgebraic
$${ { Feed.X \left( { i, z } \right) } - { Membrane.X_inlet \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
Support_Xinlet
${\mathit{Support}}_{\mathit{Xinlet}}$
eAlgebraic
$${ { Support.X_inlet \left( { i, z } \right) } - { Membrane.X_outlet \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
Permeate_X
${\mathit{Permeate}}_{\mathit{X}}$
eAlgebraic
$${ { Permeate.X \left( { i, z } \right) } - { Support.X_outlet \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
Membrane_Pinlet
${\mathit{Membrane}}_{\mathit{Pinlet}}$
eAlgebraic
$${ { Feed.P \left( { z } \right) } - { Membrane.P_inlet \left( { z } \right) } } = 0; {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
Support_Pinlet
${\mathit{Support}}_{\mathit{Pinlet}}$
eAlgebraic
$${ { Support.P_inlet \left( { z } \right) } - { Membrane.P_outlet \left( { z } \right) } } = 0; {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
Support_Poutlet
${\mathit{Support}}_{\mathit{Poutlet}}$
eAlgebraic
$${ { Permeate.P \left( { z } \right) } - { Support.P_outlet \left( { z } \right) } } = 0; {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
Feed_T
${\mathit{Feed}}_{\mathit{T}}$
eAlgebraic
$${ { Feed.T } - { T_feed } } = 0$$
Membrane_T
${\mathit{Membrane}}_{\mathit{T}}$
eAlgebraic
$${ { Membrane.T } - { T_feed } } = 0$$
Support_S
${\mathit{Support}}_{\mathit{S}}$
eAlgebraic
$${ { Support.T } - { T_feed } } = 0$$
Permeate_T
${\mathit{Permeate}}_{\mathit{T}}$
eAlgebraic
$${ { Permeate.T } - { T_feed } } = 0$$
Feed
$\mathit{Feed}$
z
Axial domain
$\mathit{z}$
eStructuredGrid
$$
Nc
Number of components
$\mathit{Nc}$
eArray
$$
Rc
$\mathit{Rc}$
1
-1
-1
$\mathit{J}{\mathit{K}}^{-1}{\mathit{mol}}^{-1}$
U
$\mathit{U}$
velocity_t
$\mathit{z}$
z
Flux
$\mathit{Flux}$
molar_flux_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
C
$\mathit{C}$
molar_concentration_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
X
$\mathit{X}$
fraction_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
T
$\mathit{T}$
temperature_t
P
$\mathit{P}$
pressure_t
$\mathit{z}$
z
Length
$\mathit{Length}$
length_t
D_z
${\mathit{D}}_{\mathit{z}}$
diffusivity_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
a_V
${\mathit{a}}_{\mathit{V}}$
specific_area_t
A_cross
${\mathit{A}}_{\mathit{cross}}$
area_t
Area
$\mathit{Area}$
area_t
X_in
${\mathit{X}}_{\mathit{in}}$
fraction_t
$\mathit{Nc}$
Nc
X_out
${\mathit{X}}_{\mathit{out}}$
fraction_t
$\mathit{Nc}$
Nc
Q_in
${\mathit{Q}}_{\mathit{in}}$
volume_flowrate_t
Q_out
${\mathit{Q}}_{\mathit{out}}$
volume_flowrate_t
C_in
${\mathit{C}}_{\mathit{in}}$
molar_concentration_t
$\mathit{Nc}$
Nc
C_out
${\mathit{C}}_{\mathit{out}}$
molar_concentration_t
$\mathit{Nc}$
Nc
P_in
${\mathit{P}}_{\mathit{in}}$
pressure_t
P_out
${\mathit{P}}_{\mathit{out}}$
pressure_t
BCinlet_C
${\mathit{BCinlet}}_{\mathit{C}}$
eAlgebraic
$${ \left( { { U \left( { z_0 } \right) } \cdot \left( { { C \left( { i, z_0 } \right) } - { C_in \left( { i } \right) } } \right) } \right) - \left( { \left( { { D_z \left( { i, z_0 } \right) } \cdot \left( { { \partial { { \left( { C \left( { i, z_0 } \right) } \right) } } } \over { \partial {z} } } \right) } \right) \over { Length } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z_0 } = { z } _{0}}$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z_0
${\mathit{z}}_{\mathit{0}}$
$\mathit{z}$
z
eLowerBound
BCinlet_U
${\mathit{BCinlet}}_{\mathit{U}}$
eAlgebraic
$${ { U \left( { z_0 } \right) } - \left( { { Q_in } \over { A_cross } } \right) } = 0; {{ z_0 } = { z } _{0}}$$
z_0
${\mathit{z}}_{\mathit{0}}$
$\mathit{z}$
z
eLowerBound
BCinlet_Xin
${\mathit{BCinlet}}_{\mathit{Xin}}$
eAlgebraic
$${ \left( { \left( { { X_in \left( { i } \right) } \cdot { P_in } } \right) \over \left( { { Rc } \cdot { T } } \right) } \right) - { C_in \left( { i } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
BCinlet_Pin
${\mathit{BCinlet}}_{\mathit{Pin}}$
eAlgebraic
$${ { P \left( { z_0 } \right) } - { P_in } } = 0; {{ z_0 } = { z } _{0}}$$
z_0
${\mathit{z}}_{\mathit{0}}$
$\mathit{z}$
z
eLowerBound
BCoutlet_C
${\mathit{BCoutlet}}_{\mathit{C}}$
eAlgebraic
$${ { \partial { { \left( { C \left( { i, z } \right) } \right) } } } \over { \partial {z} } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z } = { z } _{n}}$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eUpperBound
BCoutlet_U
${\mathit{BCoutlet}}_{\mathit{U}}$
eAlgebraic
$${ { \partial { { \left( { U \left( { z_L } \right) } \right) } } } \over { \partial {z} } } = 0; {{ z_L } = { z } _{n}}$$
z_L
${\mathit{z}}_{\mathit{L}}$
$\mathit{z}$
z
eUpperBound
BCoutlet_Xout
${\mathit{BCoutlet}}_{\mathit{Xout}}$
eAlgebraic
$${ \left( { \left( { { X_out \left( { i } \right) } \cdot { P \left( { z_L } \right) } } \right) \over \left( { { Rc } \cdot { T } } \right) } \right) - { C \left( { i, z_L } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z_L } = { z } _{n}}$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z_L
${\mathit{z}}_{\mathit{L}}$
$\mathit{z}$
z
eUpperBound
BCoutlet_Qout
${\mathit{BCoutlet}}_{\mathit{Qout}}$
eAlgebraic
$${ { U \left( { z_L } \right) } - \left( { { Q_out } \over { A_cross } } \right) } = 0; {{ z_L } = { z } _{n}}$$
z_L
${\mathit{z}}_{\mathit{L}}$
$\mathit{z}$
z
eUpperBound
BCoutlet_Cout
${\mathit{BCoutlet}}_{\mathit{Cout}}$
eAlgebraic
$${ { C_out \left( { i } \right) } - { C \left( { i, z_L } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z_L } = { z } _{n}}$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z_L
${\mathit{z}}_{\mathit{L}}$
$\mathit{z}$
z
eUpperBound
BCoutlet_Pout
${\mathit{BCoutlet}}_{\mathit{Pout}}$
eAlgebraic
$${ { P_out } - { P \left( { z_L } \right) } } = 0; {{ z_L } = { z } _{n}}$$
z_L
${\mathit{z}}_{\mathit{L}}$
$\mathit{z}$
z
eUpperBound
P
$\mathit{P}$
eAlgebraic
$${ { P \left( { z } \right) } - { P_in } } = 0; {\forall { z } \in \left( { z } _{0}, { z } _{n} \right] }$$
z
$\mathit{z}$
$\mathit{z}$
z
eOpenClosed
aV
$\mathit{aV}$
eAlgebraic
$${ { a_V } - \left( { { Area } \over \left( { { Length } \cdot { A_cross } } \right) } \right) } = 0$$
C
$\mathit{C}$
eAlgebraic
$${ \left( { \left( { \left( { \left( { \left( - { D_z \left( { i, z } \right) } \right) } \right) \cdot \left( { { \partial { ^2 { \left( { C \left( { i, z } \right) } \right) } } } \over { \partial {z ^2} } } \right) } \right) \over \left( { { Length } ^ {2 {}}} \right) } \right) + \left( { \left( { { \partial { { \left( { { U \left( { z } \right) } \cdot { C \left( { i, z } \right) } } \right) } } } \over { \partial {z} } } \right) \over { Length } } \right) } \right) + \left( { { a_V } \cdot { Flux \left( { i, z } \right) } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left( { z } _{0}, { z } _{n} \right) }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eOpenOpen
X
$\mathit{X}$
eAlgebraic
$${ \left( { { X \left( { i, z } \right) } \cdot \left( \sum { { C.array \left( { *, z } \right) } } \right) } \right) - { C \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
PVT
$\mathit{PVT}$
eAlgebraic
$${ \left( { { P \left( { z } \right) } \over \left( { { Rc } \cdot { T } } \right) } \right) - \left( \sum { { C.array \left( { *, z } \right) } } \right) } = 0; {\forall { z } \in \left( { z } _{0}, { z } _{n} \right) }$$
z
$\mathit{z}$
$\mathit{z}$
z
eOpenOpen
Membrane
$\mathit{Membrane}$
z
Axial domain
$\mathit{z}$
eStructuredGrid
$$
r
Radial domain
$\mathit{r}$
eStructuredGrid
$$
Nc
Number of components
$\mathit{Nc}$
eArray
$$
Ro
$\mathit{Ro}$
1
-3
$\mathit{kg}{\mathit{m}}^{-3}$
Rc
$\mathit{Rc}$
1
-1
-1
$\mathit{J}{\mathit{K}}^{-1}{\mathit{mol}}^{-1}$
B
$\mathit{B}$
-1
${\mathit{Pa}}^{-1}$
$\mathit{Nc}$
Nc
Q_sat
${\mathit{Q}}_{\mathit{sat}}$
-1
1
$\mathit{mol}{\mathit{kg}}^{-1}$
$\mathit{Nc}$
Nc
Flux
$\mathit{Flux}$
molar_flux_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
X_inlet
${\mathit{X}}_{\mathit{inlet}}$
fraction_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
X_outlet
${\mathit{X}}_{\mathit{outlet}}$
fraction_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
T
$\mathit{T}$
temperature_t
P_inlet
${\mathit{P}}_{\mathit{inlet}}$
pressure_t
$\mathit{z}$
z
P_outlet
${\mathit{P}}_{\mathit{outlet}}$
pressure_t
$\mathit{z}$
z
G_ij
${\mathit{G}}_{\mathit{ij}}$
gij_t
$\mathit{Nc}$
Nc
$\mathit{Nc}$
Nc
$\mathit{z}$
z
$\mathit{r}$
r
D_ij
${\mathit{D}}_{\mathit{ij}}$
diffusivity_t
$\mathit{Nc}$
Nc
$\mathit{Nc}$
Nc
$\mathit{z}$
z
$\mathit{r}$
r
D_i
${\mathit{D}}_{\mathit{i}}$
diffusivity_t
$\mathit{Nc}$
Nc
θ
$\mathit{\theta}$
fraction_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
$\mathit{r}$
r
Gij_dTheta
${\mathit{Gij}}_{\mathit{dTheta}}$
Gij_dTheta
$\mathit{Nc}$
Nc
$\mathit{Nc}$
Nc
$\mathit{z}$
z
$\mathit{r}$
r
J_theta
${\mathit{J}}_{\mathit{theta}}$
J_theta
$\mathit{Nc}$
Nc
$\mathit{Nc}$
Nc
$\mathit{z}$
z
$\mathit{r}$
r
Length
$\mathit{Length}$
length_t
Area
$\mathit{Area}$
area_t
Thickness
$\mathit{Thickness}$
length_t
BCinlet_Theta
${\mathit{BCinlet}}_{\mathit{Theta}}$
eAlgebraic
$${ { \theta \left( { i, z, r_0 } \right) } - \left( { \left( { \left( { { B \left( { i } \right) } \cdot { X_inlet \left( { i, z } \right) } } \right) \cdot { P_inlet \left( { z } \right) } } \right) \over \left( { {1 {}} + \left( \sum { { \left( { \left( { B.array \left( { * } \right) } \right) \times \left( { X_inlet.array \left( { *, z } \right) } \right) } \right) \times \left( { P_inlet \left( { z } \right) } \right) } } \right) } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {{ r_0 } = { r } _{0}}$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r_0
${\mathit{r}}_{\mathit{0}}$
$\mathit{r}$
r
eLowerBound
BCoutlet_Theta
${\mathit{BCoutlet}}_{\mathit{Theta}}$
eAlgebraic
$${ { \theta \left( { i, z, r_R } \right) } - \left( { \left( { \left( { { B \left( { i } \right) } \cdot { X_outlet \left( { i, z } \right) } } \right) \cdot { P_outlet \left( { z } \right) } } \right) \over \left( { {1 {}} + \left( \sum { { \left( { \left( { B.array \left( { * } \right) } \right) \times \left( { X_outlet.array \left( { *, z } \right) } \right) } \right) \times \left( { P_outlet \left( { z } \right) } \right) } } \right) } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {{ r_R } = { r } _{n}}$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r_R
${\mathit{r}}_{\mathit{R}}$
$\mathit{r}$
r
eUpperBound
GammaFactor
$\mathit{GammaFactor}$
eAlgebraic
$${ \left( { \left( { { G_ij \left( { i, k, z, r } \right) } - \left( { {1 {}} - \left( { \left( { { i } - { k } } \right) \over \left( { \left( { { i } - { k } } \right) + {1e-15 {}}} \right) } \right) } \right) } \right) \cdot \left( { {1 {}} - \left( \sum { { \theta.array \left( { *, z, r } \right) } } \right) } \right) } \right) - { \theta \left( { i, z, r } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { k } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
k
$\mathit{k}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eClosedClosed
Dij
$\mathit{Dij}$
eAlgebraic
$${ { D_ij \left( { i, k, z, r } \right) } - \left( { \left( { { D_i \left( { i } \right) } ^ \left( { { \theta \left( { i, z, r } \right) } \over \left( { \left( { { \theta \left( { i, z, r } \right) } + { \theta \left( { k, z, r } \right) } } \right) + {1e-10 {}}} \right) } \right) } \right) \cdot \left( { { D_i \left( { k } \right) } ^ \left( { { \theta \left( { k, z, r } \right) } \over \left( { \left( { { \theta \left( { i, z, r } \right) } + { \theta \left( { k, z, r } \right) } } \right) + {1e-10 {}}} \right) } \right) } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { k } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
k
$\mathit{k}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eClosedClosed
OperatingMode
$\mathit{OperatingMode}$
eSTN
sFickLaw
$\mathit{sFickLaw}$
Flux
$\mathit{Flux}$
eAlgebraic
$${ { Flux \left( { i, z } \right) } + \left( { \left( { \left( { \left( { { Ro } \cdot { Q_sat \left( { i } \right) } } \right) \cdot { D_i \left( { i } \right) } } \right) \cdot \left( { { \partial { { \left( { \theta \left( { i, z, r } \right) } \right) } } } \over { \partial {r} } } \right) } \right) \over { Thickness } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right) }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eClosedOpen
J_theta
${\mathit{J}}_{\mathit{theta}}$
eAlgebraic
$${ J_theta \left( { i, j, z, r } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
j
$\mathit{j}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eClosedClosed
Gij_dTheta
${\mathit{Gij}}_{\mathit{dTheta}}$
eAlgebraic
$${ Gij_dTheta \left( { i, j, z, r } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
j
$\mathit{j}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eClosedClosed
$\mathit{OperatingMode}$
OperatingMode
sMaxwellStefan_Dijoo
${\mathit{sMaxwellStefan}}_{\mathit{Dijoo}}$
Flux
$\mathit{Flux}$
eAlgebraic
$${ { Flux \left( { i, z } \right) } + \left( { \left( { \left( { { Ro } \cdot { Q_sat \left( { i } \right) } } \right) \cdot { D_i \left( { i } \right) } } \right) \cdot \left( \sum { { Gij_dTheta.array \left( { i, *, z, r } \right) } } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right) }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eClosedOpen
J_theta
${\mathit{J}}_{\mathit{theta}}$
eAlgebraic
$${ J_theta \left( { i, j, z, r } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
j
$\mathit{j}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eClosedClosed
Gij_dTheta
${\mathit{Gij}}_{\mathit{dTheta}}$
eAlgebraic
$${ { Gij_dTheta \left( { i, j, z, r } \right) } - \left( { \left( { { G_ij \left( { i, j, z, r } \right) } \cdot \left( { { \partial { { \left( { \theta \left( { j, z, r } \right) } \right) } } } \over { \partial {r} } } \right) } \right) \over { Thickness } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
j
$\mathit{j}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eClosedClosed
$\mathit{OperatingMode}$
OperatingMode
sMaxwellStefan
$\mathit{sMaxwellStefan}$
Flux
$\mathit{Flux}$
eAlgebraic
$${ \left( { { Flux \left( { i, z } \right) } + \left( { \left( { { Q_sat \left( { i } \right) } \cdot { D_i \left( { i } \right) } } \right) \cdot \left( \sum { { J_theta.array \left( { i, *, z, r } \right) } } \right) } \right) } \right) + \left( { \left( { \left( { { Q_sat \left( { i } \right) } \cdot { D_i \left( { i } \right) } } \right) \cdot { Ro } } \right) \cdot \left( \sum { { Gij_dTheta.array \left( { i, *, z, r } \right) } } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right) }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eClosedOpen
J_theta
${\mathit{J}}_{\mathit{theta}}$
eAlgebraic
$${ { J_theta \left( { i, j, z, r } \right) } - \left( { \left( { \left( { { i } - { j } } \right) \over \left( { \left( { { i } - { j } } \right) + {1e-15 {}}} \right) } \right) \cdot \left( { \left( { \left( { { Flux \left( { i, z } \right) } \cdot { \theta \left( { j, z, r } \right) } } \right) \over \left( { { Q_sat \left( { i } \right) } \cdot { D_ij \left( { i, j, z, r } \right) } } \right) } \right) - \left( { \left( { { Flux \left( { j, z } \right) } \cdot { \theta \left( { i, z, r } \right) } } \right) \over \left( { { Q_sat \left( { j } \right) } \cdot { D_ij \left( { i, j, z, r } \right) } } \right) } \right) } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
j
$\mathit{j}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eClosedClosed
Gij_dTheta
${\mathit{Gij}}_{\mathit{dTheta}}$
eAlgebraic
$${ { Gij_dTheta \left( { i, j, z, r } \right) } - \left( { \left( { { G_ij \left( { i, j, z, r } \right) } \cdot \left( { { \partial { { \left( { \theta \left( { j, z, r } \right) } \right) } } } \over { \partial {r} } } \right) } \right) \over { Thickness } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
j
$\mathit{j}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eClosedClosed
$\mathit{OperatingMode}$
OperatingMode
NULL
Support
$\mathit{Support}$
z
Axial domain
$\mathit{z}$
eStructuredGrid
$$
r
Radial domain
$\mathit{r}$
eStructuredGrid
$$
Nc
Number of components
$\mathit{Nc}$
eArray
$$
e
$\mathit{e}$
$$
Rc
$\mathit{Rc}$
1
-1
-1
$\mathit{J}{\mathit{K}}^{-1}{\mathit{mol}}^{-1}$
Flux
$\mathit{Flux}$
molar_flux_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
X_inlet
${\mathit{X}}_{\mathit{inlet}}$
fraction_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
X_outlet
${\mathit{X}}_{\mathit{outlet}}$
fraction_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
T
$\mathit{T}$
temperature_t
P
$\mathit{P}$
pressure_t
$\mathit{z}$
z
$\mathit{r}$
r
P_inlet
${\mathit{P}}_{\mathit{inlet}}$
pressure_t
$\mathit{z}$
z
P_outlet
${\mathit{P}}_{\mathit{outlet}}$
pressure_t
$\mathit{z}$
z
X
$\mathit{X}$
fraction_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
$\mathit{r}$
r
D_i
${\mathit{D}}_{\mathit{i}}$
diffusivity_t
$\mathit{Nc}$
Nc
D_ij
${\mathit{D}}_{\mathit{ij}}$
diffusivity_t
$\mathit{Nc}$
Nc
$\mathit{Nc}$
Nc
Thickness
$\mathit{Thickness}$
length_t
BCinlet_X
${\mathit{BCinlet}}_{\mathit{X}}$
eAlgebraic
$${ { X \left( { i, z, r_0 } \right) } - { X_inlet \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {{ r_0 } = { r } _{0}}$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r_0
${\mathit{r}}_{\mathit{0}}$
$\mathit{r}$
r
eLowerBound
BCoutlet_X
${\mathit{BCoutlet}}_{\mathit{X}}$
eAlgebraic
$${ { X \left( { i, z, r_R } \right) } - { X_outlet \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {{ r_R } = { r } _{n}}$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r_R
${\mathit{r}}_{\mathit{R}}$
$\mathit{r}$
r
eUpperBound
BCinlet_P
${\mathit{BCinlet}}_{\mathit{P}}$
eAlgebraic
$${ { P_inlet \left( { z } \right) } - { P \left( { z, r_0 } \right) } } = 0; {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {{ r_0 } = { r } _{0}}$$
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r_0
${\mathit{r}}_{\mathit{0}}$
$\mathit{r}$
r
eLowerBound
BCoutlet_P
${\mathit{BCoutlet}}_{\mathit{P}}$
eAlgebraic
$${ { P_outlet \left( { z } \right) } - { P \left( { z, r_R } \right) } } = 0; {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {{ r_R } = { r } _{n}}$$
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r_R
${\mathit{r}}_{\mathit{R}}$
$\mathit{r}$
r
eUpperBound
P
$\mathit{P}$
eAlgebraic
$${ { \partial { { \left( { P \left( { z, r } \right) } \right) } } } \over { \partial {r} } } = 0; {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right) }$$
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eClosedOpen
OperatingMode
$\mathit{OperatingMode}$
eSTN
sNoResistance
$\mathit{sNoResistance}$
Flux
$\mathit{Flux}$
eAlgebraic
$${ { \partial { { \left( { X \left( { i, z, r } \right) } \right) } } } \over { \partial {r} } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left( { r } _{0}, { r } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eOpenClosed
$\mathit{OperatingMode}$
OperatingMode
sFickLaw
$\mathit{sFickLaw}$
Flux
$\mathit{Flux}$
eAlgebraic
$${ { Flux \left( { i, z } \right) } + \left( { \left( { \left( { \left( { { e } \cdot { D_ij \left( { i, i } \right) } } \right) \cdot \left( { { \partial { { \left( { X \left( { i, z, r } \right) } \right) } } } \over { \partial {r} } } \right) } \right) \cdot { P \left( { z, r } \right) } } \right) \over \left( { \left( { { Rc } \cdot { T } } \right) \cdot { Thickness } } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left( { r } _{0}, { r } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eOpenClosed
$\mathit{OperatingMode}$
OperatingMode
sMaxwellStefan
$\mathit{sMaxwellStefan}$
Flux
$\mathit{Flux}$
eAlgebraic
$${ \left( { \left( { \left( { \left( - { { \partial { { \left( { { X \left( { i, z, r } \right) } \cdot { P \left( { z, r } \right) } } \right) } } } \over { \partial {r} } } \right) } \right) \over \left( { \left( { { Rc } \cdot { T } } \right) \cdot { Thickness } } \right) } \right) - \left( \sum { { \left( { \left( { \left( { Flux \left( { i, z } \right) } \right) \times \left( { X.array \left( { *, z, r } \right) } \right) } \right) - \left( { \left( { Flux.array \left( { *, z } \right) } \right) \times \left( { X \left( { i, z, r } \right) } \right) } \right) } \right) \over \left( { \left( { e } \right) \times \left( { D_ij.array \left( { i, * } \right) } \right) } \right) } } \right) } \right) - \left( { \left( { \left( { {1 {}} - \left( \sum { { X.array \left( { *, z, r } \right) } } \right) } \right) \cdot { Flux \left( { i, z } \right) } } \right) \over \left( { { e } \cdot { D_i \left( { i } \right) } } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left( { r } _{0}, { r } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
r
$\mathit{r}$
$\mathit{r}$
r
eOpenClosed
$\mathit{OperatingMode}$
OperatingMode
NULL
Permeate
$\mathit{Permeate}$
z
Axial domain
$\mathit{z}$
eStructuredGrid
$$
Nc
Number of components
$\mathit{Nc}$
eArray
$$
Rc
$\mathit{Rc}$
1
-1
-1
$\mathit{J}{\mathit{K}}^{-1}{\mathit{mol}}^{-1}$
U
$\mathit{U}$
velocity_t
$\mathit{z}$
z
Flux
$\mathit{Flux}$
molar_flux_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
C
$\mathit{C}$
molar_concentration_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
X
$\mathit{X}$
fraction_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
T
$\mathit{T}$
temperature_t
P
$\mathit{P}$
pressure_t
$\mathit{z}$
z
Length
$\mathit{Length}$
length_t
D_z
${\mathit{D}}_{\mathit{z}}$
diffusivity_t
$\mathit{Nc}$
Nc
$\mathit{z}$
z
a_V
${\mathit{a}}_{\mathit{V}}$
specific_area_t
A_cross
${\mathit{A}}_{\mathit{cross}}$
area_t
Area
$\mathit{Area}$
area_t
X_in
${\mathit{X}}_{\mathit{in}}$
fraction_t
$\mathit{Nc}$
Nc
X_out
${\mathit{X}}_{\mathit{out}}$
fraction_t
$\mathit{Nc}$
Nc
Q_in
${\mathit{Q}}_{\mathit{in}}$
volume_flowrate_t
Q_out
${\mathit{Q}}_{\mathit{out}}$
volume_flowrate_t
C_in
${\mathit{C}}_{\mathit{in}}$
molar_concentration_t
$\mathit{Nc}$
Nc
C_out
${\mathit{C}}_{\mathit{out}}$
molar_concentration_t
$\mathit{Nc}$
Nc
P_in
${\mathit{P}}_{\mathit{in}}$
pressure_t
P_out
${\mathit{P}}_{\mathit{out}}$
pressure_t
BCinlet_C
${\mathit{BCinlet}}_{\mathit{C}}$
eAlgebraic
$${ \left( { { U \left( { z_0 } \right) } \cdot \left( { { C \left( { i, z_0 } \right) } - { C_in \left( { i } \right) } } \right) } \right) - \left( { \left( { { D_z \left( { i, z_0 } \right) } \cdot \left( { { \partial { { \left( { C \left( { i, z_0 } \right) } \right) } } } \over { \partial {z} } } \right) } \right) \over { Length } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z_0 } = { z } _{0}}$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z_0
${\mathit{z}}_{\mathit{0}}$
$\mathit{z}$
z
eLowerBound
BCinlet_U
${\mathit{BCinlet}}_{\mathit{U}}$
eAlgebraic
$${ { U \left( { z_0 } \right) } - \left( { { Q_in } \over { A_cross } } \right) } = 0; {{ z_0 } = { z } _{0}}$$
z_0
${\mathit{z}}_{\mathit{0}}$
$\mathit{z}$
z
eLowerBound
BCinlet_Xin
${\mathit{BCinlet}}_{\mathit{Xin}}$
eAlgebraic
$${ \left( { \left( { { X_in \left( { i } \right) } \cdot { P_in } } \right) \over \left( { { Rc } \cdot { T } } \right) } \right) - { C_in \left( { i } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
BCinlet_Pin
${\mathit{BCinlet}}_{\mathit{Pin}}$
eAlgebraic
$${ { P \left( { z_0 } \right) } - { P_in } } = 0; {{ z_0 } = { z } _{0}}$$
z_0
${\mathit{z}}_{\mathit{0}}$
$\mathit{z}$
z
eLowerBound
BCoutlet_C
${\mathit{BCoutlet}}_{\mathit{C}}$
eAlgebraic
$${ { \partial { { \left( { C \left( { i, z } \right) } \right) } } } \over { \partial {z} } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z } = { z } _{n}}$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eUpperBound
BCoutlet_U
${\mathit{BCoutlet}}_{\mathit{U}}$
eAlgebraic
$${ { \partial { { \left( { U \left( { z_L } \right) } \right) } } } \over { \partial {z} } } = 0; {{ z_L } = { z } _{n}}$$
z_L
${\mathit{z}}_{\mathit{L}}$
$\mathit{z}$
z
eUpperBound
BCoutlet_Xout
${\mathit{BCoutlet}}_{\mathit{Xout}}$
eAlgebraic
$${ \left( { \left( { { X_out \left( { i } \right) } \cdot { P \left( { z_L } \right) } } \right) \over \left( { { Rc } \cdot { T } } \right) } \right) - { C \left( { i, z_L } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z_L } = { z } _{n}}$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z_L
${\mathit{z}}_{\mathit{L}}$
$\mathit{z}$
z
eUpperBound
BCoutlet_Qout
${\mathit{BCoutlet}}_{\mathit{Qout}}$
eAlgebraic
$${ { U \left( { z_L } \right) } - \left( { { Q_out } \over { A_cross } } \right) } = 0; {{ z_L } = { z } _{n}}$$
z_L
${\mathit{z}}_{\mathit{L}}$
$\mathit{z}$
z
eUpperBound
BCoutlet_Cout
${\mathit{BCoutlet}}_{\mathit{Cout}}$
eAlgebraic
$${ { C_out \left( { i } \right) } - { C \left( { i, z_L } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z_L } = { z } _{n}}$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z_L
${\mathit{z}}_{\mathit{L}}$
$\mathit{z}$
z
eUpperBound
BCoutlet_Pout
${\mathit{BCoutlet}}_{\mathit{Pout}}$
eAlgebraic
$${ { P_out } - { P \left( { z_L } \right) } } = 0; {{ z_L } = { z } _{n}}$$
z_L
${\mathit{z}}_{\mathit{L}}$
$\mathit{z}$
z
eUpperBound
P
$\mathit{P}$
eAlgebraic
$${ { P \left( { z } \right) } - { P_in } } = 0; {\forall { z } \in \left( { z } _{0}, { z } _{n} \right] }$$
z
$\mathit{z}$
$\mathit{z}$
z
eOpenClosed
aV
$\mathit{aV}$
eAlgebraic
$${ { a_V } - \left( { { Area } \over \left( { { Length } \cdot { A_cross } } \right) } \right) } = 0$$
C
$\mathit{C}$
eAlgebraic
$${ \left( { \left( { \left( { \left( { \left( - { D_z \left( { i, z } \right) } \right) } \right) \cdot \left( { { \partial { ^2 { \left( { C \left( { i, z } \right) } \right) } } } \over { \partial {z ^2} } } \right) } \right) \over \left( { { Length } ^ {2 {}}} \right) } \right) + \left( { \left( { { \partial { { \left( { { U \left( { z } \right) } \cdot { C \left( { i, z } \right) } } \right) } } } \over { \partial {z} } } \right) \over { Length } } \right) } \right) + \left( { { a_V } \cdot { Flux \left( { i, z } \right) } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left( { z } _{0}, { z } _{n} \right) }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eOpenOpen
X
$\mathit{X}$
eAlgebraic
$${ \left( { { X \left( { i, z } \right) } \cdot \left( \sum { { C.array \left( { *, z } \right) } } \right) } \right) - { C \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
i
$\mathit{i}$
$\mathit{Nc}$
Nc
eClosedClosed
z
$\mathit{z}$
$\mathit{z}$
z
eClosedClosed
PVT
$\mathit{PVT}$
eAlgebraic
$${ \left( { { P \left( { z } \right) } \over \left( { { Rc } \cdot { T } } \right) } \right) - \left( \sum { { C.array \left( { *, z } \right) } } \right) } = 0; {\forall { z } \in \left( { z } _{0}, { z } _{n} \right) }$$
z
$\mathit{z}$
$\mathit{z}$
z
eOpenOpen