`_
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 `_
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 `_
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
]]>
eAlgebraic
$${ { Purity_feed \left( { i } \right) } - { Feed.X_out \left( { i } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }$$
Nc
eClosedClosed
eAlgebraic
$${ { Purity_permeate \left( { i } \right) } - { Permeate.X_out \left( { i } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }$$
Nc
eClosedClosed
eAlgebraic
$${ \left( { { Selectivity \left( { i, j, z } \right) } \cdot \left( { { Feed.X \left( { i, z } \right) } \cdot { Permeate.X \left( { j, z } \right) } } \right) } \right) - \left( { { Permeate.X \left( { i, z } \right) } \cdot { Feed.X \left( { j, z } \right) } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
Nc
eClosedClosed
Nc
eClosedClosed
z
eClosedClosed
eAlgebraic
$${ { Feed.Flux \left( { i, z } \right) } - { Membrane.Flux \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
Nc
eClosedClosed
z
eClosedClosed
eAlgebraic
$${ { Support.Flux \left( { i, z } \right) } - { Membrane.Flux \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
Nc
eClosedClosed
z
eClosedClosed
eAlgebraic
$${ { Permeate.Flux \left( { i, z } \right) } + { Support.Flux \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
Nc
eClosedClosed
z
eClosedClosed
eAlgebraic
$${ { Feed.X \left( { i, z } \right) } - { Membrane.X_inlet \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
Nc
eClosedClosed
z
eClosedClosed
eAlgebraic
$${ { Support.X_inlet \left( { i, z } \right) } - { Membrane.X_outlet \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
Nc
eClosedClosed
z
eClosedClosed
eAlgebraic
$${ { Permeate.X \left( { i, z } \right) } - { Support.X_outlet \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
Nc
eClosedClosed
z
eClosedClosed
eAlgebraic
$${ { Feed.P \left( { z } \right) } - { Membrane.P_inlet \left( { z } \right) } } = 0; {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
z
eClosedClosed
eAlgebraic
$${ { Support.P_inlet \left( { z } \right) } - { Membrane.P_outlet \left( { z } \right) } } = 0; {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
z
eClosedClosed
eAlgebraic
$${ { Permeate.P \left( { z } \right) } - { Support.P_outlet \left( { z } \right) } } = 0; {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
z
eClosedClosed
eAlgebraic
$${ { Feed.T } - { T_feed } } = 0$$
eAlgebraic
$${ { Membrane.T } - { T_feed } } = 0$$
eAlgebraic
$${ { Support.T } - { T_feed } } = 0$$
eAlgebraic
$${ { Permeate.T } - { T_feed } } = 0$$
eStructuredGrid
eArray
1
-1
-1
velocity_t
z
molar_flux_t
Nc
z
molar_concentration_t
Nc
z
fraction_t
Nc
z
temperature_t
pressure_t
z
length_t
diffusivity_t
Nc
z
specific_area_t
area_t
area_t
fraction_t
Nc
fraction_t
Nc
volume_flowrate_t
volume_flowrate_t
molar_concentration_t
Nc
molar_concentration_t
Nc
pressure_t
pressure_t
eAlgebraic
$${ \left( { { U \left( { z_0 } \right) } \cdot \left( { { C \left( { i, z_0 } \right) } - { C_in \left( { i } \right) } } \right) } \right) - \left( { \left( { { D_z \left( { i, z_0 } \right) } \cdot \left( { { \partial { { \left( { C \left( { i, z_0 } \right) } \right) } } } \over { \partial {z} } } \right) } \right) \over { Length } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z_0 } = { z } _{0}}$$
Nc
eClosedClosed
z
eLowerBound
eAlgebraic
$${ { U \left( { z_0 } \right) } - \left( { { Q_in } \over { A_cross } } \right) } = 0; {{ z_0 } = { z } _{0}}$$
z
eLowerBound
eAlgebraic
$${ \left( { \left( { { X_in \left( { i } \right) } \cdot { P_in } } \right) \over \left( { { Rc } \cdot { T } } \right) } \right) - { C_in \left( { i } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }$$
Nc
eClosedClosed
eAlgebraic
$${ { P \left( { z_0 } \right) } - { P_in } } = 0; {{ z_0 } = { z } _{0}}$$
z
eLowerBound
eAlgebraic
$${ { \partial { { \left( { C \left( { i, z } \right) } \right) } } } \over { \partial {z} } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z } = { z } _{n}}$$
Nc
eClosedClosed
z
eUpperBound
eAlgebraic
$${ { \partial { { \left( { U \left( { z_L } \right) } \right) } } } \over { \partial {z} } } = 0; {{ z_L } = { z } _{n}}$$
z
eUpperBound
eAlgebraic
$${ \left( { \left( { { X_out \left( { i } \right) } \cdot { P \left( { z_L } \right) } } \right) \over \left( { { Rc } \cdot { T } } \right) } \right) - { C \left( { i, z_L } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z_L } = { z } _{n}}$$
Nc
eClosedClosed
z
eUpperBound
eAlgebraic
$${ { U \left( { z_L } \right) } - \left( { { Q_out } \over { A_cross } } \right) } = 0; {{ z_L } = { z } _{n}}$$
z
eUpperBound
eAlgebraic
$${ { C_out \left( { i } \right) } - { C \left( { i, z_L } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z_L } = { z } _{n}}$$
Nc
eClosedClosed
z
eUpperBound
eAlgebraic
$${ { P_out } - { P \left( { z_L } \right) } } = 0; {{ z_L } = { z } _{n}}$$
z
eUpperBound
eAlgebraic
$${ { P \left( { z } \right) } - { P_in } } = 0; {\forall { z } \in \left( { z } _{0}, { z } _{n} \right] }$$
z
eOpenClosed
eAlgebraic
$${ { a_V } - \left( { { Area } \over \left( { { Length } \cdot { A_cross } } \right) } \right) } = 0$$
eAlgebraic
$${ \left( { \left( { \left( { \left( { \left( - { D_z \left( { i, z } \right) } \right) } \right) \cdot \left( { { \partial { ^2 { \left( { C \left( { i, z } \right) } \right) } } } \over { \partial {z ^2} } } \right) } \right) \over \left( { { Length } ^ {2 {}}} \right) } \right) + \left( { \left( { { \partial { { \left( { { U \left( { z } \right) } \cdot { C \left( { i, z } \right) } } \right) } } } \over { \partial {z} } } \right) \over { Length } } \right) } \right) + \left( { { a_V } \cdot { Flux \left( { i, z } \right) } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left( { z } _{0}, { z } _{n} \right) }$$
Nc
eClosedClosed
z
eOpenOpen
eAlgebraic
$${ \left( { { X \left( { i, z } \right) } \cdot \left( \sum { { C.array \left( { *, z } \right) } } \right) } \right) - { C \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
Nc
eClosedClosed
z
eClosedClosed
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
eOpenOpen
eStructuredGrid
eStructuredGrid
eArray
1
-3
1
-1
-1
-1
Nc
-1
1
Nc
molar_flux_t
Nc
z
fraction_t
Nc
z
fraction_t
Nc
z
temperature_t
pressure_t
z
pressure_t
z
gij_t
Nc
Nc
z
r
diffusivity_t
Nc
Nc
z
r
diffusivity_t
Nc
fraction_t
Nc
z
r
Gij_dTheta
Nc
Nc
z
r
J_theta
Nc
Nc
z
r
length_t
area_t
length_t
eAlgebraic
$${ { \theta \left( { i, z, r_0 } \right) } - \left( { \left( { \left( { { B \left( { i } \right) } \cdot { X_inlet \left( { i, z } \right) } } \right) \cdot { P_inlet \left( { z } \right) } } \right) \over \left( { {1 {}} + \left( \sum { { \left( { \left( { B.array \left( { * } \right) } \right) \times \left( { X_inlet.array \left( { *, z } \right) } \right) } \right) \times \left( { P_inlet \left( { z } \right) } \right) } } \right) } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {{ r_0 } = { r } _{0}}$$
Nc
eClosedClosed
z
eClosedClosed
r
eLowerBound
eAlgebraic
$${ { \theta \left( { i, z, r_R } \right) } - \left( { \left( { \left( { { B \left( { i } \right) } \cdot { X_outlet \left( { i, z } \right) } } \right) \cdot { P_outlet \left( { z } \right) } } \right) \over \left( { {1 {}} + \left( \sum { { \left( { \left( { B.array \left( { * } \right) } \right) \times \left( { X_outlet.array \left( { *, z } \right) } \right) } \right) \times \left( { P_outlet \left( { z } \right) } \right) } } \right) } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {{ r_R } = { r } _{n}}$$
Nc
eClosedClosed
z
eClosedClosed
r
eUpperBound
eAlgebraic
$${ \left( { \left( { { G_ij \left( { i, k, z, r } \right) } - \left( { {1 {}} - \left( { \left( { { i } - { k } } \right) \over \left( { \left( { { i } - { k } } \right) + {1e-15 {}}} \right) } \right) } \right) } \right) \cdot \left( { {1 {}} - \left( \sum { { \theta.array \left( { *, z, r } \right) } } \right) } \right) } \right) - { \theta \left( { i, z, r } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { k } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
Nc
eClosedClosed
Nc
eClosedClosed
z
eClosedClosed
r
eClosedClosed
eAlgebraic
$${ { D_ij \left( { i, k, z, r } \right) } - \left( { \left( { { D_i \left( { i } \right) } ^ \left( { { \theta \left( { i, z, r } \right) } \over \left( { \left( { { \theta \left( { i, z, r } \right) } + { \theta \left( { k, z, r } \right) } } \right) + {1e-10 {}}} \right) } \right) } \right) \cdot \left( { { D_i \left( { k } \right) } ^ \left( { { \theta \left( { k, z, r } \right) } \over \left( { \left( { { \theta \left( { i, z, r } \right) } + { \theta \left( { k, z, r } \right) } } \right) + {1e-10 {}}} \right) } \right) } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { k } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
Nc
eClosedClosed
Nc
eClosedClosed
z
eClosedClosed
r
eClosedClosed
eSTN
eAlgebraic
$${ { Flux \left( { i, z } \right) } + \left( { \left( { \left( { \left( { { Ro } \cdot { Q_sat \left( { i } \right) } } \right) \cdot { D_i \left( { i } \right) } } \right) \cdot \left( { { \partial { { \left( { \theta \left( { i, z, r } \right) } \right) } } } \over { \partial {r} } } \right) } \right) \over { Thickness } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right) }$$
Nc
eClosedClosed
z
eClosedClosed
r
eClosedOpen
eAlgebraic
$${ J_theta \left( { i, j, z, r } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
Nc
eClosedClosed
Nc
eClosedClosed
z
eClosedClosed
r
eClosedClosed
eAlgebraic
$${ Gij_dTheta \left( { i, j, z, r } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
Nc
eClosedClosed
Nc
eClosedClosed
z
eClosedClosed
r
eClosedClosed
OperatingMode
eAlgebraic
$${ { Flux \left( { i, z } \right) } + \left( { \left( { \left( { { Ro } \cdot { Q_sat \left( { i } \right) } } \right) \cdot { D_i \left( { i } \right) } } \right) \cdot \left( \sum { { Gij_dTheta.array \left( { i, *, z, r } \right) } } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right) }$$
Nc
eClosedClosed
z
eClosedClosed
r
eClosedOpen
eAlgebraic
$${ J_theta \left( { i, j, z, r } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
Nc
eClosedClosed
Nc
eClosedClosed
z
eClosedClosed
r
eClosedClosed
eAlgebraic
$${ { Gij_dTheta \left( { i, j, z, r } \right) } - \left( { \left( { { G_ij \left( { i, j, z, r } \right) } \cdot \left( { { \partial { { \left( { \theta \left( { j, z, r } \right) } \right) } } } \over { \partial {r} } } \right) } \right) \over { Thickness } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
Nc
eClosedClosed
Nc
eClosedClosed
z
eClosedClosed
r
eClosedClosed
OperatingMode
eAlgebraic
$${ \left( { { Flux \left( { i, z } \right) } + \left( { \left( { { Q_sat \left( { i } \right) } \cdot { D_i \left( { i } \right) } } \right) \cdot \left( \sum { { J_theta.array \left( { i, *, z, r } \right) } } \right) } \right) } \right) + \left( { \left( { \left( { { Q_sat \left( { i } \right) } \cdot { D_i \left( { i } \right) } } \right) \cdot { Ro } } \right) \cdot \left( \sum { { Gij_dTheta.array \left( { i, *, z, r } \right) } } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right) }$$
Nc
eClosedClosed
z
eClosedClosed
r
eClosedOpen
eAlgebraic
$${ { J_theta \left( { i, j, z, r } \right) } - \left( { \left( { \left( { { i } - { j } } \right) \over \left( { \left( { { i } - { j } } \right) + {1e-15 {}}} \right) } \right) \cdot \left( { \left( { \left( { { Flux \left( { i, z } \right) } \cdot { \theta \left( { j, z, r } \right) } } \right) \over \left( { { Q_sat \left( { i } \right) } \cdot { D_ij \left( { i, j, z, r } \right) } } \right) } \right) - \left( { \left( { { Flux \left( { j, z } \right) } \cdot { \theta \left( { i, z, r } \right) } } \right) \over \left( { { Q_sat \left( { j } \right) } \cdot { D_ij \left( { i, j, z, r } \right) } } \right) } \right) } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
Nc
eClosedClosed
Nc
eClosedClosed
z
eClosedClosed
r
eClosedClosed
eAlgebraic
$${ { Gij_dTheta \left( { i, j, z, r } \right) } - \left( { \left( { { G_ij \left( { i, j, z, r } \right) } \cdot \left( { { \partial { { \left( { \theta \left( { j, z, r } \right) } \right) } } } \over { \partial {r} } } \right) } \right) \over { Thickness } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { j } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left[ { r } _{0}, { r } _{n} \right] }$$
Nc
eClosedClosed
Nc
eClosedClosed
z
eClosedClosed
r
eClosedClosed
OperatingMode
NULL
eStructuredGrid
eStructuredGrid
eArray
1
-1
-1
molar_flux_t
Nc
z
fraction_t
Nc
z
fraction_t
Nc
z
temperature_t
pressure_t
z
r
pressure_t
z
pressure_t
z
fraction_t
Nc
z
r
diffusivity_t
Nc
diffusivity_t
Nc
Nc
length_t
eAlgebraic
$${ { X \left( { i, z, r_0 } \right) } - { X_inlet \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {{ r_0 } = { r } _{0}}$$
Nc
eClosedClosed
z
eClosedClosed
r
eLowerBound
eAlgebraic
$${ { X \left( { i, z, r_R } \right) } - { X_outlet \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {{ r_R } = { r } _{n}}$$
Nc
eClosedClosed
z
eClosedClosed
r
eUpperBound
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
eClosedClosed
r
eLowerBound
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
eClosedClosed
r
eUpperBound
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
eClosedClosed
r
eClosedOpen
eSTN
eAlgebraic
$${ { \partial { { \left( { X \left( { i, z, r } \right) } \right) } } } \over { \partial {r} } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left( { r } _{0}, { r } _{n} \right] }$$
Nc
eClosedClosed
z
eClosedClosed
r
eOpenClosed
OperatingMode
eAlgebraic
$${ { Flux \left( { i, z } \right) } + \left( { \left( { \left( { \left( { { e } \cdot { D_ij \left( { i, i } \right) } } \right) \cdot \left( { { \partial { { \left( { X \left( { i, z, r } \right) } \right) } } } \over { \partial {r} } } \right) } \right) \cdot { P \left( { z, r } \right) } } \right) \over \left( { \left( { { Rc } \cdot { T } } \right) \cdot { Thickness } } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left( { r } _{0}, { r } _{n} \right] }$$
Nc
eClosedClosed
z
eClosedClosed
r
eOpenClosed
OperatingMode
eAlgebraic
$${ \left( { \left( { \left( { \left( - { { \partial { { \left( { { X \left( { i, z, r } \right) } \cdot { P \left( { z, r } \right) } } \right) } } } \over { \partial {r} } } \right) } \right) \over \left( { \left( { { Rc } \cdot { T } } \right) \cdot { Thickness } } \right) } \right) - \left( \sum { { \left( { \left( { \left( { Flux \left( { i, z } \right) } \right) \times \left( { X.array \left( { *, z, r } \right) } \right) } \right) - \left( { \left( { Flux.array \left( { *, z } \right) } \right) \times \left( { X \left( { i, z, r } \right) } \right) } \right) } \right) \over \left( { \left( { e } \right) \times \left( { D_ij.array \left( { i, * } \right) } \right) } \right) } } \right) } \right) - \left( { \left( { \left( { {1 {}} - \left( \sum { { X.array \left( { *, z, r } \right) } } \right) } \right) \cdot { Flux \left( { i, z } \right) } } \right) \over \left( { { e } \cdot { D_i \left( { i } \right) } } \right) } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }, {\forall { r } \in \left( { r } _{0}, { r } _{n} \right] }$$
Nc
eClosedClosed
z
eClosedClosed
r
eOpenClosed
OperatingMode
NULL
eStructuredGrid
eArray
1
-1
-1
velocity_t
z
molar_flux_t
Nc
z
molar_concentration_t
Nc
z
fraction_t
Nc
z
temperature_t
pressure_t
z
length_t
diffusivity_t
Nc
z
specific_area_t
area_t
area_t
fraction_t
Nc
fraction_t
Nc
volume_flowrate_t
volume_flowrate_t
molar_concentration_t
Nc
molar_concentration_t
Nc
pressure_t
pressure_t
eAlgebraic
$${ \left( { { U \left( { z_0 } \right) } \cdot \left( { { C \left( { i, z_0 } \right) } - { C_in \left( { i } \right) } } \right) } \right) - \left( { \left( { { D_z \left( { i, z_0 } \right) } \cdot \left( { { \partial { { \left( { C \left( { i, z_0 } \right) } \right) } } } \over { \partial {z} } } \right) } \right) \over { Length } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z_0 } = { z } _{0}}$$
Nc
eClosedClosed
z
eLowerBound
eAlgebraic
$${ { U \left( { z_0 } \right) } - \left( { { Q_in } \over { A_cross } } \right) } = 0; {{ z_0 } = { z } _{0}}$$
z
eLowerBound
eAlgebraic
$${ \left( { \left( { { X_in \left( { i } \right) } \cdot { P_in } } \right) \over \left( { { Rc } \cdot { T } } \right) } \right) - { C_in \left( { i } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }$$
Nc
eClosedClosed
eAlgebraic
$${ { P \left( { z_0 } \right) } - { P_in } } = 0; {{ z_0 } = { z } _{0}}$$
z
eLowerBound
eAlgebraic
$${ { \partial { { \left( { C \left( { i, z } \right) } \right) } } } \over { \partial {z} } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z } = { z } _{n}}$$
Nc
eClosedClosed
z
eUpperBound
eAlgebraic
$${ { \partial { { \left( { U \left( { z_L } \right) } \right) } } } \over { \partial {z} } } = 0; {{ z_L } = { z } _{n}}$$
z
eUpperBound
eAlgebraic
$${ \left( { \left( { { X_out \left( { i } \right) } \cdot { P \left( { z_L } \right) } } \right) \over \left( { { Rc } \cdot { T } } \right) } \right) - { C \left( { i, z_L } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z_L } = { z } _{n}}$$
Nc
eClosedClosed
z
eUpperBound
eAlgebraic
$${ { U \left( { z_L } \right) } - \left( { { Q_out } \over { A_cross } } \right) } = 0; {{ z_L } = { z } _{n}}$$
z
eUpperBound
eAlgebraic
$${ { C_out \left( { i } \right) } - { C \left( { i, z_L } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {{ z_L } = { z } _{n}}$$
Nc
eClosedClosed
z
eUpperBound
eAlgebraic
$${ { P_out } - { P \left( { z_L } \right) } } = 0; {{ z_L } = { z } _{n}}$$
z
eUpperBound
eAlgebraic
$${ { P \left( { z } \right) } - { P_in } } = 0; {\forall { z } \in \left( { z } _{0}, { z } _{n} \right] }$$
z
eOpenClosed
eAlgebraic
$${ { a_V } - \left( { { Area } \over \left( { { Length } \cdot { A_cross } } \right) } \right) } = 0$$
eAlgebraic
$${ \left( { \left( { \left( { \left( { \left( - { D_z \left( { i, z } \right) } \right) } \right) \cdot \left( { { \partial { ^2 { \left( { C \left( { i, z } \right) } \right) } } } \over { \partial {z ^2} } } \right) } \right) \over \left( { { Length } ^ {2 {}}} \right) } \right) + \left( { \left( { { \partial { { \left( { { U \left( { z } \right) } \cdot { C \left( { i, z } \right) } } \right) } } } \over { \partial {z} } } \right) \over { Length } } \right) } \right) + \left( { { a_V } \cdot { Flux \left( { i, z } \right) } } \right) } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left( { z } _{0}, { z } _{n} \right) }$$
Nc
eClosedClosed
z
eOpenOpen
eAlgebraic
$${ \left( { { X \left( { i, z } \right) } \cdot \left( \sum { { C.array \left( { *, z } \right) } } \right) } \right) - { C \left( { i, z } \right) } } = 0; {\forall { i } \in \left[ { Nc } _{0}, { Nc } _{n} \right] }, {\forall { z } \in \left[ { z } _{0}, { z } _{n} \right] }$$
Nc
eClosedClosed
z
eClosedClosed
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
eOpenOpen