**Appendix**

#### **A. Iterative gradient method of the identification of co-diffusion coefficients**

The methodology for solving the direct boundary problem (9)–(15), which describes the diffusion process in a heterogeneous nanoporous bed, is developed in [9, 12, 15]. According to [12] the procedure for determining the diffusion coefficients (16) requires a special technique for calculating the gradients ∇*J n D*intra*<sup>s</sup> k* ð Þ*t* and ∇*J n D*inter*<sup>s</sup> k* ð Þ*t* of the residual functional (17). This leads to the problem of optimizing the extended Lagrange function [12, 15]:

$$\Phi(D\_{\text{inter}\_{uk}}, D\_{\text{intra}\_{uk}}) = f\_s + I\_{s\_{\text{max}}} + I\_{s\_{\text{min}}},\tag{A.1}$$

*ϕsk*

*Qsk*

*ϕsk*

where L ¼

*∂ ∂t* þ *∂*

*wsk* ð Þ¼ *t*, *X*, *Z*

X*sk* ð Þ¼ *t*,*X*, *Z*

<sup>L</sup><sup>∗</sup> <sup>¼</sup>

**27**

*∂*

*DOI: http://dx.doi.org/10.5772/intechopen.88138*

Heaviside operational method in [15].

problem (A.5)–(A.10) in the operator:

*∂ ∂t* � *∂*

*<sup>∂</sup><sup>Z</sup> <sup>D</sup>*intersk

Δ*Qsk*

*∂ ∂Z*

order terms, has the form

<sup>Δ</sup>*Js <sup>D</sup>*intrask , *<sup>D</sup>*intersk � � <sup>¼</sup>

*<sup>∂</sup><sup>Z</sup> <sup>D</sup>*intersk

*∂ ∂Z* � � *<sup>e</sup>*int*erk*

<sup>0</sup> *<sup>∂</sup>*

ð Þ *t*,*X*, *Z*

Δ*D*intersk

ð *T* *L* ð*k*

*Lk*�<sup>1</sup>

ð 1

0 <sup>L</sup>�<sup>1</sup>

*L* ð*k*

*Lk*�<sup>1</sup>

0

þ ð *T*

0

Δ*Csk* ð Þ *t*, *Z*

" #

ð Þ¼ *<sup>t</sup>*, *<sup>Z</sup>* <sup>¼</sup> *Lk* 0, *<sup>ϕ</sup>sk*�<sup>1</sup>

ð Þ *<sup>t</sup>*, *<sup>Z</sup> <sup>t</sup>*¼*ttotal* <sup>j</sup> <sup>¼</sup> 0; *<sup>ψ</sup>sk* ð Þ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup> <sup>t</sup>*¼*ttotal* <sup>j</sup> <sup>¼</sup> 0 conditions at *<sup>t</sup>* <sup>¼</sup> *<sup>t</sup>*

*дϕs*<sup>1</sup>

We have obtained the solutions *ϕsk* and *ψsk* of problem (A.5)–(A.10) using

subtracting the first equations from the transformed ones and neglecting secondorder terms of smallness, we obtain the basic equations of the problem (9)–(15) in terms of increments Δ*Csk* ð Þ *t*, *Z* and Δ*Qsk* ð Þ *t*, *X*, *Z* , *s* ¼ 1, 2 in the operator form

Similarly, we write the system of the basic equations of conjugate boundary

*∂t*

*D*intrask *R*2

, <sup>Ψ</sup>*sk* ð Þ¼ *<sup>t</sup>*,*X*, *<sup>Z</sup> <sup>ϕ</sup>sk*

� *e*int*erk*

The calculated increment of the residual functional (17), neglecting second-

2 *X ∂ ∂X* � �*Qsk* ð Þ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup>*

*∂*2 *<sup>∂</sup>X*<sup>2</sup> <sup>þ</sup>

*D*intrask *R*2

<sup>L</sup><sup>∗</sup> <sup>Ψ</sup>*sk* ð Þ¼ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup> Esk* ð Þ*<sup>t</sup> <sup>δ</sup>*ð Þ *<sup>Z</sup>* � *hk* , <sup>Ψ</sup>*sk* <sup>∈</sup>ð Þ 0, 1 <sup>∪</sup> <sup>Ω</sup>*kt*, *<sup>k</sup>* <sup>¼</sup> 1, *<sup>N</sup>* <sup>þ</sup> 1 (A.12)

� *<sup>D</sup>*intrask *R*2

> *∂*2 *<sup>∂</sup>X*<sup>2</sup> <sup>þ</sup>

*∂ ∂X* j*<sup>X</sup>*¼<sup>1</sup>

L*wsk* ð Þ¼ *t*, *X*, *Z* X*sk* , *wsk* ∈ ð Þ 0, 1 ∪ Ω*kt*, *k* ¼ 1, *N* þ 1 (A.11)

*D*intrask *R*

> 2 *X ∂ ∂X*

> > ð Þ *t*, *Z*

" #

ð Þ *t*,*X*, *Z*

*∂ <sup>∂</sup><sup>X</sup> Qsk*

� �

*ψsk*

Δ*D*intrask *R*2

X*sk*,1 ð Þ� *t*, *Z Esk* ð Þ*t δ*ð Þ *Z* � *hk dZdt*

X*sk*<sup>2</sup> ð Þ� *t*, *X*, *Z Esk* ð Þ*t δ*ð Þ *Z* � *hk XdXdZdt*

*∂ ∂X* j*<sup>X</sup>*¼<sup>1</sup>

> 2 *X ∂ ∂X*

.

ð Þ *<sup>t</sup>*,*X*, *<sup>Z</sup> <sup>X</sup>*¼<sup>1</sup>

(A.13)

(A.14)

� �

*∂*2 *<sup>∂</sup>X*<sup>2</sup> <sup>þ</sup>

Substituting in the direct problem (9)–(15) *D*intersk , *D*intrask , *Сsk* ð Þ *t*, *Z* , and

*<sup>∂</sup><sup>X</sup> <sup>ψ</sup>sk* ð Þ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup>* <sup>j</sup>*X*¼<sup>0</sup> <sup>¼</sup> 0; *<sup>ψ</sup>sk*

*Competitive Adsorption and Diffusion of Gases in a Microporous Solid*

ð Þ¼ *t*, *L*<sup>1</sup> 0,

ð Þ *t*, *X*, *Z* by the corresponding values with increments *D*intersk þ Δ*D*intersk , *D*intrask þ Δ*D*intrask , *Csk* ð Þþ *t*, *Z* Δ*Csk* ð Þ *t*, *z* , and *Qsk* ð Þþ *t*, *X*, *Z* Δ*Qsk*

> *∂ ∂Z* � � *<sup>e</sup>*int*erk*

<sup>0</sup> *<sup>∂</sup>*

*∂t* þ

*∂ <sup>∂</sup><sup>Z</sup> Csk*

Δ*D*intrask *R*2

where <sup>L</sup><sup>∗</sup> is the conjugate Lagrange operator of operator <sup>L</sup>.

<sup>L</sup>�<sup>1</sup>

� �

*ϕs*1

*total* � �; (A.7)

ð Þ *t*, *Z* (A.8)

ð Þ *t*, *X*, *Z* ,

ð Þ *<sup>t</sup>*,*X*, *<sup>Z</sup>* <sup>j</sup>*X*¼<sup>1</sup> <sup>¼</sup> *<sup>φ</sup>sk*

ð Þ¼ *t*, *Z* ¼ *Lk*�<sup>1</sup> 0; *s* ¼ 1, 2, *k* ¼ *N*, 2, (A.9)

*<sup>д</sup><sup>Z</sup>* ð Þ¼ *<sup>t</sup>*, *<sup>Z</sup>* <sup>¼</sup> <sup>0</sup> <sup>0</sup> (A.10)

where *Ismacro* ,*Ismicro* are the components given by Eqs. (A.2) and (A.3), corresponding to the macro- and microporosity, respectively

$$I\_{t\_{\rm narrow}} = \int\_0^T \int\_{t\_{\rm 1}}^{L\_k} \phi\_{s\_k}(t, Z) \left( \frac{\partial \mathbf{C}\_{s\_k}}{\partial t} - \frac{D\_{\rm inter\_k}}{l^2} \frac{\partial^2 \mathbf{C}\_{s\_k}}{\partial Z^2} + e\_{\rm inter\_k} K\_{s\_k} \frac{D\_{\rm intra\_k}}{R^2} \left( \frac{\partial \mathbf{Q}(t, X, Z)\_{s\_k}}{\partial X} \right)\_{X=1} \right) d\mathbf{Z} dt \tag{A.2}$$

$$I\_{t\_{\min}} = \bigcap\_{0}^{T} \left[ \int\_{0}^{1} \int\_{t\_{k-1}}^{t\_k} \mathcal{w}\_{s\_k}(t, X, Z) \left( \frac{\partial \mathcal{Q}\_{s\_k}(t, X, Z)}{\partial t} - \frac{D\_{\text{intra}\_k}}{R^2} \left( \frac{\partial^2 \mathcal{Q}\_{s\_k}}{\partial X^2} + \frac{2}{X} \frac{\partial \mathcal{Q}\_{s\_k}}{\partial X} \right) \right) \mathbf{X} d\mathbf{X} d\mathbf{Z} dt \right] \tag{A.3}$$

*Js* is the residual functional (17), *<sup>ϕ</sup>sk* , *<sup>ψ</sup>sk* , *<sup>s</sup>*,¼1,2—unknown factors of Lagrange, to be determined from the stationary condition of the functional Φ *D*intersk , *D*intrask � � [9, 15]:

$$
\Delta\Phi\left(D\_{\text{inter}\_{ik}}, D\_{\text{intra}\_{ik}}\right) \equiv \Delta f\_s + \Delta I\_{s\_{\text{max}}} + \Delta I\_{s\_{\text{min}}} = \mathbf{0} \tag{A.4}
$$

The calculation of the components in Eq. (A.4) is carried out by assuming that the values *D*intersk , *D*intrask are incremented by Δ*D*intersk , Δ*D*intrask . As a result, concentration *Csk* ð Þ *t*, *Z* changes by increment Δ*Csk*ð Þ *t*, *Z* and concentration *Qsk* ð Þ *t*, *X*, *Z* by increment Δ*Qsk* ð Þ *t*,*X*, *Z* , *s* ¼ 1, 2.

Conjugate problem. The calculation of the increments Δ*Js*, Δ*Jsmacro* , and Δ*Jsmicro* in Eq. (A.4) (using integration by parts and the initial and boundary conditions of the direct problem (9)–(15)) leads to solving the additional conjugate problem to determine the Lagrange factors *ϕsk* and *ψsk* of the functional (A.1) [15]:

$$\frac{\partial \phi\_{s\_k}(t, Z)}{\partial t} + \frac{D\_{\text{inter}\_k}}{l^2} \frac{\partial^2 \phi\_{s\_k}}{\partial Z^2} + e\_{\text{inter}\_k} K\_{s\_k} \frac{D\_{\text{intra}\_k}}{R^2} \frac{\partial \psi\_{s\_k}(t, X, Z)}{\partial X} \bigg|\_{X=1} = E\_{s\_k}^n(t) \delta(Z - h\_k) \tag{A.5}$$

where *En sk* ðÞ¼ *t Csk D*int*rask <sup>n</sup>*, *D*int*ersk <sup>n</sup>*; *t*, *hk* � � <sup>þ</sup> *Qsk <sup>D</sup>*int*rask <sup>n</sup>*, *<sup>D</sup>*int*ersk <sup>n</sup>*; *<sup>t</sup>*, *hk* � � � *Msk*ð Þ*<sup>t</sup>* , *δ*ð Þ *Z* � *hk* (function of Dirac) [15].

$$\frac{\partial \boldsymbol{\mu}\_{s\_k}(t, \boldsymbol{X}, \boldsymbol{Z})}{\partial t} + \frac{D\_{\text{intr}\_{sk}}}{R^2} \left( \frac{\partial^2 \boldsymbol{\mu}\_{s\_k}}{\partial \mathbf{X}^2} + \frac{2}{\boldsymbol{X}} \frac{\partial \boldsymbol{\mu}\_{s\_k}}{\partial \mathbf{X}} \right) = \boldsymbol{E}\_{s\_k}^{\boldsymbol{u}}(t) \delta(\boldsymbol{Z} - h\_k) \tag{A.6}$$

*Competitive Adsorption and Diffusion of Gases in a Microporous Solid DOI: http://dx.doi.org/10.5772/intechopen.88138*

$$(\phi\_{\boldsymbol{\eta}\_{k}}(\mathbf{t}, \mathbf{Z})\_{|\boldsymbol{\eta} = \boldsymbol{\eta}^{\rm total}} = \mathbf{0}; \boldsymbol{\mu}\_{\boldsymbol{\eta}\_{k}}(\mathbf{t}, \mathbf{X}, \mathbf{Z})\_{|\boldsymbol{\eta} = \boldsymbol{\eta}^{\rm total}} = \mathbf{0} \text{ (conditions } \mathbf{at} \,\boldsymbol{t} = \boldsymbol{t}^{\rm total}\text{)}; \tag{A.7}$$

$$\frac{\partial}{\partial \mathbf{X}} \boldsymbol{\mu}\_{\boldsymbol{s}\_{k}}(\mathbf{t}, \mathbf{X}, \mathbf{Z})\_{|\mathbf{X}=\mathbf{0}} = \mathbf{0}; \boldsymbol{\mu}\_{\boldsymbol{s}\_{k}}(\mathbf{t}, \mathbf{X}, \mathbf{Z})\_{|\mathbf{X}=\mathbf{1}} = \boldsymbol{\varrho}\_{\boldsymbol{s}\_{k}}(\mathbf{t}, \mathbf{Z}) \tag{A.8}$$

$$\phi\_{s\_k}(t, Z = L\_k) = 0, \phi\_{s\_{k-1}}(t, Z = L\_{k-1}) = 0; s = \overline{1, 2}, k = \overline{N, 2}, \tag{A.9}$$

$$\phi\_{s\_1}(t, L\_1) = 0, \,\frac{\partial \phi\_{s\_1}}{\partial Z}(t, Z = \mathbf{0}) = \mathbf{0} \tag{A.10}$$

We have obtained the solutions *ϕsk* and *ψsk* of problem (A.5)–(A.10) using Heaviside operational method in [15].

Substituting in the direct problem (9)–(15) *D*intersk , *D*intrask , *Сsk* ð Þ *t*, *Z* , and *Qsk* ð Þ *t*, *X*, *Z* by the corresponding values with increments *D*intersk þ Δ*D*intersk , *D*intrask þ Δ*D*intrask , *Csk* ð Þþ *t*, *Z* Δ*Csk* ð Þ *t*, *z* , and *Qsk* ð Þþ *t*, *X*, *Z* Δ*Qsk* ð Þ *t*, *X*, *Z* , subtracting the first equations from the transformed ones and neglecting secondorder terms of smallness, we obtain the basic equations of the problem (9)–(15) in terms of increments Δ*Csk* ð Þ *t*, *Z* and Δ*Qsk* ð Þ *t*, *X*, *Z* , *s* ¼ 1, 2 in the operator form

$$\mathcal{L}w\_{\mathfrak{s}\_k}(\mathbf{t}, \mathbf{X}, \mathbf{Z}) = \mathbf{X}\_{\mathfrak{s}\_k}, w\_{\mathfrak{s}\_k} \in (\mathbf{0}, \mathbf{1}) \cup \mathfrak{Q}\_{kl}, k = \overline{\mathbf{1}, N+1} \tag{A.11}$$

Similarly, we write the system of the basic equations of conjugate boundary problem (A.5)–(A.10) in the operator:

$$\mathcal{L}^\* \Psi\_{\eta\_k}(t, X, Z) = E\_{\eta\_k}(t) \delta(Z - h\_k), \Psi\_{\eta\_k} \in (0, 1) \cup \Omega\_{\text{kl}}, k = \overline{1, N + 1} \tag{A.12}$$

where L ¼ *∂ ∂t* � *∂ <sup>∂</sup><sup>Z</sup> <sup>D</sup>*intersk *∂ ∂Z* � � *<sup>e</sup>*int*erk D*intrask *R ∂ ∂X* j*<sup>X</sup>*¼<sup>1</sup> <sup>0</sup> *<sup>∂</sup> ∂t* � *<sup>D</sup>*intrask *R*2 *∂*2 *<sup>∂</sup>X*<sup>2</sup> <sup>þ</sup> 2 *X ∂ ∂X* � � 2 6 6 6 4 3 7 7 7 5, <sup>L</sup><sup>∗</sup> <sup>¼</sup> *∂ ∂t* þ *∂ <sup>∂</sup><sup>Z</sup> <sup>D</sup>*intersk *∂ ∂Z* � � *<sup>e</sup>*int*erk D*intrask *R*2 *∂ ∂X* j*<sup>X</sup>*¼<sup>1</sup> <sup>0</sup> *<sup>∂</sup> ∂t* þ *D*intrask *R*2 *∂*2 *<sup>∂</sup>X*<sup>2</sup> <sup>þ</sup> 2 *X ∂ ∂X* � � 2 6 6 6 4 3 7 7 7 5, *wsk* ð Þ¼ *t*, *X*, *Z* Δ*Csk* ð Þ *t*, *Z* Δ*Qsk* ð Þ *t*,*X*, *Z* " #, <sup>Ψ</sup>*sk* ð Þ¼ *<sup>t</sup>*,*X*, *<sup>Z</sup> <sup>ϕ</sup>sk* ð Þ *t*, *Z ψsk* ð Þ *t*,*X*, *Z* " #. X*sk* ð Þ¼ *t*,*X*, *Z ∂ ∂Z* Δ*D*intersk *∂ <sup>∂</sup><sup>Z</sup> Csk* � � � *e*int*erk* Δ*D*intrask *R*2 *∂ <sup>∂</sup><sup>X</sup> Qsk* ð Þ *<sup>t</sup>*,*X*, *<sup>Z</sup> <sup>X</sup>*¼<sup>1</sup> Δ*D*intrask *R*2 *∂*2 *<sup>∂</sup>X*<sup>2</sup> <sup>þ</sup> 2 *X ∂ ∂X* � �*Qsk* ð Þ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup>* 2 6 6 6 4 3 7 7 7 5 (A.13)

where <sup>L</sup><sup>∗</sup> is the conjugate Lagrange operator of operator <sup>L</sup>.

The calculated increment of the residual functional (17), neglecting secondorder terms, has the form

$$\begin{split} \Delta I\_s(\mathbf{D}\_{\text{intra}\_{\text{alt}}}, \mathbf{D}\_{\text{inter}\_{\text{ik}}}) &= \int\_0^T \int\_{L\_{k-1}}^{L\_k} \mathcal{L}^{-1} \mathbf{X}\_{s\_{k,1}}(t, \mathbf{Z}) \cdot \mathbf{E}\_{s\_k}(t) \delta(\mathbf{Z} - h\_k) d\mathbf{Z} dt \\ &\quad + \int\_0^T \int\_{L\_{k1}}^{L\_k} \int\_{\mathbf{X}} \mathcal{L}^{-1} \mathbf{X}\_{s\_{k2}}(t, \mathbf{X}, \mathbf{Z}) \cdot \mathbf{E}\_{s\_k}(t) \delta(\mathbf{Z} - h\_k) \mathbf{X} d\mathbf{X} d\mathbf{Z} dt \end{split} \tag{A.14}$$

**Appendix**

*Zeolites - New Challenges*

∇*J n D*inter*<sup>s</sup> k*

*Ismacro* ¼

[9, 15]:

*Qsk*

**26**

*<sup>∂</sup>ϕsk* ð Þ *<sup>t</sup>*, *<sup>Z</sup> ∂t*

where *En sk*

ð *T* *L* ð*k*

*Lk*�<sup>1</sup>

ð *T* ð 1 *L* ð*k*

*Lk*�<sup>1</sup>

ð Þ *t*, *X*, *Z* by increment Δ*Qsk*

þ

*<sup>∂</sup>ψsk* ð Þ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup> ∂t*

*D*inter*sk l* 2

*δ*ð Þ *Z* � *hk* (function of Dirac) [15].

þ

*∂*2 *ϕsk*

*<sup>∂</sup>Z*<sup>2</sup> <sup>þ</sup> *<sup>e</sup>*inter*kKsk*

*∂*2 *ψsk <sup>∂</sup>X*<sup>2</sup> <sup>þ</sup>

ðÞ¼ *t Csk D*int*rask <sup>n</sup>*, *D*int*ersk <sup>n</sup>*; *t*, *hk*

*D*int*rask R*2

*ψsk*

ð Þ *t*,*X*, *Z*

ΔΦ *D*intersk , *D*intrask

0

0

*ϕsk*

0

*Ismicro* ¼

**A. Iterative gradient method of the identification of co-diffusion coefficients**

The methodology for solving the direct boundary problem (9)–(15), which describes the diffusion process in a heterogeneous nanoporous bed, is developed in [9, 12, 15]. According to [12] the procedure for determining the diffusion coeffi-

ð Þ*t* of the residual functional (17). This leads to the problem of optimizing

*<sup>∂</sup>Z*<sup>2</sup> <sup>þ</sup> *<sup>e</sup>*inter*kKsk*

*<sup>∂</sup><sup>t</sup>* � *<sup>D</sup>*int*rask*

*Js* is the residual functional (17), *<sup>ϕ</sup>sk* , *<sup>ψ</sup>sk* , *<sup>s</sup>*,¼1,2—unknown factors of Lagrange, to

The calculation of the components in Eq. (A.4) is carried out by assuming that

Conjugate problem. The calculation of the increments Δ*Js*, Δ*Jsmacro* , and Δ*Jsmicro* in Eq. (A.4) (using integration by parts and the initial and boundary conditions of the direct problem (9)–(15)) leads to solving the additional conjugate problem to determine the Lagrange factors *ϕsk* and *ψsk* of the functional (A.1) [15]:

> *D*intra*<sup>s</sup> k R*2

> > 2 *X <sup>∂</sup>ψsk ∂X*

!

*<sup>∂</sup>ψsk*

� � <sup>þ</sup> *Qsk <sup>D</sup>*int*rask <sup>n</sup>*, *<sup>D</sup>*int*ersk <sup>n</sup>*; *<sup>t</sup>*, *hk*

ð Þ *t*,*X*, *Z ∂X*

<sup>¼</sup> *En sk* � � � � *X*¼1

<sup>¼</sup> *En sk*

� � � *Msk*ð Þ*<sup>t</sup>* ,

ð Þ*t δ*ð Þ *Z* � *hk* (A.6)

ð Þ*t δ*ð Þ *Z* � *hk*

(A.5)

be determined from the stationary condition of the functional Φ *D*intersk , *D*intrask

the values *D*intersk , *D*intrask are incremented by Δ*D*intersk , Δ*D*intrask . As a result, concentration *Csk* ð Þ *t*, *Z* changes by increment Δ*Csk*ð Þ *t*, *Z* and concentration

ð Þ *t*,*X*, *Z* , *s* ¼ 1, 2.

*R*2

! !

� � � <sup>Δ</sup>*Js* <sup>þ</sup> <sup>Δ</sup>*Ismacro* <sup>þ</sup> <sup>Δ</sup>*Ismicro* <sup>¼</sup> 0 (A.4)

� � <sup>¼</sup> *Js* <sup>þ</sup> *Ismacro* <sup>þ</sup> *Ismicro* , (A.1)

*D*intra*<sup>s</sup> k R*2

> *∂*2 *Qsk <sup>∂</sup>X*<sup>2</sup> <sup>þ</sup>

� �

*n D*intra*<sup>s</sup> k* ð Þ*t* and

*<sup>∂</sup>Q t*ð Þ , *<sup>X</sup>*, *<sup>Z</sup> sk ∂X* � �

> 2 *X*

*<sup>∂</sup>Qsk ∂X*

*X*¼1

*XdXdZdt*

� �

(A.2)

(A.3)

*dZdt*

cients (16) requires a special technique for calculating the gradients ∇*J*

where *Ismacro* ,*Ismicro* are the components given by Eqs. (A.2) and (A.3),

*∂*2 *Csk*

*<sup>∂</sup>Qsk* ð Þ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup>*

Φ *D*intersk , *D*intrask

corresponding to the macro- and microporosity, respectively

*<sup>∂</sup><sup>t</sup>* � *<sup>D</sup>*inter*sk l* 2

the extended Lagrange function [12, 15]:

ð Þ *<sup>t</sup>*, *<sup>Z</sup> <sup>∂</sup>Csk*

wher<sup>е</sup> *wsk* ¼ L�<sup>1</sup> <sup>X</sup>*sk* and <sup>L</sup>�<sup>1</sup> is the inverse operator of operator <sup>L</sup>. Defining the scalar product

$$\left(\mathcal{L}w\_{\boldsymbol{\varsigma}\_{k}}(\boldsymbol{t},\boldsymbol{X},\boldsymbol{Z}),\Psi\_{\boldsymbol{\varsigma}\_{k}}(\boldsymbol{t},\boldsymbol{X},\boldsymbol{Z})\right) = \begin{bmatrix} \int\limits\_{\Omega\_{\Gamma}} \mathcal{L}\Delta \mathcal{L}\_{\boldsymbol{\varsigma}\_{k}}(\boldsymbol{t},\boldsymbol{Z})\phi\_{\boldsymbol{\varsigma}\_{k}}(\boldsymbol{t},\boldsymbol{Z})d\boldsymbol{Z}d\boldsymbol{t} \\\\ \iint\limits\_{(0,R)\cup\Omega\_{\Gamma}} \mathcal{L}\Delta Q\_{\boldsymbol{\varsigma}\_{k}}(\boldsymbol{t},\boldsymbol{X},\boldsymbol{Z})\psi\_{\boldsymbol{\varsigma}\_{k}}(\boldsymbol{t},\boldsymbol{X},\boldsymbol{Z})\mathbf{X}d\boldsymbol{X}d\boldsymbol{Z}d\boldsymbol{t} \end{bmatrix} \tag{A.15}$$

and taking into account (A.19) Lagrange's identity [12, 15]

$$\left( \mathcal{L}w\_{\flat\_k}(\mathbf{t}, \mathbf{X}, \mathbf{Z}), \Psi\_{\flat\_k}(\mathbf{t}, \mathbf{X}, \mathbf{Z}) \right) = \left( w\_{\flat\_k}(\mathbf{t}, \mathbf{X}, \mathbf{Z}), \mathcal{L}^\* \Psi\_{\flat\_k}(\mathbf{t}, \mathbf{X}, \mathbf{Z}) \right) \tag{A.16}$$

and the equality <sup>L</sup>�<sup>1</sup> <sup>∗</sup> *Esk* ð Þ*<sup>t</sup> <sup>δ</sup>*ð Þ *<sup>Z</sup>* � *hk* � � <sup>¼</sup> <sup>Ψ</sup>*sk* , we obtain the increment of the residual functional expressed by the solution of conjugate problem (A.6)–(A.10) and the vector of the right-hand parts of Eq. (A.13):

$$
\Delta J\_s \left( D\_{\text{inter}\_{ik}}, D\_{\text{intra}\_{ik}} \right) = \left( \Psi\_{s\_k}(\mathbf{t}, \mathbf{X}, \mathbf{Z}), \mathbf{X}\_{\ast\_k}(\mathbf{t}, \mathbf{X}, \mathbf{Z}) \right) \tag{A.17}
$$

(A.5)–(A.10). They provide high performance of computing process, avoiding a large number of inner loop iterations by using exact analytical methods [2, 15].

**B. The linearization schema of the nonlinear co-adsorption model: system of**

*∂*2 *Сs <sup>∂</sup>Z*<sup>2</sup> � *<sup>e</sup>*inter*K*~*<sup>s</sup>*

*R*2

1 þ *K*1*C*1ð Þþ *t*, *Z K*2*C*2ð Þ *t*, *Z*

The linearization schema of nonlinear co-adsorption (1)–(8) is shown in order to demonstrate the simplicity of implementation for the case of two diffusing components (*m* ¼ 2) and isothermal adsorption. The simplified model (1)–(8) for the case

> *∂*2 *Qs <sup>∂</sup>X*<sup>2</sup> <sup>þ</sup>

*Cs*ð Þ¼ *t* ¼ 0, *Z* 0; *Qs*ð Þ¼ *t* ¼ 0,*X*, *Z* 0;*X* ∈ð Þ 0, 1 , *s* ¼ 1, 2 (A.23)

*D*intra*<sup>s</sup> R*2

2 *X ∂Qs ∂X*

*∂Qs ∂X* � �

� �, *<sup>s</sup>* <sup>¼</sup> 1, 2 (A.22)

*<sup>∂</sup><sup>X</sup> Qs*ð Þ¼ *<sup>t</sup>*,*<sup>X</sup>* <sup>¼</sup> 0, *<sup>Z</sup>* <sup>0</sup> (A.24)

<sup>1</sup> <sup>þ</sup> *<sup>K</sup>*1*C*1ð Þþ *<sup>t</sup>*, *<sup>Z</sup> <sup>K</sup>*2*C*2ð Þ *<sup>t</sup>*, *<sup>Z</sup>* ð Þ Langmuir equilibrium ,

*<sup>д</sup><sup>Z</sup>* ð Þ¼ *<sup>t</sup>*, *<sup>Z</sup>* <sup>¼</sup> <sup>0</sup> 0, *<sup>t</sup>*<sup>∈</sup> 0, *<sup>t</sup>*

*Cs*ð Þ *t*, *Z*

*C*1*C*1þ

*∂φs ∂C*<sup>2</sup> � � � � ð Þ 0,0 C2

constants and partial pressure of the gas phase for 1-th and 2-th component and *θ*1, *θ*2, are the intra-crystallite spaces occupied by the corresponding adsorbed mol-

C1 þ

!

*X*¼1

(A.21)

(A.25)

1

CA <sup>þ</sup> …

(A.27)

*total* � � (A.26)

, where *p*1, *p*<sup>2</sup> are the co-adsorption equilibrium

<sup>1</sup>þ*K*1*C*1ð Þþ *<sup>t</sup>*, *<sup>Z</sup> <sup>K</sup>*2*C*2ð Þ *<sup>t</sup>*, *<sup>Z</sup>* is represented by the series of

<sup>þ</sup>*<sup>∂</sup>*2*φ<sup>s</sup> ∂*C2 2 � � � ð Þ 0,0 C2 2

**linearized problems and construction of solutions**

*Competitive Adsorption and Diffusion of Gases in a Microporous Solid*

*<sup>д</sup><sup>t</sup>* <sup>¼</sup> *<sup>D</sup>*inter*<sup>s</sup> l* 2

*<sup>д</sup><sup>t</sup>* <sup>¼</sup> *<sup>D</sup>*intra*<sup>s</sup>*

boundary conditions for coordinate X of the crystallite

*∂*

*дQs*ð Þ *t*, *X*, *Z*

*<sup>Q</sup>*1ð Þ¼ *<sup>t</sup>*, *<sup>X</sup>* <sup>¼</sup> 1, *<sup>Z</sup> <sup>K</sup>*1*C*2ð Þ *<sup>t</sup>*, *<sup>Z</sup>*

*<sup>Q</sup>*2ð Þ¼ *<sup>t</sup>*,*<sup>X</sup>* <sup>¼</sup> 1, *<sup>Z</sup> <sup>K</sup>*2*C*2ð Þ *<sup>t</sup>*, *<sup>Z</sup>*

boundary and interface conditions for coordinate Z

*Cs*ð Þ¼ *<sup>t</sup>*, 1 1, *дС<sup>s</sup>*

*<sup>p</sup>*2ð Þ <sup>1</sup>�*θ*1�*θ*<sup>2</sup>

*∂C*<sup>1</sup> � � � � ð Þ 0,0

*∂*2*φ<sup>s</sup> ∂C*<sup>2</sup> 1 � � � ð Þ 0,0 *C*2 <sup>1</sup> <sup>þ</sup> <sup>2</sup> *<sup>∂</sup>*2*φ<sup>s</sup> ∂C*1*C*<sup>2</sup> � � � ð Þ 0,0

0

B@

,*K*<sup>2</sup> <sup>¼</sup> *<sup>θ</sup>*<sup>2</sup>

of *m* ¼ 2 is converted into the form

*DOI: http://dx.doi.org/10.5772/intechopen.88138*

with initial conditions

*<sup>K</sup>*<sup>1</sup> <sup>¼</sup> *<sup>θ</sup>*<sup>1</sup>

Tailor [5]:

**29**

*<sup>p</sup>*1ð Þ <sup>1</sup>�*θ*1�*θ*<sup>2</sup>

ecules. The expression *φs*ð Þ¼ *C*1,*C*<sup>2</sup>

*<sup>φ</sup>s*ð Þ¼ *<sup>C</sup>*1,*C*<sup>2</sup> *<sup>φ</sup>s*ð Þþ 0, 0 *<sup>∂</sup>φ<sup>s</sup>*

þ 1 2!

*дСs*ð Þ *t*, *Z*

where *ϕsk* ð Þ *t*, *Z* and *ψsk* ð Þ *<sup>t</sup>*,*X*, *<sup>Z</sup>* belong to <sup>Ω</sup>*kt* and 0, 1 ½ � <sup>∪</sup> <sup>Ω</sup>*kt*, respectively, <sup>L</sup>�<sup>1</sup> <sup>∗</sup> is the conjugate operator to inverse operator L�<sup>1</sup> , and Ψ*sk* is the solution of conjugate problem (A.5)–(A.10).

Reporting in Eq. (A.17) the components X*sk* ð Þ *t*, *X*, *Z* taking into account Eq. (A.18), we obtain the formula which establishes the relationship between the direct problem (9)–(15) and the conjugate problem (A.6)–(A.10) which makes it possible to obtain the analytical expressions of components of the residual functional gradient:

$$
\Delta f\_s(D\_{\mathrm{intr}\_{\mathrm{dr}}}, D\_{\mathrm{intr}\_{\mathrm{dr}}}) = \begin{pmatrix}
\left(\phi\_{\iota\_k}(\mathbf{t}, \mathbf{Z}), \frac{\partial}{\partial \mathbf{Z}} \left(\Delta D\_{\mathrm{intr}\_{\mathrm{dr}}} \frac{\partial}{\partial \mathbf{Z}} C\_{\iota\_k}\right) - e\_{\mathrm{intr}\_{\mathrm{dr}}} \frac{\Delta D\_{\mathrm{intr}\_{\mathrm{dr}}}}{R^2} \frac{\partial}{\partial \mathbf{X}} Q\_{\iota\_k}(\mathbf{t}, \mathbf{X}, \mathbf{Z})\_{\mathbf{X}=1}\right) \\
+ \left(\psi\_{\iota\_k}(\mathbf{t}, \mathbf{X}, \mathbf{Z}), \frac{\Delta D\_{\mathrm{intr}\_{\mathrm{dr}}}}{R^2} \left(\frac{\partial^2}{\partial \mathbf{X}^2} + \frac{2}{\mathbf{X}} \frac{\partial}{\partial \mathbf{X}}\right) Q\_{\iota\_k}(\mathbf{t}, \mathbf{X}, \mathbf{Z})\right)
\end{pmatrix} \tag{A.18}
$$

Differentiating expression (A.18), by Δ*D*intrask and Δ*D*intersk , respectively, and calculating the scalar products according to Eq. (A.15), we obtain the required analytical expressions for the gradient of the residual functional in the intra- and inter-crystallite spaces, respectively:

$$\begin{split} \nabla f\_{D\_{\text{inner}\_{ik}}}(t) &= -\frac{e\_{\text{inter}}}{R^2} \int\_{L\_{k-1}}^{L\_k} \frac{\partial}{\partial X} \mathbf{Q}\_{\mathbf{k}}(t, \mathbf{1}, Z) \boldsymbol{\phi}\_{\boldsymbol{\upkappa}}(t, Z) dZ \\ &+ \frac{1}{R^2} \int\_{L\_{k-1}}^{L\_k} \int\_0^1 \left( \frac{\partial^2}{\partial X^2} + \frac{2}{X} \frac{\partial}{\partial X} \right) \mathbf{Q}\_{\boldsymbol{\upkappa}}(t, X, Z) \boldsymbol{\upmu}\_{\boldsymbol{\upkappa}}(t, X, Z) \mathbf{X} d\mathbf{X} dZ \qquad \text{(A.19)} \end{split} \tag{A.19}$$

$$\nabla U\_{D\_{\text{inter}\_{\text{ik}}}}(t) = \int\_{L\_{k-1}}^{L\_k} \frac{\partial^2 \mathcal{L}\_{s\_k}(t, Z)}{\partial Z^2} \phi\_{s\_k}(t, Z) dZ \tag{A.20}$$

The formulas of gradients ∇*J n D*intra*<sup>s</sup> k* ð Þ*t* and ∇*J n D*inter*<sup>s</sup> k* ð Þ*t* include analytical expressions of the solutions of the direct problem (9)–(14) and inverse problem (A.5)–(A.10). They provide high performance of computing process, avoiding a large number of inner loop iterations by using exact analytical methods [2, 15].
