**B. The linearization schema of the nonlinear co-adsorption model: system of linearized problems and construction of solutions**

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 of *m* ¼ 2 is converted into the form

$$\frac{\partial C\_s(t, Z)}{\partial t} = \frac{D\_{\text{inter}\_t}}{l^2} \frac{\partial^2 C\_s}{\partial Z^2} - \varepsilon\_{\text{inter}} \tilde{K}\_s \frac{D\_{\text{intra}\_t}}{R^2} \left(\frac{\partial Q\_s}{\partial X}\right)\_{X=1} \tag{A.21}$$

$$\frac{\partial Q\_s(t, X, Z)}{\partial t} = \frac{D\_{\text{intra}\_\epsilon}}{R^2} \left( \frac{\partial^2 Q\_s}{\partial X^2} + \frac{2}{X} \frac{\partial Q\_s}{\partial X} \right), \mathbf{s} = \overline{\mathbf{1}, \mathbf{2}} \tag{A.22}$$

with initial conditions

wher<sup>е</sup> *wsk* ¼ L�<sup>1</sup>

*Zeolites - New Challenges*

Defining the scalar product

<sup>L</sup>*wsk* ð Þ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup>* , <sup>Ψ</sup>*sk* ð Þ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup>* � � <sup>¼</sup>

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*

the conjugate operator to inverse operator L�<sup>1</sup>

*ϕsk*

inter-crystallite spaces, respectively:

þ 1 *R*2

The formulas of gradients ∇*J*

*e*inter*<sup>k</sup> R*2

*L* ð*k*

*∂ <sup>∂</sup><sup>X</sup> Qsk*

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

*Lk*�<sup>1</sup>

ð 1

0

∇*JD*intersk

*L* ð*k*

*Lk*�<sup>1</sup>

ðÞ¼� *t*

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

þ *ψsk* ð Þ *t*,*X*, *Z* ,

where *ϕsk* ð Þ *t*, *Z* and *ψsk*

problem (A.5)–(A.10).

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

∇*JD*intrask

**28**

and the vector of the right-hand parts of Eq. (A.13):

Δ*Js D*intersk , *D*intrask

<sup>X</sup>*sk* and <sup>L</sup>�<sup>1</sup> is the inverse operator of operator <sup>L</sup>.

Ð Ð Ω*kT*

<sup>L</sup>*wsk* ð Þ *<sup>t</sup>*,*X*, *<sup>Z</sup>* , <sup>Ψ</sup>*sk* ð Þ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup>* � � <sup>¼</sup> *wsk* ð Þ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup>* ,L<sup>∗</sup> <sup>Ψ</sup>*sk* ð Þ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup>* � � (A.16)

residual functional expressed by the solution of conjugate problem (A.6)–(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

> *∂ <sup>∂</sup><sup>Z</sup> Csk* � �

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

� �

the analytical expressions of components of the residual functional gradient:

Δ*D*intersk

Δ*D*intrask *R*2

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

ð Þ *t*, 1, *Z ϕsk*

*Qsk*

*Csk* ð Þ *t*, *Z*

*n D*inter*<sup>s</sup> k*

2 *X ∂ ∂X*

> *L* ð*k*

*∂*2

ð Þ*t* and ∇*J*

expressions of the solutions of the direct problem (9)–(14) and inverse problem

*Lk*�<sup>1</sup>

� �

ðÞ¼ *t*

*n D*intra*<sup>s</sup> k*

ÐÐÐ ð Þ 0, *R* ∪ Ω*kT*

2 6 4

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

LΔ*Csk* ð Þ *t*, *Z ϕsk*

� � <sup>¼</sup> <sup>Ψ</sup>*sk* , we obtain the increment of the

� � <sup>¼</sup> <sup>Ψ</sup>*sk* ð Þ *<sup>t</sup>*,*X*, *<sup>Z</sup>* , X*sk* ð Þ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup>*<sup>Þ</sup> � (A.17)

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

� *e*int*erk*

� �

2 *X ∂ ∂X*

� �

ð Þ *t*, *Z dZ*

ð Þ *t*, *Z dZdt*

, and Ψ*sk* is the solution of conjugate

Δ*D*intrask *R*2

*Qsk* ð Þ *t*,*X*, *Z*

ð Þ *t*, *X*, *Z ψsk* ð Þ *t*, *X*, *Z XdXdZ* (A.19)

*<sup>∂</sup>Z*<sup>2</sup> *<sup>ϕ</sup>sk* ð Þ *<sup>t</sup>*, *<sup>Z</sup> dZ* (A.20)

ð Þ*t* include analytical

*∂*

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

(A.18)

3 7 5

(A.15)

LΔ*Qsk* ð Þ *t*, *X*, *Z ψsk* ð Þ *t*, *X*, *Z XdXdZdt*

$$\mathbf{C}\_{\mathbf{s}}(t=0,Z) = \mathbf{0}; \mathbf{Q}\_{\mathbf{s}}(t=0,X,Z) = \mathbf{0}; X \in (0,1), \mathbf{s} = \overline{1,2} \tag{A.23}$$

boundary conditions for coordinate X of the crystallite

$$\frac{\partial}{\partial X}Q\_{\varepsilon}(t,X=0,Z)=0\tag{A.24}$$

$$\begin{aligned} Q\_1(t, X = 1, Z) &= \frac{K\_1 \mathcal{C}\_2(t, Z)}{1 + K\_1 \mathcal{C}\_1(t, Z) + K\_2 \mathcal{C}\_2(t, Z)} \quad (\text{Langmuir}),\\ Q\_2(t, X = 1, Z) &= \frac{K\_2 \mathcal{C}\_2(t, Z)}{1 + K\_1 \mathcal{C}\_1(t, Z) + K\_2 \mathcal{C}\_2(t, Z)} \end{aligned} \quad (\text{Langmonic equilibrium}),\tag{A.25}$$

boundary and interface conditions for coordinate Z

$$\mathbf{C}\_{\mathbf{s}}(t, \mathbf{1}) = \mathbf{1}, \frac{\partial C\_{\mathbf{s}}}{\partial \mathbf{Z}}(t, Z = \mathbf{0}) = \mathbf{0}, t \in \left(\mathbf{0}, t^{total}\right) \tag{A.26}$$

*<sup>K</sup>*<sup>1</sup> <sup>¼</sup> *<sup>θ</sup>*<sup>1</sup> *<sup>p</sup>*1ð Þ <sup>1</sup>�*θ*1�*θ*<sup>2</sup> ,*K*<sup>2</sup> <sup>¼</sup> *<sup>θ</sup>*<sup>2</sup> *<sup>p</sup>*2ð Þ <sup>1</sup>�*θ*1�*θ*<sup>2</sup> , where *p*1, *p*<sup>2</sup> are the co-adsorption equilibrium 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 molecules. The expression *φs*ð Þ¼ *C*1,*C*<sup>2</sup> *Cs*ð Þ *t*, *Z* <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 Tailor [5]:

$$\begin{split} \rho\_{s}(\mathbf{C}\_{1}, \mathbf{C}\_{2}) &= \rho\_{s}(\mathbf{0}, \mathbf{0}) + \left( \frac{\partial \rho\_{s}}{\partial \mathbf{C}\_{1}} \Big|\_{(0,0)} \mathbf{C}\_{1} + \frac{\partial \rho\_{s}}{\partial \mathbf{C}\_{2}} \Big|\_{(0,0)} \mathbf{C}\_{2} \right) \\ &+ \frac{1}{2!} \Bigg( \begin{array}{c} \frac{\delta^{2} \rho\_{s}}{\delta \mathbf{C}\_{1}^{2}} \Big|\_{(0,0)} \mathbf{C}\_{1}^{2} + 2 \frac{\delta^{2} \rho\_{s}}{\delta \mathbf{C}\_{1} \mathbf{C}\_{2}} \Big|\_{(0,0)} \mathbf{C}\_{1} \mathbf{C}\_{1} + \\ & \qquad \qquad \qquad + \frac{\delta^{2} \rho\_{s}}{\delta \mathbf{C}\_{2}^{2}} \Big|\_{(0,0)} \mathbf{C}\_{2}^{2} \end{array} \Bigg) + \dots \end{split} \tag{A.27}$$

As a result of transformations, limiting to the series not higher than the second order, we obtain

*Cs*<sup>0</sup> ð Þ¼ *<sup>t</sup>*, 1 1, *дСs*<sup>0</sup>

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

*∂*2 *Сsn*

boundary conditions for coordinate X of the crystallite. *∂*

*ν*¼0

*ν*¼0 *C*2*<sup>s</sup>*

boundary and interface conditions for coordinate Z.

*Csn* ð Þ¼ *<sup>t</sup>*, 1 0, *дС<sup>s</sup>*<sup>0</sup>

*дQsn* ð Þ *t*,*X*, *Z*

*дСsn* ð Þ *t*, *Z*

with initial conditions.

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

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

any number of gases.

**31**

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

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

The problem *An*; *n* ¼ 1, ∞ (*n*th *approximation with zero initial and boundary conditions*): to construct in the domain *D* a solution of a system of equations

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

*Csn* ð Þ¼ *t* ¼ 0, *Z* 0; *Qsn* ð Þ¼ *t* ¼ 0,*X*, *Z* 0; *s* ¼ 1, 2 (A.40)

*<sup>C</sup>*1*<sup>v</sup>* ð Þ *<sup>t</sup>*, Z *<sup>C</sup>*1,*n*�1�*ν*ð Þþ *<sup>t</sup>*, Z <sup>1</sup>

*<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>*

The problems *As*<sup>0</sup> , *s* ¼ 1, 2 are linear with respect to zero approximation *Cs*<sup>0</sup> , *Qs*<sup>0</sup> ; the problems *Asn* ; n ¼ 1, ∞ are linear with respect to the *n*-th approximation *Csn* , *Qsn*

As demonstrated for the two-component adsorption model (A.21)–(A.26), our proposed methodology can easily be developed and applied to the co-adsorption of

and nonlinear with respect to all previous *n-1* approximations *Cs*<sup>0</sup> , … ,*Csn*�<sup>1</sup> .

ð Þ *<sup>t</sup>*, Z <sup>1</sup> 2 *K*2 *K*1

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

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

*R*2

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

*<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>ð Þ 0, *<sup>T</sup>* (A.37)

*X*¼1

� � (A.39)

(A.38)

(A.42)

*<sup>∂</sup>Qsn ∂X* � �

> 2 *X*

*<sup>∂</sup><sup>X</sup> Qsn* ð Þ¼ *<sup>t</sup>*,*<sup>X</sup>* <sup>¼</sup> 0, *<sup>Z</sup>* <sup>0</sup> (A.41)

2 *K*2 *K*1 *<sup>C</sup>*2,*n*�1�*ν*ð Þ *<sup>t</sup>*, Z � �,

*K*1 � �<sup>2</sup> *<sup>C</sup>*2,*n*�1�*ν*ð Þ *<sup>t</sup>*, Z !

*total* � � (A.43)

*<sup>C</sup>*1,*n*�1�*ν*ð Þþ *<sup>t</sup>*, Z *<sup>K</sup>*<sup>2</sup>

*<sup>∂</sup>Qsn ∂X*

$$\begin{aligned} \frac{K\_1 C\_1(t, Z)}{1 + K\_1 C\_1(t, Z) + K\_2 C\_2(t, Z)} &= K\_1 C\_1(t, Z) - \left( K\_1^2 C\_1^2(t, Z) + \frac{1}{2} K\_1 K\_2 C\_1(t, Z) C\_2(t, Z) \right), \\\ \frac{K\_2 C\_2(t, Z)}{1 + K\_1 C\_1(t, Z) + K\_2 C\_2(t, Z)} &= K\_2 C\_2(t, Z) - \left( \frac{1}{2} K\_1 K\_2 C\_1^2(t, Z) C\_2^1(t, Z) + K\_2^2 C\_2^2(t, Z) \right). \end{aligned} \tag{A.28}$$

Substituting the expanded expression (A.28) in Eq. (A.25) of nonlinear systems (A.20)–(A.26), we obtain

$$\begin{aligned} Q\_1(t, \mathbf{X}, Z)\_{\mathbf{X}=1} &= K\_1 \mathbf{C}\_1(t, Z) - \varepsilon \left( \mathbf{C}\_1^2(t, Z) + \frac{1}{2} \mathbf{K}\_2 \mathbf{C}\_1(t, Z) \mathbf{C}\_2(t, Z) \right), \\ Q\_2(t, \mathbf{X}, Z)\_{\mathbf{X}=1} &= K\_2 \mathbf{C}\_2(t, Z) - \varepsilon \left( \frac{1}{2} \mathbf{K}\_2 \mathbf{C}\_1(t, Z) \mathbf{C}\_2(t, Z) + \left( \frac{\mathbf{K}\_2}{K\_1} \right)^2 \mathbf{C}\_2^2(t, Z) \right) \end{aligned} \tag{A.29}$$

where *<sup>ε</sup>* <sup>¼</sup> *<sup>K</sup>*<sup>2</sup> <sup>1</sup> < <1 is the small parameter.

Taking into account the approximate equations of the kinetics of co-adsorption (A.29) containing the small parameter *ε*, we search for the solution of the problem (A.21)–(A.26) by using asymptotic series with a parameter *ε* in the form [6, 7]

$$\mathbf{C}\_{\mathbf{t}}(\mathbf{t}, \mathbf{Z}) = \mathbf{C}\_{\mathbf{t}\_0}(\mathbf{t}, \mathbf{Z}) + \varepsilon \mathbf{C}\_{\mathbf{t}\_1}(\mathbf{t}, \mathbf{Z}) + \varepsilon^2 \mathbf{C}\_{\mathbf{t}\_2}(\mathbf{t}, \mathbf{Z}) + \dots,\tag{A.30}$$

$$\mathcal{Q}\_{\mathfrak{s}}(t,X,Z) = \mathcal{Q}\_{\mathfrak{s}\mathfrak{l}}(t,X,Z) + \varepsilon \mathcal{Q}\_{\mathfrak{s}\mathfrak{l}}(t,X,Z) + \varepsilon^2 \mathcal{Q}\_{\mathfrak{s}\mathfrak{l}}(t,X,Z) + \dots, \mathfrak{s} = \overline{1,2} \tag{A.31}$$

As the result of substituting the asymptotic series (A.30)–(A.31) into the equations of the nonlinear boundary problem (A.21)–(A.26) considering Eq. (A.28), the problem (A.21)–(A.26) will be parallelized into two types of linearized boundary problems [6]:

The problem *As*<sup>0</sup> , *s* ¼ 1, 2 (*zero approximation with initial and boundary conditions of the initial problem*): to find a solution in the domain *D* ¼ fð Þ *t*, X, Z : *t*>0,*X* ∈ ð Þ 0, 1 , *Z* ∈ð Þg 0, 1 of a system of partial differential equations

$$\frac{\partial C\_{s\_0}(t, Z)}{\partial t} = \frac{D\_{\text{inter}\_{\text{r}}}}{l^2} \frac{\partial^2 C\_{s\_0}}{\partial Z^2} - \varepsilon\_{\text{inter}} \tilde{K}\_s \frac{D\_{\text{intra}\_{\text{r}}}}{R^2} \left(\frac{\partial Q\_{s\_0}}{\partial X}\right)\_{X=1} \tag{A.32}$$

$$\frac{\partial Q\_{\epsilon\_0}(t,X,Z)}{\partial t} = \frac{D\_{\text{intra}\_{\epsilon}}}{R^2} \left( \frac{\partial^2 Q\_{\epsilon\_0}}{\partial X^2} + \frac{2}{X} \frac{\partial Q\_{\epsilon\_0}}{\partial X} \right) \tag{A.33}$$

with initial conditions.

$$\mathbf{C}\_{\circ\_0}(\mathbf{t}=\mathbf{0}, Z) = \mathbf{0}; \mathbf{Q}\_{\circ\_0}(\mathbf{t}=\mathbf{0}, X, Z) = \mathbf{0}; X \in (\mathbf{0}, \mathbf{1}), \mathbf{s} = \overline{\mathbf{1}, \mathbf{2}} \tag{A.34}$$

boundary conditions for coordinate X of the crystallite

$$\frac{\partial}{\partial X}Q\_{s\_0}(t,X=0,Z) = 0\tag{A.35}$$

$$\mathcal{Q}\_{\mathfrak{s}\_0}(t, X=1, Z) = K\_\mathfrak{s} \mathcal{C}\_{\mathfrak{s}\_0}(t, Z), \mathfrak{s} = \overline{\mathbf{1}, \mathbf{2}} \tag{A.36}$$

boundary and interface conditions for coordinate Z.

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

$$\mathbf{C}\_{\mathbf{s}\_0}(t, \mathbf{1}) = \mathbf{1}, \frac{\partial C\_{\mathbf{s}\_0}}{\partial \mathbf{Z}}(t, Z = \mathbf{0}) = \mathbf{0}, t \in (\mathbf{0}, T) \tag{A.37}$$

The problem *An*; *n* ¼ 1, ∞ (*n*th *approximation with zero initial and boundary conditions*): to construct in the domain *D* a solution of a system of equations

$$\frac{\partial C\_{\boldsymbol{\epsilon}\_{\pi}}(\mathbf{t}, \mathbf{Z})}{\partial \mathbf{t}} = \frac{D\_{\text{inter}\_{\boldsymbol{\epsilon}}}}{l^{2}} \frac{\partial^{2} C\_{\boldsymbol{\epsilon}\_{\pi}}}{\partial \mathbf{Z}^{2}} - \boldsymbol{\varepsilon}\_{\text{inter}} \boldsymbol{\tilde{K}}\_{\boldsymbol{\epsilon}} \frac{D\_{\text{intra}\_{\boldsymbol{\epsilon}}}}{R^{2}} \left(\frac{\partial Q\_{\boldsymbol{\epsilon}\_{\pi}}}{\partial \mathbf{X}}\right)\_{\boldsymbol{X}=\mathbf{1}} \tag{A.38}$$

$$\frac{\partial Q\_{\star}(t,X,Z)}{\partial t} = \frac{D\_{\text{intra}\_{\text{r}}}}{R^{2}} \left( \frac{\partial^{2} Q\_{\star\_{\text{r}}}}{\partial X^{2}} + \frac{2}{X} \frac{\partial Q\_{\star\_{\text{r}}}}{\partial X} \right) \tag{A.39}$$

with initial conditions.

As a result of transformations, limiting to the series not higher than the second

1*C*<sup>2</sup>

*K*1*K*2*C*<sup>2</sup>

2

Substituting the expanded expression (A.28) in Eq. (A.25) of nonlinear systems

2 *K*2 *K*1

Taking into account the approximate equations of the kinetics of co-adsorption (A.29) containing the small parameter *ε*, we search for the solution of the problem (A.21)–(A.26) by using asymptotic series with a parameter *ε* in the form [6, 7]

ð Þþ *<sup>t</sup>*, *<sup>X</sup>*, *<sup>Z</sup> <sup>ε</sup>*<sup>2</sup>

As the result of substituting the asymptotic series (A.30)–(A.31) into the equations of the nonlinear boundary problem (A.21)–(A.26) considering Eq. (A.28), the problem (A.21)–(A.26) will be parallelized into two types of linearized boundary

The problem *As*<sup>0</sup> , *s* ¼ 1, 2 (*zero approximation with initial and boundary conditions*

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

*∂*2 *Qs*<sup>0</sup> *<sup>∂</sup>X*<sup>2</sup> <sup>þ</sup>

*Cs*<sup>0</sup> ð Þ¼ *t* ¼ 0, *Z* 0; *Qs*<sup>0</sup> ð Þ¼ *t* ¼ 0,*X*, *Z* 0;*X* ∈ð Þ 0, 1 , *s* ¼ 1, 2 (A.34)

*<sup>∂</sup>Qs*<sup>0</sup> *∂X* � �

> 2 *X*

*<sup>∂</sup><sup>X</sup> Qs*<sup>0</sup> ð Þ¼ *<sup>t</sup>*, *<sup>X</sup>* <sup>¼</sup> 0, *<sup>Z</sup>* <sup>0</sup> (A.35)

*Qs*<sup>0</sup> ð Þ¼ *t*,*X* ¼ 1, *Z KsCs*<sup>0</sup> ð Þ *t*, *Z* , *s* ¼ 1, 2 (A.36)

� �

*<sup>∂</sup>Qs*<sup>0</sup> *∂X*

*X*¼1

� �

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

<sup>1</sup>ð Þþ *<sup>t</sup>*, *<sup>Z</sup>* <sup>1</sup>

2 *K*2 *K*1

*of the initial problem*): to find a solution in the domain *D* ¼ fð Þ *t*, X, Z : *t*>0,*X* ∈ ð Þ 0, 1 , *Z* ∈ð Þg 0, 1 of a system of partial differential equations

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

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

*R*2

*∂*2 *С<sup>s</sup>*<sup>0</sup>

*дQs*<sup>0</sup> ð Þ *t*,*X*, *Z*

boundary conditions for coordinate X of the crystallite *∂*

boundary and interface conditions for coordinate Z.

<sup>1</sup> < <1 is the small parameter.

*Cs*ð Þ¼ *<sup>t</sup>*, Z *Cs*<sup>0</sup> ð Þþ *<sup>t</sup>*, Z *<sup>ε</sup>Cs*<sup>1</sup> ð Þþ *<sup>t</sup>*, Z *<sup>ε</sup>*<sup>2</sup>

<sup>1</sup>ð Þþ *<sup>t</sup>*, *<sup>Z</sup>* <sup>1</sup>

2

<sup>1</sup>ð Þ *<sup>t</sup>*, *<sup>Z</sup> <sup>C</sup>*<sup>1</sup>

*C*1ð Þ *t*, *Z C*2ð Þ *t*, *Z*

! (A.29)

*K*1 � �<sup>2</sup>

� �

� �

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

2*C*<sup>2</sup> <sup>2</sup>ð Þ *t*, *Z*

(A.28)

(A.32)

(A.33)

<sup>2</sup>ð Þþ *<sup>t</sup>*, *<sup>Z</sup> <sup>K</sup>*<sup>2</sup>

,

*C*2 <sup>2</sup>ð Þ *t*, *Z*

*Cs*<sup>2</sup> ð Þþ *t*, Z … , (A.30)

*Qs*<sup>2</sup> ð Þþ *t*, *X*, *Z* … , *s* ¼ 1, 2 (A.31)

,

order, we obtain

*Zeolites - New Challenges*

*K*1*C*1ð Þ *t*, *Z*

*K*2*C*2ð Þ *t*, *Z*

(A.20)–(A.26), we obtain

where *<sup>ε</sup>* <sup>¼</sup> *<sup>K</sup>*<sup>2</sup>

problems [6]:

**30**

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

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

*Qs*ð Þ¼ *t*, *X*, *Z Qs*<sup>0</sup> ð Þþ *t*, *X*, *Z εQs*<sup>1</sup>

*дС<sup>s</sup>*<sup>0</sup> ð Þ *t*, *Z*

with initial conditions.

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

<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>* <sup>¼</sup> *<sup>K</sup>*1*C*1ð Þ� *<sup>t</sup>*, *<sup>Z</sup> <sup>K</sup>*<sup>2</sup>

<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>* <sup>¼</sup> *<sup>K</sup>*2*C*2ð Þ� *<sup>t</sup>*, *<sup>Z</sup>* <sup>1</sup>

$$\mathcal{C}\_{\mathfrak{t}\_{\mathfrak{t}}}(\mathfrak{t}=\mathbf{0}, Z) = \mathbf{0}; \mathcal{Q}\_{\mathfrak{t}\_{\mathfrak{t}}}(\mathfrak{t}=\mathbf{0}, X, Z) = \mathbf{0}; \mathfrak{s} = \overline{\mathbf{1}, \mathbf{2}} \tag{A.40}$$

boundary conditions for coordinate X of the crystallite.

$$\frac{\partial}{\partial \mathbf{X}} \mathbf{Q}\_{\mathbf{t}\_n}(\mathbf{t}, \mathbf{X} = \mathbf{0}, \mathbf{Z}) = \mathbf{0} \tag{\text{A.41}}$$

$$\mathbf{Q}\_{\mathbf{t}\_n}(\mathbf{t}, \mathbf{X}, \mathbf{Z})\_{\mathbf{X} = 1} = K\_1 \mathbf{C}\_{\mathbf{1}\_n}(\mathbf{t}, \mathbf{Z}) - \sum\_{\nu = 0}^{n - 1} \mathbf{C}\_{\mathbf{1}\_\nu}(\mathbf{t}, \mathbf{Z}) \left( \mathbf{C}\_{\mathbf{1}\_\nu - 1 - \nu}(\mathbf{t}, \mathbf{Z}) + \frac{1 \mathbf{K}\_2}{2 \mathbf{K}\_1} \mathbf{C}\_{\mathbf{2}\_\nu - 1 - \nu}(\mathbf{t}, \mathbf{Z}) \right), \tag{\text{A.42}}$$

$$\mathbf{Q}\_{\mathbf{2}\_{\nu}}(\mathbf{t}, \mathbf{X}, \mathbf{Z})\_{\mathbf{X}=1} = \mathbf{K}\_{\mathbf{2}} \mathbf{C}\_{\mathbf{2}\_{\nu}}(\mathbf{t}, \mathbf{Z}) - \sum\_{\nu=0}^{n-1} \mathbf{C}\_{\mathbf{2}}(\mathbf{t}, \mathbf{Z}) \left(\frac{\mathbf{1} \mathbf{K}\_{\mathbf{2}}}{2\mathbf{K}\_{1}} \mathbf{C}\_{\mathbf{1}, \nu - 1-\nu}(\mathbf{t}, \mathbf{Z}) + \left(\frac{\mathbf{K}\_{2}}{\mathbf{K}\_{1}}\right)^{2} \mathbf{C}\_{\mathbf{2}, \nu - 1-\nu}(\mathbf{t}, \mathbf{Z})\right) \tag{A.42}$$

boundary and interface conditions for coordinate Z.

$$\mathbf{C}\_{s\_0}(t, \mathbf{1}) = \mathbf{0}, \frac{\partial \mathbf{C}\_{s\_0}}{\partial \mathbf{Z}}(t, \mathbf{Z} = \mathbf{0}) = \mathbf{0}, t \in \left(\mathbf{0}, t^{total}\right) \tag{A.43}$$

The problems *As*<sup>0</sup> , *s* ¼ 1, 2 are linear with respect to zero approximation *Cs*<sup>0</sup> , *Qs*<sup>0</sup> ; the problems *Asn* ; n ¼ 1, ∞ are linear with respect to the *n*-th approximation *Csn* , *Qsn* and nonlinear with respect to all previous *n-1* approximations *Cs*<sup>0</sup> , … ,*Csn*�<sup>1</sup> .

As demonstrated for the two-component adsorption model (A.21)–(A.26), our proposed methodology can easily be developed and applied to the co-adsorption of any number of gases.

*Zeolites - New Challenges*
