**4. A mathematical model of competitive co-adsorption and co-diffusion in microporous solids**

#### **4.1 Co-adsorption model in general formulation**

The model presented is similar to the bipolar model [2, 3, 8, 9]. By developing the approach described by Ruthven and Kärger [10, 11] and Petryk et al. [5]

#### *Zeolites - New Challenges*

concerning the elaboration of a complex process of co-adsorption and co-diffusion, it is necessary to specify the most important hypotheses limiting the process.

boundary and interface conditions for coordinate Z

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

� � � �

*RgT* � �.

� � � *RgT*, where *Ugs* � *Uadss*

aged over the pore volume of the adsorbent [11].

and the position of the particle in the zeolite bed.

**to the benzene-hexane mixture**

*дСsk* ð Þ *t*, *Z*

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

*дQsk* ð Þ *t*, *X*, *Z*

boundary and interface conditions for coordinate Z

<sup>¼</sup> 0, *<sup>∂</sup>*

boundary conditions for coordinate X in the particle

*<sup>∂</sup><sup>Z</sup> <sup>D</sup>*inter*<sup>s</sup>*

¼ 1, *N*, *t*∈ 0, *t*

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

*Z*¼*Lk*

with *Ks*ð Þ¼ *<sup>T</sup> <sup>k</sup>*0*<sup>s</sup>* exp � <sup>Δ</sup>*Hs*

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

Δ*Hs* ¼ *ϕ* � *Ugs* � *Uadss*

zeolite bed.

in domains: Ω*kt* ¼ 0, *t*

with initial conditions

*Csk* ð Þ¼ *t* ¼ 0, *Z* 0; *Qsk*

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

**17**

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

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

Here the activation energy is the heat of adsorption defined as

kinetic energies of the molecule of the *i*th component of the adsorbate in the gaseous and adsorbed states is the magnitude of the Lennard-Jones potential, aver-

The non-isothermal model (1)–(8) can easily be transformed into isothermal model, removing the temperature Eq. (2) and condition (8) and replacing the functions *Ks*ð Þ *T* with the corresponding equilibrium constants *Ks*. The competitive diffusion coefficients *D*intra*<sup>s</sup>* and *D*inter*<sup>s</sup>* can be considered as functions of the time

**4.2 The inverse model of co-diffusion coefficient identification: application**

On the basis of a developed nonlinear co-adsorption model (1)–(8), we

diffusion coefficients *D*intra*<sup>s</sup>* and *D*inter*<sup>s</sup>* as a function of time and coordinate in the

The mathematical model of gas diffusion kinetics in the zeolite bed is defined

*<sup>∂</sup>Z*<sup>2</sup> � *<sup>e</sup>*inter*kK*~*sk*

*k R*2

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

*k*�1

*total* � �;

*total* � � � <sup>Ω</sup>*k*, <sup>Ω</sup>*<sup>k</sup>* <sup>¼</sup> *Lk*�<sup>1</sup> ð Þ , *Lk* , *<sup>k</sup>* <sup>¼</sup> 1, *<sup>N</sup>* <sup>þ</sup> 1, *<sup>L</sup>*<sup>0</sup> <sup>¼</sup> <sup>0</sup><*L*<sup>1</sup> <sup>&</sup>lt; … �

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

!

ð Þ¼ *t* ¼ 0,*X*, *Z* 0;*X* ∈ ð Þ 0, 1 , *Z* ∈ Ω*k*, *k* ¼ 1, *N* þ 1, (11)

*Csk*�<sup>1</sup> ð Þ� *t*, *Z D*inter*<sup>s</sup>*

h i

2 *X* *<sup>∂</sup>Qsk ∂X*

*<sup>∂</sup>Qsk ∂X* � �

*X*¼1

*total* � �; (12)

*Z*¼*Lk*

¼ 0, *k*

(13)

*Csk* ð Þ *t*, *Z*

*k*

(9)

(10)

construct an inverse model for the identification of the competitive

< *LN*þ<sup>1</sup> ¼ 1Þ by the solutions of the system of differential equations

*∂*2 *Сsk*

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

*k l* 2

*∂ ∂z*

*T t*ð , ZÞj

*<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>0</sup> (7)

—the difference between the

(8)

*<sup>Z</sup>*¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>*:*

The general hypothesis adopted to develop the model presented in the most general formulation is that the interaction between the co-adsorbed molecules of several gases and the adsorption centers on the surface in the nanoporous crystallites is determined by the nonlinear competitive equilibrium function of the Langmuir type, taking into account physical assumptions [10]:


Taking into account these hypotheses, we have developed a nonlinear coadsorption model. The meaning of the symbols is given in the nomenclature:

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

$$-H\frac{\partial T(t,z)}{\partial t} - \imath h\_{\mathcal{S}} \frac{\partial T}{\partial \mathbf{z}} - \sum\_{s=1}^{m} \Delta \overline{H}\_{s} \frac{\partial \overline{Q}\_{s}}{\partial t} - 2\frac{a\_{h}}{R\_{\text{column}}}T + \Lambda \frac{\partial^{2}T}{\partial \mathbf{z}^{2}} = \mathbf{0} \tag{2}$$

$$\frac{\partial Q\_s(t, X, Z)}{\partial t} = \frac{D\_{\text{intra}\_\bullet}}{R^2} \left( \frac{\partial^2 Q\_s}{\partial X^2} + \frac{2}{X} \frac{\partial Q\_s}{\partial X} \right) \tag{3}$$

with initial conditions

$$C\_s(t=0, Z) = 0; Q\_s(t=0, X, Z) = 0; Z \in (0, 1), X \in (0, 1), s = \overline{1, m} \tag{4}$$

boundary conditions for coordinate X of the crystallite

$$\frac{\partial}{\partial X}Q\_s(t, X=0, Z) = 0 \text{ (symmetry conditions)},\tag{5}$$

$$Q\_s(t, X=1, Z) = \frac{K\_s(T)\mathcal{C}\_t(t, Z)}{\mathbf{1} + \sum\_{t\_1=1}^m K\_{t\_1}(T)\mathcal{C}\_{t\_1}(t, Z)}, s = \overline{1, \mathfrak{m}} \text{ (Langmuir eguilibrium)}, \quad \text{(6)}$$

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

boundary and interface conditions for coordinate Z

$$\mathbf{C}\_{\mathbf{s}}(t, \mathbf{1}) = \mathbf{1}, \frac{\partial C\_{\mathbf{s}}}{\partial \mathbf{Z}}(t, Z = \mathbf{0}) = \mathbf{0}, t > \mathbf{0} \tag{7}$$

$$T(\mathbf{t}, \mathbf{Z})\Big|\_{\mathbf{Z}=\mathbf{1}} = T\_{\text{initial}}, \frac{\partial}{\partial \mathbf{z}} T(\mathbf{t}, \mathbf{Z})\Big|\_{\mathbf{Z}=\mathbf{0}} = \mathbf{0}.\tag{8}$$

with *Ks*ð Þ¼ *<sup>T</sup> <sup>k</sup>*0*<sup>s</sup>* exp � <sup>Δ</sup>*Hs RgT* � �.

concerning the elaboration of a complex process of co-adsorption and co-diffusion, it is necessary to specify the most important hypotheses limiting the process. The general hypothesis adopted to develop the model presented in the most general formulation is that the interaction between the co-adsorbed molecules of several gases and the adsorption centers on the surface in the nanoporous crystallites is determined by the nonlinear competitive equilibrium function of the Lang-

1.Co-adsorption is caused by the dispersion forces whose interaction is established by Lennard-Jones and the electrostatic forces of gravity and

3.During the evolution of the system toward equilibrium, there is a concentration gradient in the macropores and/or in the micropores.

4.Co-adsorption occurs on active centers distributed over the entire inner surface of the nanopores (intra-crystallite space) [10, 11]. All crystallites are spherical and have the same radius R; the crystallite bed is uniformly packed.

5.Active adsorption centers adsorb molecules of the *i*th adsorbate, forming

Taking into account these hypotheses, we have developed a nonlinear coadsorption model. The meaning of the symbols is given in the nomenclature:

> Δ*Hs ∂Qs*

> > *R*2

*Cs*ð Þ¼ *t* ¼ 0, *Z* 0; *Qs*ð Þ¼ *t* ¼ 0,*X*, *Z* 0; *Z* ∈ð Þ 0, 1 , *X* ∈ ð Þ 0, 1 , *s* ¼ 1, m (4)

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

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

6.Adsorbed molecules are held by active centers for a certain time, depending on

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

*Rcolumn*

2 *X ∂Qs ∂X*

� �

*<sup>∂</sup><sup>t</sup>* � <sup>2</sup> *<sup>α</sup><sup>h</sup>*

*<sup>∂</sup><sup>X</sup> Qs*ð Þ¼ *<sup>t</sup>*,*<sup>X</sup>* <sup>¼</sup> 0, *<sup>Z</sup>* 0 symmetry conditions � �, (5)

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

*∂Qs ∂X* � �

*X*¼1

*T*

*<sup>∂</sup>z*<sup>2</sup> <sup>¼</sup> 0 (2)

*<sup>T</sup>* <sup>þ</sup> <sup>Λ</sup>*∂*<sup>2</sup>

, *s* ¼ 1, m Langmuir equilibrium ð Þ, (6)

(1)

(3)

2.The co-diffusion process involves two types of mass transfer: diffusion in the macropores (inter-crystallite space) and diffusion in the micropores of

muir type, taking into account physical assumptions [10]:

repulsion described by van der Waals [11].

molecular layers of adsorbate on their surfaces.

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

> *∂T <sup>∂</sup><sup>z</sup>* �X*<sup>m</sup> s*¼1

*дQs*ð Þ *t*, *X*, *Z*

boundary conditions for coordinate X of the crystallite

*Ks*<sup>1</sup> ð Þ *T Cs*<sup>1</sup> ð Þ *t*, *Z*

crystallites (intra-crystallite space).

*Zeolites - New Challenges*

the temperature of the process.

*дСs*ð Þ *t*, *Z*

*<sup>∂</sup><sup>t</sup>* � *uhg*

�*<sup>H</sup> <sup>∂</sup>T t*ð Þ , *<sup>z</sup>*

with initial conditions

**16**

*∂*

*Qs*ð Þ¼ *<sup>t</sup>*, *<sup>X</sup>* <sup>¼</sup> 1, *<sup>Z</sup> Ks*ð Þ *<sup>T</sup> Cs*ð Þ *<sup>t</sup>*, *<sup>Z</sup>* <sup>1</sup> <sup>þ</sup> <sup>P</sup>*<sup>m</sup> s*1¼1

Here the activation energy is the heat of adsorption defined as Δ*Hs* ¼ *ϕ* � *Ugs* � *Uadss* � � � *RgT*, where *Ugs* � *Uadss* —the difference between the kinetic energies of the molecule of the *i*th component of the adsorbate in the gaseous and adsorbed states is the magnitude of the Lennard-Jones potential, averaged over the pore volume of the adsorbent [11].

The non-isothermal model (1)–(8) can easily be transformed into isothermal model, removing the temperature Eq. (2) and condition (8) and replacing the functions *Ks*ð Þ *T* with the corresponding equilibrium constants *Ks*. The competitive diffusion coefficients *D*intra*<sup>s</sup>* and *D*inter*<sup>s</sup>* can be considered as functions of the time and the position of the particle in the zeolite bed.

## **4.2 The inverse model of co-diffusion coefficient identification: application to the benzene-hexane mixture**

On the basis of a developed nonlinear co-adsorption model (1)–(8), we construct an inverse model for the identification of the competitive diffusion coefficients *D*intra*<sup>s</sup>* and *D*inter*<sup>s</sup>* as a function of time and coordinate in the zeolite bed.

The mathematical model of gas diffusion kinetics in the zeolite bed is defined in domains: Ω*kt* ¼ 0, *t total* � � � <sup>Ω</sup>*k*, <sup>Ω</sup>*<sup>k</sup>* <sup>¼</sup> *Lk*�<sup>1</sup> ð Þ , *Lk* , *<sup>k</sup>* <sup>¼</sup> 1, *<sup>N</sup>* <sup>þ</sup> 1, *<sup>L</sup>*<sup>0</sup> <sup>¼</sup> <sup>0</sup><*L*<sup>1</sup> <sup>&</sup>lt; … � < *LN*þ<sup>1</sup> ¼ 1Þ by the solutions of the system of differential equations

$$\frac{\partial C\_{s\_k}(t, Z)}{\partial t} = \frac{D\_{\text{inter}\_k}}{l^2} \frac{\partial^2 C\_{s\_k}}{\partial Z^2} - c\_{\text{inter}\_k} \tilde{K}\_{s\_k} \frac{D\_{\text{intra}\_k}}{R^2} \left(\frac{\partial Q\_{s\_k}}{\partial X}\right)\_{X=1} \tag{9}$$

$$\frac{\partial Q\_{\mathbf{s}\_k}(t, X, Z)}{\partial t} = \frac{D\_{\text{intra}\_{\mathbf{s}\_k}}}{R^2} \left( \frac{\partial^2 Q\_{\mathbf{s}\_k}}{\partial X^2} + \frac{2}{X} \frac{\partial Q\_{\mathbf{s}\_k}}{\partial X} \right) \tag{10}$$

with initial conditions

$$C\_{\mathfrak{s}\_k}(t=0, Z) = 0; Q\_{\mathfrak{s}\_k}(t=0, X, Z) = 0; X \in (0, 1), Z \in \Omega\_k, k = \overline{1, N+1}, \tag{11}$$

boundary and interface conditions for coordinate Z

$$C\_{t\_1}(t, L\_1) = 1, \frac{\partial C\_{t\_1}}{\partial Z}(t, Z = 0) = 0, t \in \left(0, t^{total}\right);\tag{12}$$

$$\begin{aligned} \left[\mathbf{C}\_{\mathbf{t}\_k}(\mathbf{t}, \mathbf{Z}) - \mathbf{C}\_{\mathbf{t}\_k}(\mathbf{t}, \mathbf{Z})\right]\_{Z=L\_k} &= \mathbf{0}, \frac{\partial}{\partial \mathbf{Z}} \left[D\_{\text{inter}\_{\mathbf{t}\_{k-1}}} \mathbf{C}\_{\mathbf{t}\_{k-1}}(\mathbf{t}, \mathbf{Z}) - D\_{\text{inter}\_{\mathbf{t}\_k}} \mathbf{C}\_{\mathbf{t}\_k}(\mathbf{t}, \mathbf{Z})\right]\_{Z=L\_k} = \mathbf{0}, \mathbf{k} \end{aligned} \tag{13}$$

boundary conditions for coordinate X in the particle

$$\begin{split} \frac{\partial}{\partial \mathbf{X}} Q\_{\mathbf{e}\_k}(t, \mathbf{X} = \mathbf{0}, \mathbf{Z}) &= \mathbf{0}, \mathbf{Q}\_{\mathbf{e}\_k}(t, \mathbf{X} = \mathbf{1}, \mathbf{Z}) \\ &= K\_t \mathbf{C}\_{\mathbf{e}}(t, \mathbf{Z}) \text{ (equilibrium conditions)}, \mathbf{Z} \in \Omega\_k, k = \overline{\mathbf{1}, N+1}. \end{split} \tag{14}$$

Additional condition (NMR-*experimental data*)

$$\left[\left[\mathbf{C}\_{\mathbf{z}}(\mathbf{t},\mathbf{Z}) + \overline{\mathbf{Q}}\_{\mathbf{z}}(\mathbf{t},\mathbf{Z})\right]\big|\_{\mathbf{h}\_{\mathbf{k}}} = M\_{\mathbf{z}}(\mathbf{t},\mathbf{Z})\big|\_{\mathbf{h}\_{\mathbf{k}}}, \ \mathbf{s} = \overline{\mathbf{1},\mathbf{2}}; \mathbf{h}\_{\mathbf{k}} \in \Omega\_{\mathbf{k}}.\tag{15}$$

*Csk D<sup>n</sup>* inter*<sup>s</sup> k* , *D<sup>n</sup>* intra*<sup>s</sup> k* ; *t*, *hk* � � <sup>þ</sup> *Qsk Dn*

*J D*inters*<sup>k</sup>*

∇*J n D*intter*<sup>s</sup> k* ð Þ*t* , ∇*J n D*intra*<sup>s</sup> k*

� � �

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

> 2 4

**19**

observation surfaces:

*Csk*ð Þþ *t*, *Z*

ð 1

*Qsk*ð Þ *t*,*X*, *Z dX*

3 5 *Z*¼*hi*

0

∇*J n D*inter*sk* ð Þ*t*

inter*<sup>s</sup> k* , *D<sup>n</sup>* intra*<sup>s</sup> k* ; *t*, *hk* � � � *Msk* ð Þ*<sup>t</sup>*

the model solution from the experimental data on *hk* ∈ Ω*k*, which is written as

, *D*intras*<sup>k</sup>* � � <sup>þ</sup> *Qs <sup>t</sup>*, *<sup>Z</sup>*, *<sup>D</sup>*inters

> *dt*, ∇*J n D*intra*sk* ð Þ*t*

**4.4 Analytical method of co-diffusion coefficient identification**

� � �

With the help of iterative gradient methods on the basis of the minimization of the residual functional, very precise and fast analytical methods have been developed making it possible to express the diffusion coefficients in the form of timedependent analytic functions (16). For their efficient use, it is necessary to have an extensive experimental database, with at least two experimental observation condi-

imental studies were carried out for five Z positions of the swept zeolite layer for each of the adsorbed components. The data were not sufficient to fully apply this simultaneous identification method to these five sections. We therefore used a combination of the analytical method and the iterative gradient method for deter-

using the experimental data obtained by NMR scanning. In particular, in Eqs. (9) and (10), the co-diffusion coefficients can be set directly as functions of the time *t*:

*Cs*<sup>1</sup> ð Þ¼ *<sup>t</sup>*, 1 *<sup>C</sup>initial*

Experimental NMR scanning conditions are defined simultaneously for all *P*

¼ *Mski*

For simplicity we design *u t*ð Þ¼ , *Z Csk*ð Þ *t*, *Z* , *v t*ð Þ¼ , X, *Z Qsk*ð Þ *t*, X, *Z* ,

ð Þ *<sup>t</sup>*, *<sup>Z</sup>* � � *hi*

ð Þ*t* , *D*inter*sk* ð Þ*t* . In this case, the boundary condition (11) can be given in a more

� � �

� � is the error functional, which describes the deviation of

h i<sup>2</sup>

ð Þ*t* (the gradients of the error functional), *J D*inters

� � � 2 ¼ Ð *T* 0 ∇*J n D*intra*sk* ð Þ*t* h i<sup>2</sup>

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

*k*

<sup>2</sup> , *t*∈ 0, *t*

*k* , *D*intras*<sup>k</sup>* � � � *Msk* ð Þ*<sup>t</sup>*

*hk* ∈ Ω*k*, *k* ¼ 1, *N* þ 1, (17)

*total* � �

(16)

*hk dτ*,

*k* , *D*intras *k*

*dt*.

*<sup>k</sup>* coefficients. Our exper-

, *D*inter*sk* as a function of time

*<sup>s</sup>* ð Þ*t :* (18)

, *i* ¼ 1, *P*, *s* ¼ 1, 2; *hi* ∈ ∪

*N*þ1 *k*¼1 Ω*<sup>k</sup>*

(19)

� �.

� � �

h i<sup>2</sup>

� � � 2 þ ∇*J n D*inter*sk* ð Þ*t*

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

*Cs τ*, *Z*, *D*inters*<sup>k</sup>*

∇*J n D*intra*<sup>s</sup> k* ð Þ*t*

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

2 ð *T*

0

tions for the simultaneous calculation of *D*intra*<sup>s</sup>*

Using Eqs. (9)–(15), it is possible to calculate *D*intra*<sup>s</sup>*

mining the co-diffusion coefficients.

general form—also as a function of time:

, *D*intras*<sup>k</sup>*

� � �

where *J D*inters*<sup>k</sup>*

, *D*intras*<sup>k</sup>* � � <sup>¼</sup> <sup>1</sup>

> � � � 2 ¼ Ð *T* 0 ∇*J n D*inter*sk* ð Þ*t* h i<sup>2</sup>

The problem of the calculation (9)–(15) is to find unknown functions *D*intras ∈ Ω*t*, *D*inters ∈ Ω*<sup>t</sup>* (*D*intras>0, *D*inters>0, *s* ¼ 1, 2), when absorbed masses *Csk* ð Þþ *t*, *Z Qsk* ð Þ *t*, *Z* satisfy the condition (15) for every point *hk* ⊂ Ω*<sup>k</sup>* of the *k*th layer [8, 12].

Here

$$\mathbf{e}\_{\mathrm{inter}\_{k}} = \frac{\mathbf{e}\_{\mathrm{inter}\_{k}} \mathbf{c}\_{sk}}{\mathbf{e}\_{\mathrm{inter}\_{k}} \mathbf{c}\_{sk} + \left(\mathbf{1} - \mathbf{e}\_{\mathrm{inter}\_{k}}\right) q\_{sk}} \approx \frac{\mathbf{e}\_{\mathrm{inter}\_{k}}}{\left(\mathbf{1} - \mathbf{e}\_{\mathrm{inter}\_{k}}\right) \tilde{\mathbf{K}}\_{sk}},\\\tilde{\mathbf{K}}\_{\mathbb{K}} = \frac{\mathbf{q}\_{\mathbb{K}\infty}}{\mathbf{c}\_{\mathbb{K}\infty}},\end{bmatrix}$$

where *Qs*ð Þ¼ *<sup>t</sup>*, *<sup>Z</sup>* <sup>Ð</sup> 1 0 *Qs*ð Þ *t*,*X*, *Z XdX* is the average concentration of adsorbed

component *s* in micropores and *Ms*ð Þj *t*, *Z hk* is the experimental distribution of the mass of the *s*th component absorbed in macro- and micropores at *hk* ⊂ Ω*<sup>k</sup>* (results of NMR data, **Figure 2**).

#### **4.3 Iterative gradient method of co-diffusion coefficient identification**

The calculation of *D*intra*<sup>s</sup> <sup>k</sup>* and *D*inter*<sup>s</sup> <sup>k</sup>* is a complex mathematical problem. In general, it is not possible to obtain a correct formulation of the problem (9)–(15) and to construct a unique analytical solution, because of the complexity of taking into account all the physical parameters (variation of temperature and pressure, crystallite structures, nonlinearity of Langmuir isotherms, etc.), as well as the insufficient number of reliable experimental data, measurement errors, and other factors.

Therefore, according to the principle of Tikhonov and Arsenin [13], later developed by Lions [14] and Sergienko et al. [15], the calculation of diffusion coefficients requires the use of the model for each iteration, by minimizing the difference between the calculated values and the experimental data.

The calculation of the diffusion coefficients (9)–(15) is reduced to the problem of minimizing the functional of error (16) between the model solution and the experimental data, the solution being refined incrementally by means of a special calculation procedure which uses fast high-performance gradient methods [6, 8, 12, 15].

According to [12, 15], and using the error minimization gradient method for the calculation of *D*intra*<sup>s</sup> <sup>k</sup>* and *D*inter*<sup>s</sup> <sup>k</sup>* of the *s*th diffusing component, we obtain the iteration expression for the *n* + 1th calculation step:

$$D\_{\text{intra}\_{k}}^{n+1}(t) = D\_{\text{intra}\_{k}}^{n}(t) - \nabla f\_{D\_{\text{intra}\_{k}}}^{n}(t)$$

$$\begin{array}{l} \left[\mathbf{C}\_{\text{t}\_{k}}\left(D\_{\text{inter}\_{k}}^{n}, D\_{\text{intra}\_{k}}^{n}; t, h\_{k}\right) + \overline{\mathbf{Q}}\_{\text{s}\_{k}}\left(D\_{\text{inter}\_{k}}^{n}, D\_{\text{intra}\_{k}}^{n}; t, h\_{k}\right) - \mathbf{M}\_{\text{s}\_{k}}(t)\right]^{2}, \\ \left\|\nabla f\_{D\_{\text{inter}\_{k}}}^{n}(t)\right\|\_{2}^{2} + \left\|\nabla f\_{D\_{\text{inter}\_{k}}}^{n}(t)\right\|\_{2}^{2} \\ D\_{\text{inter}\_{k}}^{n+1}(t) = D\_{\text{inter}\_{\text{na}}}^{n}(t) - \nabla f\_{D\_{\text{intra}\_{k}}}^{n}(t) \end{array}$$

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

*∂ <sup>∂</sup><sup>X</sup> Qsk*

*Zeolites - New Challenges*

*Csk* ð Þþ *t*, *Z Qsk*

where *Qs*ð Þ¼ *<sup>t</sup>*, *<sup>Z</sup>* <sup>Ð</sup>

NMR data, **Figure 2**).

[6, 8, 12, 15].

**18**

calculation of *D*intra*<sup>s</sup>*

*Csk Dn* inter*<sup>s</sup> k* , *D<sup>n</sup>* intra*<sup>s</sup> k* ; *t*, *hk*

The calculation of *D*intra*<sup>s</sup>*

layer [8, 12]. Here

ð Þ¼ *t*,*X* ¼ 0, *Z* 0, *Qsk*

Additional condition (NMR-*experimental data*)

*Csk* ð Þþ *t*, *Z Qsk* ð Þ *<sup>t</sup>*, *<sup>Z</sup>* � ��

einter*<sup>k</sup>* <sup>¼</sup> <sup>ε</sup>inter*<sup>k</sup> csk*

1 0

εinter*<sup>k</sup> csk* þ 1 � εinter*<sup>k</sup>*

ð Þ *t*,*X* ¼ 1, *Z*

� *hk*

� �*qsk*

component *s* in micropores and *Ms*ð Þj *t*, *Z hk* is the experimental distribution of the mass of the *s*th component absorbed in macro- and micropores at *hk* ⊂ Ω*<sup>k</sup>* (results of

general, it is not possible to obtain a correct formulation of the problem (9)–(15) and to construct a unique analytical solution, because of the complexity of taking into account all the physical parameters (variation of temperature and pressure, crystallite structures, nonlinearity of Langmuir isotherms, etc.), as well as the insufficient number of reliable experimental data, measurement errors, and other factors.

Therefore, according to the principle of Tikhonov and Arsenin [13], later developed by Lions [14] and Sergienko et al. [15], the calculation of diffusion coefficients requires the use of the model for each iteration, by minimizing the difference

The calculation of the diffusion coefficients (9)–(15) is reduced to the problem of minimizing the functional of error (16) between the model solution and the experimental data, the solution being refined incrementally by means of a

According to [12, 15], and using the error minimization gradient method for the

ðÞ�*t* ∇*J n D*intra*<sup>s</sup> k* ð Þ*t*

� �

*n D*inter*<sup>s</sup> k* ð Þ*t*

� � �

inter*<sup>s</sup> k* , *D<sup>n</sup>* intra*<sup>s</sup> k* ; *t*, *hk*

inter*sm* ðÞ�*t* ∇*J*

*<sup>k</sup>* of the *s*th diffusing component, we obtain the

� *Msk* ð Þ*t*

<sup>2</sup> ,

special calculation procedure which uses fast high-performance gradient methods

int*ras k*

h i<sup>2</sup>

� � �

<sup>þ</sup> *Qsk <sup>D</sup><sup>n</sup>*

� � � 2 þ ∇*J n D*inter*<sup>s</sup> k* ð Þ*t*

**4.3 Iterative gradient method of co-diffusion coefficient identification**

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

between the calculated values and the experimental data.

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

iteration expression for the *n* + 1th calculation step:

� �

*D<sup>n</sup>*þ<sup>1</sup> intra*<sup>s</sup> k*

> ∇*J n D*intra*<sup>s</sup> k* ð Þ*t*

� � �

*D<sup>n</sup>*þ<sup>1</sup> inter*<sup>s</sup> k* ðÞ¼ *<sup>t</sup> <sup>D</sup><sup>n</sup>*

ðÞ¼ *<sup>t</sup> Dn*

The problem of the calculation (9)–(15) is to find unknown functions *D*intras ∈ Ω*t*, *D*inters ∈ Ω*<sup>t</sup>* (*D*intras>0, *D*inters>0, *s* ¼ 1, 2), when absorbed masses

<sup>¼</sup> *Msk* ð Þ *<sup>t</sup>*, *<sup>Z</sup>* �

ð Þ *t*, *Z* satisfy the condition (15) for every point *hk* ⊂ Ω*<sup>k</sup>* of the *k*th

¼ *KsCsk* ð Þ *t*, *Z* ð Þ equilibrium conditions , *Z* ∈ Ω*k*, *k* ¼ 1, *N* þ 1*:*

� *hk*

<sup>≈</sup> <sup>ε</sup>inter*<sup>k</sup>* 1 � εinter*<sup>k</sup>* � �K~*sk*

*Qs*ð Þ *t*,*X*, *Z XdX* is the average concentration of adsorbed

*<sup>k</sup>* is a complex mathematical problem. In

(14)

, *s* ¼ 1, 2; *hk* ∈ Ω*k:* (15)

,K~*sk* <sup>¼</sup> <sup>q</sup>*sk*<sup>∞</sup> c*sk*<sup>∞</sup> ,

$$\frac{\left[\nabla\_{\nu\_{k}}\left(D^{\mathrm{u}}\_{\mathrm{intern}\_{k}},D^{\mathrm{u}}\_{\mathrm{in tra}\_{k}};t,h\_{k}\right)+\overline{\mathsf{Q}}\_{\nu\_{k}}\left(D^{\mathrm{u}}\_{\mathrm{inmet}\_{k}},D^{\mathrm{u}}\_{\mathrm{in tra}\_{k}};t,h\_{k}\right)-\mathsf{M}\_{\flat\_{k}}(t)\right]^{2}}{\left\|\nabla f^{\mathrm{u}}\_{D\_{\mathrm{in tra}\_{k}}}(t)\right\|^{2}+\left\|\nabla f^{\mathrm{u}}\_{D\_{\mathrm{in tra}\_{k}}}(t)\right\|^{2}}, \qquad t\in\left(0,t^{\mathrm{total}}\right)\tag{16}$$

where *J D*inters*<sup>k</sup>* , *D*intras*<sup>k</sup>* � � is the error functional, which describes the deviation of the model solution from the experimental data on *hk* ∈ Ω*k*, which is written as

$$J\left(D\_{\text{intra}\_{k}}, D\_{\text{intra}\_{k}}\right) = \frac{1}{2} \left[ \left( C\_{\text{t}} \left( \tau, Z, D\_{\text{inter}\_{k}}, D\_{\text{intra}\_{k}} \right) + \overline{Q}\_{\text{t}} \left( \mathbf{t}, Z, D\_{\text{inter}\_{k}}, D\_{\text{intra}\_{k}} \right) - M\_{\text{bi}} (\mathbf{t}) \right) \right]\_{h\_{k}}^{2} d\tau,\tag{17}$$
 
$$h\_{k} \in \Omega\_{k}, k = \overline{\mathbf{1}, N+1},\tag{17}$$

$$\begin{aligned} &\left\|\nabla\_{D\_{\mathrm{intr}\_{k}}}^{n}(t),\nabla\_{D\_{\mathrm{intr}\_{k}}}^{n}(t)\right\|\text{the gradients of the error functional},\newline\left(D\_{\mathrm{intr}\_{k}},D\_{\mathrm{intr}\_{k}}\right).\\ &\left\|\nabla f\_{D\_{\mathrm{intr}\_{k}}}^{n}(t)\right\|^{2}=\int\_{0}^{T}\left\|\nabla f\_{D\_{\mathrm{intr}\_{k}}}^{n}(t)\right\|^{2}dt,\left\|\nabla f\_{D\_{\mathrm{intr}\_{k}}}^{n}(t)\right\|^{2}=\int\_{0}^{T}\left[\nabla f\_{D\_{\mathrm{intr}\_{k}}}^{n}(t)\right]^{2}dt.\end{aligned}$$

#### **4.4 Analytical method of co-diffusion coefficient identification**

With the help of iterative gradient methods on the basis of the minimization of the residual functional, very precise and fast analytical methods have been developed making it possible to express the diffusion coefficients in the form of timedependent analytic functions (16). For their efficient use, it is necessary to have an extensive experimental database, with at least two experimental observation conditions for the simultaneous calculation of *D*intra*<sup>s</sup> <sup>k</sup>* and *D*inter*<sup>s</sup> <sup>k</sup>* coefficients. Our experimental studies were carried out for five Z positions of the swept zeolite layer for each of the adsorbed components. The data were not sufficient to fully apply this simultaneous identification method to these five sections. We therefore used a combination of the analytical method and the iterative gradient method for determining the co-diffusion coefficients.

Using Eqs. (9)–(15), it is possible to calculate *D*intra*<sup>s</sup> k* , *D*inter*sk* as a function of time using the experimental data obtained by NMR scanning. In particular, in Eqs. (9) and (10), the co-diffusion coefficients can be set directly as functions of the time *t*: *D*intra*<sup>s</sup> k* ð Þ*t* , *D*inter*sk* ð Þ*t* . In this case, the boundary condition (11) can be given in a more general form—also as a function of time:

$$\mathbf{C}\_{\mathfrak{s}\_1}(t, \mathbf{1}) = \mathbf{C}\_{\mathfrak{s}}^{\text{initial}}(t). \tag{18}$$

Experimental NMR scanning conditions are defined simultaneously for all *P* observation surfaces:

$$\left[\mathbf{C}\_{kl}(t,\mathbf{Z}) + \int\_{0}^{1} \mathbf{Q}\_{kl}(t,\mathbf{X},\mathbf{Z})d\mathbf{X}\right]\_{\mathbf{Z}=h\_{i}} = M\_{ki\_{i}}(t,\mathbf{Z})|\_{h\_{i}}, i = \overline{\mathbf{1},P}, s = \overline{\mathbf{1},\mathbf{Z}}; h\_{i} \in \overset{N+1}{\cup}\boldsymbol{\Omega}\_{k} \tag{19}$$

For simplicity we design *u t*ð Þ¼ , *Z Csk*ð Þ *t*, *Z* , *v t*ð Þ¼ , X, *Z Qsk*ð Þ *t*, X, *Z* ,

$$b(t) = D\_{\text{intra}\_{\text{sk}}}(t) / \mathbb{R}^2, \chi\_i(t) = \mathcal{M}\_{\text{sk}\_i}(t), i = \overline{\mathbf{1}, \mathbf{P}}\_\*$$

and considering Eq. (10) in flat form, its solution can be written as [16]:

$$w(t, \mathbf{X}, \mathbf{Z}) = -\int\_{0}^{t} \mathcal{H}\_{\mathfrak{sl}}^{(2)}(t, \tau, \mathbf{X}, \mathbf{1}) |b(\tau)u(\tau, \mathbf{Z})d\tau| \tag{20}$$

*u t*ð Þ¼ , *hi χi*ð Þ� t

ð*t*

Hð Þ<sup>2</sup>

0

Hð Þ<sup>2</sup>

*b*ð Þ*τ μsP*ð Þ*τ dτ* ¼

ð*t*

0

¼ 1, P

ð*t*

Hð Þ<sup>2</sup>

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

Let us first put *u t*ð Þ¼ , *hP <sup>μ</sup>sP*ðÞ¼ *<sup>t</sup> <sup>C</sup>initial*

<sup>3</sup> ð Þ *t*, *τ*, 1, 1 *b*ð Þ*τ u*ð Þ *τ*, *hi dτ* þ

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

surface, approaching the point of entry into the work area Z = 1.

*τ*

<sup>2</sup> ð Þ *t*, *σ*, 0, 0 *b*ð Þ *σ χ*ð Þ *sP*ð Þ� *σ μsP*ð Þ *σ dσ*

ð *σ* ð1

0 Hð Þ<sup>2</sup>

0

So we obtain the expression for calculating the co-diffusion coefficient in the

*sP*ðÞ�*t μ*<sup>0</sup>

Using calculated *D*intra*sP* ð Þ*t* with the formula (28) on the observation limit *hP*, we define the gradient method *D*inter*sP* ð Þ*t* in the same way. With *D*intra*sP* ð Þ*t* and *D*inter*sP* ð Þ*t* in *hP*, we calculate *<sup>С</sup>sk*ð Þ *<sup>t</sup>*, *hP* , substituting it in *<sup>μ</sup>sP*�<sup>1</sup>ðÞ¼ *<sup>t</sup> <sup>С</sup>sk*ð Þ *<sup>t</sup>*, *hP* for the next

<sup>2</sup> ð Þ *t*, *τ*, 0, 0Þ *μ*<sup>0</sup>

Hð Þ<sup>2</sup>

*sP*ð Þ*t*

*sP*ð Þ*τ dτ* � *μsP*ð Þ*t*

0

Differentiating Eq. (26) by *t*, after the transformations series, we obtain

*<sup>b</sup>*ð ÞH*<sup>σ</sup>* ð Þ<sup>2</sup>

<sup>3</sup> ð Þ *t*, *τ*, 1, 1 *b*ð Þ*τ μsP*ð Þ*t dτ* ¼ *χsP*ðÞ�*t μsP*ðÞþ*t*

obtained by Ivanchov [16], and taking into account.

<sup>4</sup> ð Þ *t*, *σ*; 0, 0 , we obtain

<sup>2</sup> ð Þ *t*, *σ*, 0, 0 *b*ð Þ *σ dσ*

ð*t*

ð 1

Hð Þ<sup>2</sup>

*<sup>s</sup>* ð Þ*t* , where *Z* ¼ *hP* is the observation

<sup>3</sup> ð Þ *t*, *τ*, X, 1 *b*ð Þ*τ u*ð Þ *τ*, *hi dXdτ*, *i*

<sup>3</sup> ð Þ *t*, *τ*, x, 1 *b*ð Þ*τ μsP*ð Þ*t dxdτ*

<sup>4</sup> ð Þ *t*, *σ*, 0, 0Þ *dσ* ¼ 1,

<sup>3</sup> ð Þ *σ*, *τ*, x, 1 *b*ð Þ*τ μsP*ð Þ*t dXdτ*, *t*∈ 0, *t*

<sup>4</sup> ð Þ *t*, *σ*, 0, 0Þ *b*ð Þ *σ* , the integration

*sP*ð ÞÞ *τ dτ:*

, *t*∈ 0, *t*

*total* � � (28)

(24)

(25)

*total* � �

(26)

0

ð*t*

ð 1

Hð Þ<sup>2</sup>

0

<sup>2</sup> ð ÞH *<sup>t</sup>*, *<sup>σ</sup>*, 0, 0<sup>Þ</sup> ð Þ<sup>2</sup>

<sup>2</sup> ð Þ *t*, *σ*, 0, 0 *b*ð Þ *σ μsP*ð Þþ *σ χ* ð Þ *sP*ð Þ� *σ μsP*ð Þ *σ dσ* (27)

<sup>2</sup> ð Þ *t*, *τ*, 0, 0Þ *μ*<sup>0</sup>

0

0

0

Then Eq. (24) for *i* = *P* will be

Applying to Eq. (25) the formula <sup>Ð</sup>*<sup>t</sup>*

<sup>3</sup> ð Þ¼H *<sup>t</sup>*, *<sup>σ</sup>*; 1, 1 ð Þ<sup>2</sup>

ð*t*

Hð Þ<sup>2</sup>

0

þ ð*t*

*μsP*ðÞ¼ *t*

by *τ* and the differentiation by *t*

intra-crystallite space:

*<sup>D</sup>*int*rasP* ðÞ� *<sup>t</sup> <sup>R</sup>*<sup>2</sup>

**21**

0 Hð Þ<sup>2</sup>

ð*t*

Hð Þ<sup>2</sup>

After multiplying Eq. (27) on the expression Hð Þ<sup>2</sup>

*b t*ð Þ*μsP*ðÞþ*<sup>t</sup> <sup>χ</sup>sP*ð Þ� *<sup>σ</sup> <sup>μ</sup>sP*ð Þ¼ *<sup>σ</sup> b t*ð Þ <sup>ð</sup>*<sup>t</sup>*

*b t*ðÞ¼ *<sup>R</sup>*<sup>2</sup> *<sup>χ</sup>*<sup>0</sup>

Ð*t* 0 Hð Þ<sup>2</sup>

0

where Hð Þ<sup>2</sup> <sup>4</sup>*<sup>ξ</sup>* ð Þ¼� *<sup>t</sup>*, *<sup>τ</sup>*, X, *<sup>ξ</sup>* <sup>2</sup> <sup>P</sup><sup>∞</sup> *m*¼0 *e*�*η*<sup>2</sup> *<sup>m</sup>*ð Þ *<sup>θ</sup>*2ð Þ�<sup>t</sup> *<sup>θ</sup>*2ð Þ*<sup>τ</sup> <sup>η</sup><sup>m</sup>* cos *<sup>η</sup>mX* � �ð Þ<sup>1</sup> *<sup>m</sup>*.

Here the Green influence function of the particle Hð Þ<sup>2</sup> *<sup>k</sup>* , *k* ¼ 1, 4 is used; it has the form [17].

$$\begin{split} \mathcal{H}\_{4}^{(2)}(t,\tau,\mathbf{X},\boldsymbol{\xi}) &= 2\sum\_{m=0}^{\infty} e^{-\eta\_{m}^{2}(\theta\_{2}(\mathbf{t})-\theta\_{2}(\tau))} \cos \eta\_{m} \mathbf{X} \cos \eta\_{m} \boldsymbol{\xi}, \boldsymbol{\eta}\_{m} = \frac{2m+1}{2}\pi, \\ \mathcal{H}\_{3}^{(2)}(t,\tau,\mathbf{X},\boldsymbol{\xi}) &= 2\sum\_{m=0}^{\infty} e^{-\eta\_{m}^{2}(\theta\_{2}(\mathbf{t})-\theta\_{2}(\tau))} \sin \eta\_{m} \mathbf{X} \sin \eta\_{m} \boldsymbol{\xi}, \boldsymbol{\eta}\_{m} = \frac{2m+1}{2}\pi, \\ \mathcal{H}\_{2}^{(2)}(t,\tau,\mathbf{X},\boldsymbol{\xi}) &= \mathbf{1} + 2\sum\_{m=1}^{\infty} e^{-\eta\_{m}^{2}(\theta\_{2}(\mathbf{t})-\theta\_{2}(\tau))} \cos \eta\_{m} \mathbf{X} \cdot \cos \eta\_{m} \boldsymbol{\xi}, \boldsymbol{\eta}\_{m} = m\pi, \end{split}$$

where *<sup>θ</sup>*2ðÞ¼ *<sup>t</sup>* <sup>Ð</sup>*<sup>t</sup>* 0 *b s*ð Þ*ds*.

The notation <sup>H</sup>ð Þ<sup>2</sup> <sup>4</sup>*τξ*, Hð Þ<sup>2</sup> <sup>4</sup>*ξξ* means partial derivatives of the influence function Hð Þ<sup>2</sup> 4 relative to the definite variables *τ* and *ξ*, respectively.

Based on formula (20), we calculate

$$v\_X(t, \mathbf{X}, \mathbf{Z}) = -\int\_0^t \mathcal{H}\_{4\xi X}^{(2)}(t, \tau, \mathbf{X}, \mathbf{1}) b(\tau) u(\tau, \mathbf{Z}) d\tau \tag{21}$$

Integrating parts (21), taking into account the relations.

$$\mathcal{H}\_{4X}^{(2)}(\mathbf{t},\tau,\mathbf{X},\xi) = -\mathcal{H}\_{3\xi}^{(2)}(\mathbf{t},\tau,\mathbf{X},\xi),\\\mathcal{H}\_{3\tau}^{(2)}(\mathbf{t},\tau,\mathbf{X},\xi) = -b(\tau)\mathcal{H}\_{3\xi\xi}^{(2)}(\mathbf{t},\tau,\mathbf{X},\xi),$$

and the initial condition *u*j*t*¼<sup>0</sup> ¼ 0, we find

$$w\_X(t, \mathbf{X}, \mathbf{Z}) = \int\_0^t \mathcal{H}\_3^{(2)}(t, \tau, \mathbf{X}, \mathbf{1}) u\_\tau(\tau, \mathbf{Z}) d\tau \tag{22}$$

We substitute the expression *v t*ð Þ , X, Z (20) in the observation conditions (19):

$$u(t, h\_i) - \int\_0^1 \mathbf{X} d\mathbf{X} \int\_0^t \mathcal{H}\_{4\xi}^{(2)}(t, \tau, \mathbf{X}, \mathbf{1}) b(\tau) u(\tau, h\_i) d\tau = \chi\_i(\mathbf{t}), i = \overline{\mathbf{1}, \mathbf{P}} \tag{23}$$

Integrating parts (23) and taking into account equality

$$
\mathcal{H}\_{4\xi}^{(2)}(t,\tau,X,\mathbf{1}) = -\mathcal{H}\_{3\mathbf{X}}^{(2)}(t,\tau,X,\mathbf{1}),
$$

we obtain [16]

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

$$\begin{split} u(\mathbf{t}, h\_i) &= \chi\_i(\mathbf{t}) - \int\_0^t \mathcal{H}\_3^{(2)}(\mathbf{t}, \tau, \mathbf{1}, \mathbf{1}) b(\tau) u(\tau, h\_i) d\tau + \int\_0^t \int\_0^1 \mathcal{H}\_3^{(2)}(\mathbf{t}, \tau, \mathbf{X}, \mathbf{1}) b(\tau) u(\tau, h\_i) d\mathbf{X} d\tau, \text{in} \\ &= \overline{\mathbf{1}, \mathbf{P}} \end{split} \tag{24}$$

Let us first put *u t*ð Þ¼ , *hP <sup>μ</sup>sP*ðÞ¼ *<sup>t</sup> <sup>C</sup>initial <sup>s</sup>* ð Þ*t* , where *Z* ¼ *hP* is the observation surface, approaching the point of entry into the work area Z = 1. Then Eq. (24) for *i* = *P* will be

$$\int\_{0}^{t} \mathcal{H}\_{3}^{(2)}(t,\tau,\mathbf{1},\mathbf{1}) b(\tau) \mu\_{\mathcal{P}}(t) d\tau = \chi\_{\mathcal{P}}(t) - \mu\_{\mathcal{P}}(t) + \int\_{0}^{t} \left[ \mathcal{H}\_{3}^{(2)}(t,\tau,\mathbf{x},\mathbf{1}) b(\tau) \mu\_{\mathcal{P}}(t) d\mathbf{x} d\tau + \int\_{0}^{t} \mathcal{H}\_{3}^{(2)}(t,\tau,\mathbf{x},\mathbf{1}) b(\tau) \mu\_{\mathcal{P}}(t) d\mathbf{x} d\tau + \int\_{0}^{t} \mathcal{H}\_{3}^{(2)}(t,\tau,\mathbf{x},\mathbf{1}) b(\tau) \mu\_{\mathcal{P}}(t) d\tau \right] \tag{25}$$

Applying to Eq. (25) the formula <sup>Ð</sup>*<sup>t</sup> τ <sup>b</sup>*ð ÞH*<sup>σ</sup>* ð Þ<sup>2</sup> <sup>2</sup> ð ÞH *<sup>t</sup>*, *<sup>σ</sup>*, 0, 0<sup>Þ</sup> ð Þ<sup>2</sup> <sup>4</sup> ð Þ *t*, *σ*, 0, 0Þ *dσ* ¼ 1, obtained by Ivanchov [16], and taking into account. Hð Þ<sup>2</sup> <sup>3</sup> ð Þ¼H *<sup>t</sup>*, *<sup>σ</sup>*; 1, 1 ð Þ<sup>2</sup> <sup>4</sup> ð Þ *t*, *σ*; 0, 0 , we obtain

$$\begin{split} \int\_{0}^{t} b(\tau) \mu\_{iP}(\tau) d\tau &= \int\_{0}^{t} \mathcal{H}\_{2}^{(2)}(t, \sigma, \mathbf{0}, \mathbf{0}) b(\sigma) (\chi\_{iP}(\sigma) - \mu\_{iP}(\sigma)) d\sigma \\ &+ \left\| \int\_{0}^{t} \mathcal{H}\_{2}^{(2)}(t, \sigma, \mathbf{0}, \mathbf{0}) b(\sigma) d\sigma \right\| \left\| \int\_{0}^{1} \mathcal{H}\_{3}^{(2)}(\sigma, \tau, \mathbf{x}, \mathbf{1}) b(\tau) \mu\_{iP}(\mathbf{t}) d\mathbf{X} d\tau, t \in \left[ 0, t^{total} \right] \right\| \end{split} \tag{26}$$

Differentiating Eq. (26) by *t*, after the transformations series, we obtain

$$\mu\_{sP}(t) = \int\_0^t \mathcal{H}\_2^{(2)}(t, \sigma, \mathbf{0}, \mathbf{0}) (b(\sigma)\mu\_{sP}(\sigma) + \chi\_{sP}(\sigma) - \mu\_{sP}(\sigma)) d\sigma \tag{27}$$

After multiplying Eq. (27) on the expression Hð Þ<sup>2</sup> <sup>4</sup> ð Þ *t*, *σ*, 0, 0Þ *b*ð Þ *σ* , the integration by *τ* and the differentiation by *t*

$$b(t)\mu\_{s\mathbb{P}}(t) + \chi\_{s\mathbb{P}}(\sigma) - \mu\_{s\mathbb{P}}(\sigma) = b(t) \int\_0^t \mathcal{H}\_2^{(2)}(t, \tau, \mathbf{0}, \mathbf{0}) (\mu\_{s\mathbb{P}}'(\tau)) d\tau.$$

So we obtain the expression for calculating the co-diffusion coefficient in the intra-crystallite space:

$$D\_{\text{intr}\_{\text{P}}}(t) \equiv R^2 b(t) = R^2 \frac{\chi\_{\text{sP}}'(t) - \mu\_{\text{sP}}'(t)}{\int \mathcal{H}\_2^{(2)}(t, \tau, 0, 0) )\mu\_{\text{sP}}'(\tau)d\tau - \mu\_{\text{sP}}(t)}, t \in \{0, t^{total}\} \tag{28}$$

Using calculated *D*intra*sP* ð Þ*t* with the formula (28) on the observation limit *hP*, we define the gradient method *D*inter*sP* ð Þ*t* in the same way. With *D*intra*sP* ð Þ*t* and *D*inter*sP* ð Þ*t* in *hP*, we calculate *<sup>С</sup>sk*ð Þ *<sup>t</sup>*, *hP* , substituting it in *<sup>μ</sup>sP*�<sup>1</sup>ðÞ¼ *<sup>t</sup> <sup>С</sup>sk*ð Þ *<sup>t</sup>*, *hP* for the next

*b t*ðÞ¼ *<sup>D</sup>*intra*sk* ð Þ*<sup>t</sup> <sup>=</sup>*R<sup>2</sup>

*v t*ð Þ¼� , X, Z

<sup>4</sup>*<sup>ξ</sup>* ð Þ¼� *<sup>t</sup>*, *<sup>τ</sup>*, X, *<sup>ξ</sup>* <sup>2</sup> <sup>P</sup><sup>∞</sup>

*m*¼0

*m*¼0

<sup>4</sup>*τξ*, Hð Þ<sup>2</sup>

Based on formula (20), we calculate

<sup>4</sup>*<sup>X</sup>*ð Þ ¼ �H *<sup>t</sup>*, *<sup>τ</sup>*, X, *<sup>ξ</sup>* ð Þ<sup>2</sup>

*e*�*η*<sup>2</sup>

*e*�*η*<sup>2</sup>

*m*¼1

relative to the definite variables *τ* and *ξ*, respectively.

*vX*ð Þ¼� *t*, X, Z

and the initial condition *u*j*t*¼<sup>0</sup> ¼ 0, we find

*vX*ð Þ¼ *t*, X, Z

ð*t*

Hð Þ<sup>2</sup>

0

<sup>3</sup>*<sup>ξ</sup>* ð Þ *<sup>t</sup>*, *<sup>τ</sup>*,*X*, *<sup>ξ</sup>* , Hð Þ<sup>2</sup>

ð*t*

Hð Þ<sup>2</sup>

We substitute the expression *v t*ð Þ , X, Z (20) in the observation conditions (19):

0

<sup>4</sup>*<sup>ξ</sup>* ð Þ ¼ �H *<sup>t</sup>*, *<sup>τ</sup>*, *<sup>X</sup>*, 1 ð Þ<sup>2</sup>

Integrating parts (21), taking into account the relations.

*e*�*η*<sup>2</sup>

where Hð Þ<sup>2</sup>

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<sup>4</sup> ð Þ¼ *<sup>t</sup>*, *<sup>τ</sup>*, X, *<sup>ξ</sup>* <sup>2</sup> <sup>P</sup><sup>∞</sup>

<sup>3</sup> ð Þ¼ *<sup>t</sup>*, *<sup>τ</sup>*, X, *<sup>ξ</sup>* <sup>2</sup> <sup>P</sup><sup>∞</sup>

0

The notation <sup>H</sup>ð Þ<sup>2</sup>

<sup>2</sup> ð Þ¼ *<sup>t</sup>*, *<sup>τ</sup>*, X, *<sup>ξ</sup>* <sup>1</sup> <sup>þ</sup> <sup>2</sup> <sup>P</sup><sup>∞</sup>

*b s*ð Þ*ds*.

form [17].

Hð Þ<sup>2</sup>

Hð Þ<sup>2</sup>

Hð Þ<sup>2</sup>

where *<sup>θ</sup>*2ðÞ¼ *<sup>t</sup>* <sup>Ð</sup>*<sup>t</sup>*

Hð Þ<sup>2</sup>

*u t*ð Þ� , *hi*

we obtain [16]

**20**

ð 1 *XdX* ð*t*

0

Hð Þ<sup>2</sup>

Hð Þ<sup>2</sup>

Integrating parts (23) and taking into account equality

0

, *χi*ðÞ¼ *t Mski*

and considering Eq. (10) in flat form, its solution can be written as [16]:

Hð Þ<sup>2</sup>

ð*t*

0

*e*�*η*<sup>2</sup>

*m*¼0

Here the Green influence function of the particle Hð Þ<sup>2</sup>

ð Þ*t* , *i* ¼ 1, P,

<sup>4</sup>*<sup>ξ</sup>* ð Þ *t*, *τ*, X, 1Þ *b*ð Þ*τ u*ð Þ *τ*, Z *dτ* (20)

*<sup>k</sup>* , *k* ¼ 1, 4 is used; it has the

4

<sup>2</sup> *π*,

<sup>2</sup> *<sup>π</sup>*,

*<sup>m</sup>*ð Þ *<sup>θ</sup>*2ð Þ�<sup>t</sup> *<sup>θ</sup>*2ð Þ*<sup>τ</sup> <sup>η</sup><sup>m</sup>* cos *<sup>η</sup>mX* � �ð Þ<sup>1</sup> *<sup>m</sup>*.

*<sup>m</sup>*ð Þ *<sup>θ</sup>*2ð Þ�<sup>t</sup> *<sup>θ</sup>*2ð Þ*<sup>τ</sup>* cos *<sup>η</sup>mX* cos *<sup>η</sup>mξ*, *<sup>η</sup><sup>m</sup>* <sup>¼</sup> <sup>2</sup>*m*þ<sup>1</sup>

*<sup>m</sup>*ð Þ *<sup>θ</sup>*2ð Þ�<sup>t</sup> *<sup>θ</sup>*2ð Þ*<sup>τ</sup>* sin *<sup>η</sup>mX* sin *<sup>η</sup>mξ*, *<sup>η</sup><sup>m</sup>* <sup>¼</sup> <sup>2</sup>*<sup>m</sup>* <sup>þ</sup> <sup>1</sup>

*<sup>m</sup>*ð Þ *<sup>θ</sup>*2ð Þ�<sup>t</sup> *<sup>θ</sup>*2ð Þ*<sup>τ</sup>* cos *<sup>η</sup>mX* � cos *<sup>η</sup>mξ*, *<sup>η</sup><sup>m</sup>* <sup>¼</sup> *<sup>m</sup>π*,

<sup>4</sup>*ξξ* means partial derivatives of the influence function Hð Þ<sup>2</sup>

<sup>3</sup>*<sup>τ</sup>* ð Þ¼� *<sup>t</sup>*, *<sup>τ</sup>*,*X*, *<sup>ξ</sup> <sup>b</sup>*ð ÞH*<sup>τ</sup>* ð Þ<sup>2</sup>

<sup>4</sup>*<sup>ξ</sup>* ð Þ *t*, *τ*, X, 1 *b*ð Þ*τ u*ð Þ *τ*, *hi dτ* ¼ *χi*ð Þt , *i* ¼ 1, P (23)

<sup>3</sup>*<sup>X</sup>*ð Þ *t*, *τ*, *X*, 1 ,

<sup>4</sup>*ξX*ð Þ *t*, *τ*, X, 1 *b*ð Þ*τ u*ð Þ *τ*, Z *dτ* (21)

<sup>3</sup> ð Þ *t*, *τ*, X, 1Þ *uτ*ð Þ *τ*, Z *dτ* (22)

<sup>3</sup>*ξξ*ð Þ *t*, *τ*, *X*, *ξ* ,

coefficient *D*inter*si*ð Þ*t* , *i* ¼ *P* � 1, 1 calculations. All subsequent coefficients *D*int*rasi*ð Þ*t* will be calculated by the formula

$$D\_{\text{intr}\_{\text{si}}}(t) \equiv R^2 b\_{\text{si}}(t) = R^2 \frac{\chi\_{\text{si}}'(t) - \mu\_{\text{si}}'(t)}{\int \mathcal{H}\_2^{(2)}(t, \tau, \mathbf{0}, \mathbf{0})) \mu\_{\text{si}}'(\tau) d\tau - \mu\_{\text{si}}(t)} \tag{29}$$

**Figure 5.**

**Figure 6.**

**23**

*time and at different positions in the bed.*

*Variation of the inter-crystallite concentration (a.u.) calculated for benzene (left) and hexane (right) against*

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

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

*Distribution of the benzene (left) and hexane (right) concentrations in the intra-crystallite space from the surface (abscissa 1) to the center (abscissa 0) of the crystallites, at different times: (1) dark blue, t = 25 min;*

*(2) green, t = 50 min; (3) brown, t = 100 min; and (4) red, t = 200 min.*

with parallel computing *D*inter*si*ð Þ*t* , *i* ¼ *P* � 1, 1.
