**5. Numerical simulation and analysis: co-diffusion coefficients: concentration profiles in inter- and intra-crystallite spaces**

The variation against time of the benzene and hexane intra-crystallite diffusion coefficients, *D*intra1,k аnd *D*intra2,k , respectively, is presented in **Figure 3** for the five coordinate positions: 6, 8, 10, 12, and 14 mm, defined now from the top of the bed. The curves for benzene *D*intra1,k are pseudo exponentials. *D*intra1,k decreases from 9.0

*Variation of intra-crystallite diffusion coefficients (arbitrary units) for benzene Dintra1*,*<sup>k</sup> (left) and hexane Dintra2*,*<sup>k</sup> (right) against time, at different positions in the bed. (Top) time range 6–240 mn, (bottom) time range 100–240 mn.*

#### **Figure 4.**

*Variation of inter-crystallite diffusion coefficients (a.u.) for benzene (left) and hexane (right) against time at different positions in the bed.*

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

**Figure 5.**

coefficient *D*inter*si*ð Þ*t* , *i* ¼ *P* � 1, 1 calculations. All subsequent coefficients *D*int*rasi*ð Þ*t*

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

**5. Numerical simulation and analysis: co-diffusion coefficients: concentration profiles in inter- and intra-crystallite spaces**

*si*ðÞ�*t μ*<sup>0</sup>

The variation against time of the benzene and hexane intra-crystallite diffusion coefficients, *D*intra1,k аnd *D*intra2,k , respectively, is presented in **Figure 3** for the five coordinate positions: 6, 8, 10, 12, and 14 mm, defined now from the top of the bed. The curves for benzene *D*intra1,k are pseudo exponentials. *D*intra1,k decreases from 9.0

*Variation of intra-crystallite diffusion coefficients (arbitrary units) for benzene Dintra1*,*<sup>k</sup> (left) and hexane Dintra2*,*<sup>k</sup> (right) against time, at different positions in the bed. (Top) time range 6–240 mn, (bottom) time range*

*Variation of inter-crystallite diffusion coefficients (a.u.) for benzene (left) and hexane (right) against time at*

*si*ð Þ*t*

*si*ð Þ*τ dτ* � *μsi*ð Þ*t*

, *i* ¼ *P* � 1, 1 (29)

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

Ð*t* 0 Hð Þ<sup>2</sup>

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

will be calculated by the formula

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

*Zeolites - New Challenges*

**Figure 3.**

**Figure 4.**

**22**

*different positions in the bed.*

*100–240 mn.*

*Variation of the inter-crystallite concentration (a.u.) calculated for benzene (left) and hexane (right) against time and at different positions in the bed.*

#### **Figure 6.**

*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.*

E�13 to about 1.0 E�14 a.u. (equilibrium) depending on the position of the crystallite and the time, as well as on the amount of adsorbed gas. The shapes of the variations of *D*intra2,k for hexane are roughly the same, but the diffusion coefficients are higher, from about 9.0 E�12 to 3.0 E�13 a.u.

Dinter diffusion coefficient in macropores, m<sup>2</sup>

Dintra diffusion coefficient in micropores, m<sup>2</sup>

<sup>Δ</sup>*<sup>l</sup>* <sup>¼</sup> *lk* � *lk*�1; *<sup>k</sup>* <sup>¼</sup> 1, *<sup>N</sup>* <sup>þ</sup> <sup>1</sup> layer thickness (all layers have the same thickness)

q<sup>∞</sup> equilibrium adsorbate concentration in micropores *Q* = *q*/q<sup>∞</sup> dimensionless concentration of adsorbate in micro-

*T* temperature of gas phase flow, K, and time total, s

Λ coefficient of thermal diffusion along the columns

*K*~ adsorption equilibrium constant

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

L dimensionless bed length (L = 1) *q* adsorbate concentration in micropores

pores

μ molecular mass of adsorbate, kg/mol *H* total heat capacity of the adsorbent and gas, kJ/(kg K)

<sup>Δ</sup>*Hi* activation energy (Δ*Hi* <sup>¼</sup> <sup>Δ</sup>*Hi=μ*), kJ/mol

*x* distance from crystallite center, m

*total* total duration of co-adsorption, mn *Lk* dimensionless position of the *k*th layer

*k*0*<sup>i</sup>* empirical equilibrium coefficient for the *i* adsorbate,

R mean crystallite radius, m (we assume that the

*X* = x/R dimensionless distance from crystallite center *z* distance from the bottom of the bed for mathematical simulation, m *Z=z*/*l* dimensionless distance from the bottom of the bed

crystallites are spherical)

depending on the adsorbent properties and the diffusing adsorbate component (*k*0*<sup>i</sup>* equal to the ratio of the desorption and adsorption rate constants)

*u* velocity of gas phase flow, m/s2

*hg* gas heat capacity, kJ/(kg K)

*α<sup>h</sup>* heat transfer coefficient *Rcolumn* column radius, m

Δ*Hi* adsorption heat, kJ/kg

*τ*, *ξ* variables of integration

εinter inter-crystallite bed porosity einter value utilized in Eq. (9)

*n* iteration number of identification m number of adsorbed components P number of NMR observation surfaces s index of adsorbate component i index of NMR observation surface initial index of initial value (concentrations, temperature) macro index of extended Lagrange functional component for inter-crystallite space micro index of extended Lagrange functional component for intra-crystallite space

t time

*hk* (*Lk* –*Lk*-1)/2

*t*

**25**

*Rg* gas constant, kJ mol/(m<sup>3</sup> K)

*l* bed length, m

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

M mass total

/s

/s

**Figure 4** presents the variation against time of the benzene and hexane diffusion coefficients in inter-crystallite space, *D*inter1,k аnd *D*inter2,k , for the same positions. These coefficients decrease with time from 2.0 E�6 to 1.0 E�7 a.u. (equilibrium) for benzene and from 3 E�5 to 1.0 E�6 a.u. for hexane, depending on the bed position, and increase adsorbed concentrations.

**Figure 5** shows the variation against time of the calculated concentrations C for benzene and hexane in the inter-crystallite space. As can be seen, these concentrations approach the equilibrium values for a diffusion time around 250 min. But the variations of the concentrations with time are rather different for the two gases.

**Figure 6** shows the variation of the concentrations *Q*(*t*,*X*,*z*) of adsorbed benzene (left) and hexane (right) in the micropores of the intra-crystallite space from the surface (abscissa-1) to the center (abscissa-0) of the crystallites located between 6 and 14 mm from the top of the bed and after 25–200 min. of diffusion (a, b, c, and d, respectively). The gradients increase, and the mean concentrations decrease with the increasing distance of the particles from the arrival of the gases. The particles at 6 and 8 mm are saturated with benzene after 100 min, but not yet with hexane.

## **6. Conclusion**

The main result of this work is the possibility, from a single experiment, of simultaneously distributing several co-diffusing gases in a porous solid and of using the methods of mathematical modeling to analyze for each of them the distribution of their concentrations in the intra- and inter-crystallite spaces.

Using the experimental NMR data and proposed co-adsorption models, the identification procedures for calculating the co-diffusion coefficients for two or more components in intra- and inter-crystallite spaces are developed. These procedures use the iterative gradual identification methods on minimizing of the Lagrange error function and rapid analytic methods based on the influence function. The co-diffusion coefficients were obtained as a function of time for different positions along the catalyst bed. In particular, those in the intra-crystallite space were computed by the analytical method which allowed a calculation with a relatively high degree of discretization over time and to reduce practically twice the volume of iterative calculations. Using these results, the concentrations of codiffusing benzene and hexane in the inter- and intra-crystallite spaces were calculated for each time and each position in the bed.

### **Nomenclature**


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


E�13 to about 1.0 E�14 a.u. (equilibrium) depending on the position of the crystallite and the time, as well as on the amount of adsorbed gas. The shapes of the variations of *D*intra2,k for hexane are roughly the same, but the diffusion coefficients

**Figure 4** presents the variation against time of the benzene and hexane diffusion coefficients in inter-crystallite space, *D*inter1,k аnd *D*inter2,k , for the same positions. These coefficients decrease with time from 2.0 E�6 to 1.0 E�7 a.u. (equilibrium) for benzene and from 3 E�5 to 1.0 E�6 a.u. for hexane, depending on the bed

**Figure 5** shows the variation against time of the calculated concentrations C

**Figure 6** shows the variation of the concentrations *Q*(*t*,*X*,*z*) of adsorbed benzene (left) and hexane (right) in the micropores of the intra-crystallite space from the surface (abscissa-1) to the center (abscissa-0) of the crystallites located between 6 and 14 mm from the top of the bed and after 25–200 min. of diffusion (a, b, c, and d, respectively). The gradients increase, and the mean concentrations decrease with the increasing distance of the particles from the arrival of the gases. The particles at 6 and 8 mm are saturated with benzene after 100 min, but not yet with hexane.

The main result of this work is the possibility, from a single experiment, of simultaneously distributing several co-diffusing gases in a porous solid and of using the methods of mathematical modeling to analyze for each of them the distribution

Using the experimental NMR data and proposed co-adsorption models, the identification procedures for calculating the co-diffusion coefficients for two or more components in intra- and inter-crystallite spaces are developed. These procedures use the iterative gradual identification methods on minimizing of the Lagrange error function and rapid analytic methods based on the influence function. The co-diffusion coefficients were obtained as a function of time for different positions along the catalyst bed. In particular, those in the intra-crystallite space were computed by the analytical method which allowed a calculation with a relatively high degree of discretization over time and to reduce practically twice the volume of iterative calculations. Using these results, the concentrations of codiffusing benzene and hexane in the inter- and intra-crystallite spaces were calcu-

*<sup>k</sup>* <sup>¼</sup> 1, *<sup>N</sup>* <sup>þ</sup> <sup>1</sup> layer number, subscript *<sup>k</sup>* will be added to all the

*c* adsorbate concentration in macropores

*C* = *c*/c<sup>∞</sup> dimensionless adsorbate concentration in macropores

teristic of the *k*th layer

c<sup>∞</sup> equilibrium adsorbate concentration in macropores

following symbols to specify that they are charac-

of their concentrations in the intra- and inter-crystallite spaces.

lated for each time and each position in the bed.

for benzene and hexane in the inter-crystallite space. As can be seen, these concentrations approach the equilibrium values for a diffusion time around 250 min. But the variations of the concentrations with time are rather different for

are higher, from about 9.0 E�12 to 3.0 E�13 a.u.

position, and increase adsorbed concentrations.

the two gases.

*Zeolites - New Challenges*

**6. Conclusion**

**Nomenclature**

**24**
