The theoretical bases of mathematical modeling of nonisothermal adsorption and desorption in nanoporous Catalyst Zeolite for nonlinear Langmuir's isotherm are given. An effective linearization scheme for the nonlinear model is realized. High-speed analytical solutions of the system of linearized boundary-value problems of adsorption and desorption in nanoporous media are substantiated and obtained.
The quality of mathematical models for adsorption and
desorption processes of hydrocarbons in nanoporous catalytic
media determines the effectiveness of technological solutions
for neutralizing and reducing exhaust emissions of internal
combustion engines, the number of which is rapidly
increasing, contributing to global warming crisis. [
At present, many experimental and theoretical studies of
such processes are conducted, especially studies on the
improvement of mathematical models, taking into account the
influence of various factors that limit the internal kinetics of
adsorption and desorption in nanopores of catalytic media. A
detailed analysis of these works was made in [
This paper describes the theoretical foundations for modeling non-isothermal adsorption and desorption in nanoporous catalysts for a nonlinear isotherm obtained by the American physicist, Nobel Prize winner Erwin Langmuir, who most fully determines the mechanism of adsorption equilibrium for micro- and nanoporous systems of ZSM-5 zeolites.
II. DESCRIPTION OF KINETIC PROCESSES AND
MATHEMATICAL MODEL
A general description of the interaction of a diffusing gas
stream in a biporous pore system of a catalytic medium of
nanoporous particles, taking into account the main limiting
factors of internal kinetics of mass transfer, including the
interaction of micro- and macro transfer, is given in [
The main hypothesis assumed for the model is adsorption
interaction between adsorbent molecules and active adsorption
centers on the phase separation surface in micro- and
nanopores of crystallites is determined on the basis of
Langmuir's non-linear adsorption equilibrium function taking
into account the physical prerequisites [
Proceeding from these, the adsorption equilibrium function
(adsorption isotherm) of Lengmuir type, which describes the
adsorbent phase transition from gas flow to the micro- and
nanopores of catalytic medium, will be determined by a
nonlinear dependence establishing relationship between
equilibrium concentration and adsorption value [
Here a full , 0 < b < 1 are the empirical coefficients that depend on properties of nanoporous medium and diffused substance: a full - the concentration (amount) of adsorbate in micro- and nanopores of zeolite with complete filling of the adsorption centers.
The refined kinetics of nonisothermal adsorption and
desorption for exhaust gas neutralization systems in
nanoporous catalysts, taking into account the nonlinear
function of adsorption equilibrium and the given physical
justifications, can be described by the following system of
nonlinear partial differential equations [
=β c − 1 a
) .
b a full − a with initial conditions: a) adsorption c(t, z) |t=o = 0 ,
T (t, z) |t=o = T0 , and boundary condition:
a) adsorption c(t, z) |z=o = cin , ∂ ∂z T (t, z) |z=0 = Tin , T (t, z) |z=0 = Tin (t) , c(t, z) |t=o = c0 , T (t, z) |t=o = T0 , a a full Taking into account that < 1 , the Maclaurin’s series, we obtain:
1 a / a full ceq (a) ≡ ϕ (a) = b 1− a / a full
≈ γ a (t, z ) + ε a2 (t, z ) , (9) 1 ba full where γ =
is adsorption constant, which describes linear
component of the adsorption equilibrium function ceq (a)
(according to Henry's law), ε =
- is a small
1
b(a full )
2
parameter that takes into account the nonlinear component of
the adsorption isotherm.
( T! < T2 < T3 < T4 ) [
Substituting the expanded expression (9) instead of the
dependence in the third equation of system (3), we obtain
As a result of the substitution of asymptotic sums (11) into
equations (2) and taking into account (10), the initial nonlinear
problem (2) - (8) is parallelized into two types of linear
problems [
The problem A0 (zero approximation): to find a solution for
the system of partial differential equations:
∂a
∂t
=β (c −γ a(z, t) −ε a2 (z, t))
(10)
III. THE LINEARIZATION OF A NONLINEAR MODEL
The problem (2) - (8), taking into account the approximated
kinetic equation of phase transformation (10) containing a
small parameter ε , is a mixed boundary-value problem for a
nonlinear system of second-order partial differential equations.
The solution of problem (2) - (8) will be obtained using
asymptotic expansions in small parameter ε in the form of
following power series [
∂2c0 , =Dint er ∂z2
2 − uhg ∂∂Tz0 − Q ∂∂at0 − Χ2T0 + Λ ∂∂zT20 =0, (13) =β (c0 −γ a0 ) , ∂t with initial and boundary conditions of initial problem.
The problem An (n-th approximation with zero initial and boundary conditions): to find a solution for system of equations: ∂cn (t, z) + ∂an (t, z) + u ∂t ∂t ∂cn ∂z
2 =Dinter ∂∂zc2n , −H ∂Tn∂(tt, z) − uhg ∂∂Tzn − Q ∂∂atn − Χ2Tn + Λ ∂∂2zT2n =0, (16) n−1 ∂an = β cn −γ an − ∑ ai (t, z)an−1−i (t, z) (17) ∂t i=0 with zero initial and boundary conditions.
We construct analytic solutions of problems A0 and
An ; n = 1, ∞ using the Heaviside’s operation method [
The problem A0 is linear concerning to zero approximation a0 ; The problem An ; n = 1, ∞ is linear concerning to the nth approximation an and nonlinear concerning to all previous n1 approximations. All equations of problems are obtained by linearizing the nonlinear differential equation of the internal adsorption kinetics with asymptotic sums (11), grouping the terms in the left and right sides of the equations and the conditions of the original boundary value problem for equal powers of a small parameter.
Having determined,
∞ L[c(t, z)] ≡ c* ( p, z) =(t, ∫ c z) e− pt dt , 0 ∞ L[T (t, z)] ≡ T * ( p, z) =T(t, ∫ z) e− pt dt , (18) 0 ∞ L[a(t, z)] ≡ a* ( p, z) =a(t, ∫ z) e− pt dt 0 where p is a complex Laplace transform parameter, we obtain in the Laplace images A0∗ and An∗ the above boundary value problems.
The problem A0∗ d 2с0* ( p, z) dz2
dс0* − q12 ( p) c0* − u1 dz d 2 d dz2 T0∗ − u2 dz T0∗ − q22 ( p)T0∗
1 a0* ( p, z) = β p + βγ c0* ( p, z) , =−c* ( p) ,
0 =−T * ( p) , 0 The problem An∗ dd2zc2n* − u1 dz
dcn* − q12 ( p) cn* d 2 d dz2 Tn∗ − u2 dz Tn∗ − q22 ( p)Tn∗ =−c * ( p, z) ,
n =−T * ( p, z ) ,
n 1 n−1 * an* ( p, z ) β p + βγ cn* − ∑i=0 ai an−1−i ( p, z ) , = (19) (20) (21) (22) (23) (24) IV. SOLUTIONS FOR ZERO AND N-TH
The distributions of adsorption concentration in gas phase c0 (t, z) , the temperature of the layer T0 (t, z) and the concentration of the adsorbate (absorbed substance) in the nanopores of the adsorbent a0 (t, z) are looks like:
0
The solutions сn (t, z ) , Tn (t, z ) , an (t, z ) for problems
(15)(17) are the functions describing the temporal spatial
distributions of adsorbent concentration in gas phase,
temperature and adsorption concentration in micro and
nanopores of the adsorbent particles [
β cn (t, z ) = ×
Dint er τ∫ ∞∫ c t(−tτ −τ , z,ξ ) − ∑n−1 aian−1−i (τ ,ξ ) dξ dτ 0 0 βγ ∫ e−βγ (τ −s)c (s, z,ξ ) ds i=0 0 (25) (26) (27) (28) Tn (t, z ) = an (t, z)
T (t −τ ; z,ξ ) − ∞ Qβ t ∫ t−τ 0 βγ ∫ e−βγ (t−τ −s)T ( s; z,ξ )ds dξ dτ Λ ∫0 n−1 ∑ ai ( s,ξ ) an−1−i ( s,ξ ) − cn (τ ,ξ ) i=0 (29) =(t−τ β ∫t e−βγ ) n−1 cn (τ , z) − ∑ ai (τ , z) an−1−i (τ , z) dτ 0 i=0 (30) π Φc0 (t, z ) =)z 1 ⌠ e−ϕ1(ν π ⌡ ν
0 Φс (t, z) =
sin (ν t − zϕ2 (ν )2 ) 1 ∞∫ ϕ1(ν ) co s (ν t −ϕ 2 (ν ) z) +φ2 (ν ) sin (ν t −ϕ 2 (ν ) z) 2π 0 (Γ12 (ν ) +ν 2Γ2 (ν ))1/2 2
dν − u z dν + e 2Dinter ΦT0 (t, z ) =)z 1 ⌠ e−φ1(ν π ⌡ ν
0 ΦT (t, z) = sin (ν t − zφ2 (ν )2 )
− u z dν + e 2Dinter , 21π ∞∫0 φ1(ν ) co s (ν t −φ 2 (ν ) z) +φ2 (ν ) si1n/2(ν t −φ 2 (ν ) z) (ΓT2 (ν ) +ν 2ΓT2 (ν )) 1 2 dν , ϕ1,2 (ν ) = (Γ12 (ν ) +ν 2Γ22 (ν ))1/2 ± Γ12 (ν ) 1/2 2 , Γ1 (ν ) =
u2
+
Di2nter (ν 2 + β 2γ 2 ) ; Γ2 (ν ) = Dinter (ν 2 + β 2γ 2 ) ν 3 +νβ 2 (γ +1)γ (ΓT2 (ν ) +ν 2ΓT2 (ν )) , ,φ1,2 (ν ) = 1 2 2 1/2 , , ΓT2 (ν )
Hν =, Λ T (τ ; z,ξ ) c (τ ; z,ξ ) 4Λ2
−u2 (z−ξ ) =e2
−u1 (z−ξ ) =e2 (ΦT (τ , z −ξ ) − ΦT (τ , z +ξ )) .
(Φc (τ , z −ξ ) − Φc (τ , z +ξ )) .
V. NOMENCLATURE c - concentration of moisture in the gas phase in the column; a - concentration of moisture adsorbed in the solid phase; T - temperature of gas phase flow, °C; u - velocity of gas phase flow, m/s2; Dint er - effective longitudinal diffusion coefficient; Λ - coefficient of thermal diffusion along the columns; hg gas heat capacity; Q - heat sorption effect; H - total heat capacity of the adsorbent and gas; Χ2 =2α n / R - coefficient of heat loss through the wall of the adsorbent; R - radius of adsorbent of solid particles, m ; α h - heat transfer coefficient; γ - Henry’s constant; β - mass transfert coefficient; z - distance from the top of the bed for mathematical simulation, m;
In paper proposed theoretical foundations of mathematical modeling of nonisothermal adsorption and desorption in nanoporous catalysts for exhaust gas neutralization systems for the nonlinear Langmuir isotherm. Such approach in our opinion most fully describes the mechanism of adsorption equilibrium for micro- and nanopore systems of the ZSM-5 zeolite. An effective linearization scheme for the nonlinear model is realized. High-speed analytical solutions of the system of linearized boundary-value problems of adsorption and desorption in nanoporous media was substantiated and obtained using Heaviside’s operational method.