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Introduction
The modeling and simulation are the main approach for the quantitative description of the mass transfer processes in the chemical, power, biotechnological and other industries. The modelsof the mass transfer processes are possible to be created [1, 2] on the basis of the physical approximations of the mechanics of continua, where the mathematical point is equivalent to an elementary physical volume, which is sufficiently small with respect to the apparatus volume, but at the same time sufficiently large with respect to the intermolecular volumes in the medium.
The big part of the industrial mass transfer processes are realized in one, two or three phase systems [3, 4, 5] as a result of volume (homogeneous) and surface (heterogeneous) reactions, i.e. mass appearance (disappearance) of the reagents (phase components) in the elementary volumes in the phases or on the its inter phase surfaces. As a result,the reactions are mass sources (sinks) in the volume (homogeneous chemical reactions [6]) and on the surface (inter phase mass transfer in the cases of absorption [6,7], adsorption [8], catalytic reactions [9]) of the elementary physical volumes.
The volume reactions lead to diﬀerent concentrations of the reagents in the phase volumes and as a result two mass transfer processes are realized ? convective transfer (caused by the movement of the phases)and diﬀusion transfer (caused by the concentration gradients in the phases). The mass transfer models are a mass balance in the phases, where components are convective transfer, diﬀusion transfer and volume reactions (volume mass sources or sings). The surface reactions participate as mass sources or sings in the boundary conditions of the model equations.
The theoretical analysis of the mass transfer theories shows, that the predictions of the diﬀusion boundary layer theories [1]
are more accurate than the model theories conclusions, but they are useful for modeling of the inter phase mass transfer processes (absorption, adsorption and catalytic reactions) in column apparatuses [2], where the faces of the phases are unknown.
The use of the physical approximations of the mechanics of continua for the inter phase mass transfer process modeling in industrial column apparatuses is possible if the mass appearance (disappearance) of the reagents on the inter phase surfaces of the elementary physical volumes (as a result of the heterogeneous reactions) are replaced by the mass appearance (disappearance) of the reagents in the same elementary physical volumes(as a result of the equivalent homogeneous reactions) [2], i.e. the surface mass sources (sinks), caused by absorption [7], adsorption [8] or catalytic reactions [9] must be replaced with equivalent volume mass sources (sinks).
The new approach to modeling the mass transfer processes in industrial column apparatuses is the creation of the convection diﬀusion and average-concentration models [2].
The convection-diﬀusion models [3, 4, 5] permit the qualitative analysis of the processes only, because the velocity distribution in the column is unknown. On this base is possible to be obtained the role of the diﬀerent physical eﬀect in the process and to reject those processes, whose relative inﬂuence is less than 1%, i.e. to be made process mechanism identification.
The average-concentration models are obtained from the convection-diﬀusion models, where average velocities and concentrations are introduced. The velocity distributions are introduced by the parameters in the model, which must to be determined experimentally [2].
The models of the homogeneous chemical reactions [6, 7], co-current and counter-current physical (chemical) absorption [6, 7], non-stationary physical (chemical) adsorption [8], gas (liquid) ? solid catalytic reactions in the cases of physical (chemical)
adsorption mechanism [9], are presented.
1. Convection-Diﬀusion Type Models
In the general case [3-5] a multi component (i =1,2,...,i0 ) and multi phase ( j =1,2,3 for gas, liquid and solid phases) ﬂow in a cylindrical column with radius r0 [m] and active zone height l [m] will be considered. If F0 is the ﬂuid ﬂow rate in the column and F j , j = 1,2,3 are the phase ﬂow rates [m3.s-1], the parts of the column volume occupied by the gas, liquid and solid phase, respectively, i.e. the phase volumes [m3] in 1 m3 of the column volume (hold-up coefficients of the phases), are:
(1.1)
The input velocities of the phases in the column u0j [m.s-1] , j = 1,2,3 are possible to be defned as:
(1.2)
The physical elementary column volumes contain the elementary phase volumes ԑj , j = 1,2,3 and will be presented as mathematical points M (r , z) in a cylindrical coordinate system (r , z ), where r and z [m] are radial and axial coordinates. As a result, the mathematical point M (r , z) is equivalent to the elementary phase volumes, too.
The concentrations [kg-mol.m-3] of the reagents (components of the phases) are c_{ij} , i = 1,2,...,i_{0} , j = 1,2,3 i.e. the quantities of the reagents (kg-mol) in 1 m^{3} of the phase volumes in the column.
In the cases of a stationary motion of ﬂuids in cylindrical column apparatus, uj(r , z), vj(r , z) , j= 1,2,3 [m.s-1] are the axial and radial velocity components of the phases in the elementary phase volumes.
The volume reactions [kg-mol.m^{-}^{3}.s-1] in the phases (homogeneous chemical reactions and heterogeneous reactions, as a volume mass source or sink in the phase volumes in the column) are Q_{ij}(c_{ij}), j= 1,2,3, i = 1,2,...,i_{0} . The reagent concentrations in the elementary phase volumes increase ( Q_{ij} > 0) or decrease (Q_{ij} < 0 ) and the reaction rates Q_{ij} are determined by these concentrations
c_{ij }(r , z) [ kg-mol . m^{-3} ].
The volume reactions lead to diﬀerent values of the reagent concentrations in the elementary phase volumes and as a result, two mass transfer eﬀects exist ? convective transfer (caused by the ﬂuid motion) and diﬀusion transfer (caused by the concentration gradient).
i.e., diﬀusive transfer rate (kg-mol.s^{-1}) in 1 m^{3} of the phase volume and Dij [m2.s-1] are the diﬀusivities of the reagents (i =1,2,...,i0) in the phases ( j =1,2,3 ) .
The mathematical model of the processes in the column apparatuses, in the physical approximations of the mechanics of continua, represents the mass balances in the phase volumes (phase parts in the elementary column volume) between the convective transfer, the diﬀusive transfer and the volume mass sources (sinks) [2]. The sum total of these three eﬀects (in the cases of stationary processes) is equal to zero:
(1.3)
The axial and radial velocity components uj (r , z) and vj (r , z) , j= 1,2,3 satisfy the continuity equations
The model of the mass transfer processes in the column apparatuses (1.3) includes boundary conditions, which express a symmetric concentrations distributions (r = 0), impenetrability of the column wall ( r = r_{0} ) , constant input concentrations c_{ij}^{0 }and mass balances at the column input ( z = 0 ):
1.1. Column Chemical Reactor
In the case of one-phase ﬂuid motion [6, 7] in the column chemical reactors,the phase index j =1,2,3 is possible to be ignored. For a two component chemical reaction in the column, the convection-diﬀusion model has the form:
where
1.2. Column Physical (chemical) Absorber
The convection-diﬀusion type models of the absorption processes in the gas-liquid systems [7] is possible to be obtained from(1.3)if j=1,2
in the liquid phase, as volume sources or sinks of the reagents in the phase parts of the elementary (column) volume. As a result:
where k_{0} [s^{-1}] is the volume interphase mass transfer coefcient, χ - the Henry?s number, k - the chemical reaction rate constant. The same models is possible to be used for modeling of the extraction processes if χ is the redistribution factor. In the countercurrent absorber z = z^{j} , j = 1,2 (z_{1 }+ z_{2}= l ).
1.3. Column Physical (chemical) Adsorber
In the adsorption process [8] participate two reagents ( i_{0 }= 2 ), where the frst (active component) is in the gas or liquid phase ( i =1, j = 1, 2 ) and the second (active site) is in the solid phase ( i = 2, j = 3). The adsorption is a process of interphase mass transfer of an active component (AC) from the gas (liquid) volume to the solid interface and a heterogeneous reaction with an active site (AS), due to a physical (Van der Vaals?s) or chemical (valence) force.
The convection-diﬀusion type models of the adsorption processes in the gas (liquid)-solid systems are possible to be obtained from (1.3) if j = 1, 3 = 2,3 i0 = 2, where i = 1 is for (AC) in the gas (liquid) phase, i = 2 is for (AS) in the adsorbent (solid phase). The volume adsorption rate in the case of a solid adsorbent is Q3 = b0 Q03 [kg-mol.m-3.s-1], where b 0[m2 .m-3] is m2 of the inner surface in the solid phase (the surface of the capillaries in the solid phase) in 1 m3 of the solid phase (adsorbent), Q03 [kg-mol.m-3.s-1] ? the surface adsorption rate. A gas adsorption will be considered for convenience, where c 11 [kg-mol.m-3] is the volume concentration of the AC in the gas phase (elementary) volume, c13 [kg-mol.m-3] ? the volume concentration of the AC in the void volume of the solid phase (adsorbent), c23 [kg-eq.m-3] -the volume concentration of the AS in the solid phase (elementary) volume (1 kg-eq AS in the adsorbent combine 1 kg-mol AC), u1= u1(r) ? velocity of the gas phase [m.s-1], u3 = 0 (solid phase is immobile). All concentrations are in kg-mol (kg-eq) in 1 m3 of the phase (elementary) volume.
1.3.1 Physical Adsorption
In the cases of physical adsorption the surface adsorption rate is:
(1.8)
Let us consider a non-stationary gas adsorption in a column apparatus, where the solid phase (adsorbent) is immobile. The convection-diﬀusion model of this process is possible to be obtained from (1.3), where the diﬀusivity of the free AS in the solid phase (adsorbent) volume is equal to zero. If the rate of the interphase mass transfer of the AC from the gas phase to the solid phase is k ( c11? c13) and the process is non-stationary as a result of the free AS concentration decrease, i.e. the convection-diﬀusion model has the form:
(1.9)
where t [s] is the time.
1.3.2 Chemical Adsorption
In the cases of chemical adsorption the model is
(1.10)
where k is the chemical reaction rate constant.
1.4. Heterogeneous Catalytic Reactor
1.4.1 Physical Adsorption Mechanism
For a long duration process in the case of physical adsorption mechanism the model has the form: