5 MARCO CONTEXTUAL
7.3 REVISIÓN AMBIENTAL INICIAL
7.3.4 Informe final de la RAI
where a small am ount of nitrogen had necessarily to be added to methane. This can be explainetJ considering th at methane diffusivities in chapter 3 are calculated according to Wilke’s formula therefore methane cannot be pure, or rather the molar fractions of all the other components cannot be zero otherwise the equation would not be defined (see equation 4.29). 4.3 Governing equations
Air
R e ac to r shell M e m b ran eMethane
4. Mathematical modelling of membrane reactor 76 A ir
CH
4 » Shell side Tube side Shell side M em brane rFig. 4.3: 2D schem atic of m em brane reactor
Various models for the study of a membrane reactor were developed in order to over come the numerical difficulties. Models with different levels of complexity and approxi mations were assessed trying to achieve a satisfactory numerical solution and at the same time physically acceptable results. Chapter 3 deals with the easiest and more simplified model in which kinetics is assumed infinitely fast (no information about the kinetics is needed) and Pick’s law is used to calculate diffusive fluxes. However, at relatively low tem peratures, when the kinetic rate cannot be considered infinitely fast, a more rigorous approach is needed. Radial tem perature and concentration profiles within the membrane must be calculated in order to evaluate more accurate conversions, heat produced in the reactor and outlet tem peratures of gas and solid.
A kinetic expression with zero order with respect to oxygen (as widely employed in literature) cannot be used since the configuration with separate feed of reactants allows a large range of O2/C H 4 ratios (0 ; 00). Such kinetics would result in finite reaction rate
even in areas where the corresponding reactant is not present. This is physically unrealistic and it also leads to difficulties in the numerical solution.
As shown by figure 4.2 and 4.3 the membrane separates two fluid phases which, from now on will be named tube and shell side. The two reactants diffuse through the mem brane and react inside the pores, producing carbon dioxide and water th a t counter-diffuse towards the fluid phases.
The assumptions made to ease the com putation while keeping a realistic physical approach are:
• Plug flow both in tube and shell side.
• Negligible axial mass diffusion and heat conduction in the fluid phases (tube and shell side).
• Negligible radiation. • A diabatic operation.
• Negligible pressure drop across the reactor’s fluid phases.
W ith such assumptions, the system of differential equations th a t describes the mem brane reactor can be split in three ID problems linked together. More specifically, mass and heat generation term s relative to the axial ODEs of the two fluid phases correspond to the radial mass and heat fluxes in the boundary conditions of the ODE of the membrane. This concept will be explained more extensively in the following section 4.3.1.
The choice of a plug flow regime in the fluid phases is questionable because it definitely introduces a source of inaccuracy especially when the reactor operates in laminar regime. This issue has been discussed by Hayes and Kolaczkowski (1997) and Groppi et al. (1995b) who compare lum ped and distributed models for the study of a monolithic system in catalytic combustion. They show how the Nusselt number, calculated from the results of a 2D distributed model varies along the axial coordinate of th e monolith and has a peak in correspondence to the light off position. This behaviour cannot be reproduced by any of the correlations available in literature which instead exhibit a continuously decreasing trend of Nu. Groppi et al. (1995b) also show th a t the light-off position is anticipated in lumped models, however they give a good prediction of the outlet tem perature of the gas phase which is a prim ary objective when combustors are assessed in gas turbine applications. In a more rigorous model the radial profile of concentration and tem perature in the fluid phases should be evaluated with a 2D approach instead of assuming plug conditions. The solution of such a system is more complicated th a n the one of a plug flow model, especially in case a pressure gradient is applied across th e membrane. W hen a pressure difference is applied on the membrane a radial convective flux is generated and a radial component of the velocity in the fluid phases has to be taken into account. A 2D approach implies the solution of coupled mass, heat and m om entum balances in the fluid phases and consequently a much larger size of the numerical problem. On the other hand, in lumped model the contribution of radial fluxes in the fluid phase is assumed to occur only in a film next to the interface gas-porous medium followed by instantaneous mixing in the bulk.
As for the assum ption of negligible axial heat conduction, this can be justified by the studies of Groppi et al (1996). They state th a t the “wall conductivity can be neglected in ceramic monoliths despite the presence of steep axial gradients” . This is true in case the gas flow rate is high (as in gas turbines) so th a t “heat transfer w ith low wall con duction is negligible w ith respect to the convective contribution” (Groppi et al. 1996). These considerations have been verified in monolithic system s but they can be extended to m embrane reactors because even in this case a porous wall is in contact w ith a fluid and exchanges heat and mass with it. It can be safely deduced th a t the same concepts apply for monoliths and membranes if the respective fluid pheises are in the same flow regimes.
4. Mathematical modelling of membrane reactor 78
Molecular and Knudsen diffusion are the main mass tran sp o rt mechanisms in the mem brane of the system studied in this work, convection playing a minor role unless a relevant pressure gradient is applied. In a correct approach, th e dusty gas model (DGM) has to be preferred to the Pick’s law because, as pointed out by m any authors (K rishna and Wessel- ingh, 1997 and Veldsink et ah, 1994), it gives a b etter representation of multicomponent mass tran spo rt in porous media w ith or w ithout a convective contribution. In particular, differently from Pick’s law the DGM accounts for the contribution of molar fraction gra dients of all the species in the m ixture and resistances due to friction between molecules of gas and solid (see §2.5).
Unlike Pick’s law, the radial molar flux of a generic com ponent in the DGM equation (equation 4.21) is not directly dependent on the molar fraction gradient of the same component. This implies th a t when the DGM is used, the system of equations defining the mass balances cannot be expressed as 2"^^ order system in term s of molar fractions but as a order in term s of molar fiuxes. It is obvious th a t when the order of the problem is lowered the number of equations must be doubled. The solution of such system is complicated by the strong non-linearity of the DGM equations. In the following sections two models are reported which respectively use Pick’s law and DGM equations for the calculation of diffusive fiuxes within the membrane.
4.3.1 Fluid phases balances
The equations of the two fiuid phases (tube and shell side) are common for the two models described in this chapter and they are based on the assum ption of plug fiow regime. Axial convection is the only transp ort mechanism considered and, although reactions are not taking place in the fluid phases it is appropriate to consider generation term s to account for heat and mass transfer to or from the membrane. Molar and heat balances for tube and shell sides and relative boundary conditions are indicated from equation 4.2 to 4.9.
Shell side molar balance can be defined according to equation 4.2:
« -
" Where:N = number of components
Fis — axial molar flux of component i. [mol s“ ^m
= Shell side generation term corresponding to the radial molar flux of component i in the interface between shell side gas phase and membrane. See following sections 4.3.2 and 4.3.3. [mol s“ ^m
Rs — radius of external reactor shell, [m] W ith boundary condition:
z = 0 ^ Fisl^^Q = Vi = 1 , . . . (4.3) W here the value of is the inlet molar flux of com ponent i which is known if inlet velocity, tem perature, pressure and molar fractions are given.
W ith the same notation, the molar balance in the tu be side can be expressed according to equation 4.4:
= ° Vi = l , . . . , A f (4.4)
Where:
R m t — inner membrane radius [m]
= Tube side generation term corresponding to the radial molar flux of component i in the interface between tu be side gas phase and membrane, [mol s~^m
Fit = tube side molar flux of component i. [mol s~^m W ith boundary condition:
z = 0 ^ Fitl^^Q = Fit Vi = 1 , . . . , jV (4.5) H eat balance of the shell side can be evaluated considering the assum ption of plug flow and neglecting axial dispersion:
Where:
Ts = tem perature of the shell side bulk gas. [K]
= shell side heat generation term th a t corresponds to the radial heat flux in the interface between shell side gas phase and membrane. [J m~^s
Cpi = heat capacity of component i. [J mol~^K W ith boundary conditions:
Ts\^^q = Tso (4.7)
W here Tsqis the inlet tem perature of the shell side.
4. M athematical modelling of membrane reactor 80
^
j r p n- g = 0 (4.8)
W here:
q^, — tu b e side heat generation term th a t corresponds to the radial h eat flux in the interface between tube side gas phase and membrane. [J m “ ^s ~^]
Tt = tem perature of the tube side bulk gas [K] And with boundary conditions:
z = 0 = Tjo (4.9)
In th e following paragraphs only the equations for th e m embrane will be considered. The equations and the solution methods for the fluid phase one always th e same for all the programmes developed.
4.3.2 Membrane model equations for isobaric systems using P ick’s law
W hen molar fluxes are calculated with the Stefan-Maxwell or the DGM equations, the numerical solution of the resulting mass balances is rath er complex due to their strong non-linearity. Hence a simpler approach using the Pick’s law can be attem pted in order to obtain an approxim ate solution th a t would provide a t least a qualitative idea of the behaviour of the reactor. The rationale is to test the num erical m ethod w ith a simplified model and gain information on the system th a t subsequently can be used to develop a more detailed model.
Under the assumptions of isobaric operation of the reactor, only concentrations and fluxes of the 2 reactants are calculated in the model herein described, hence the system is not considered a proper multicomponent one. As it can be noted, this model is ju st an extension of the one reported in chapter 3, dropping the assum ption of instantaneous reaction. The resulting m aterial balances are 2 second order differential equations with variables the reactant molar fractions. Radial molar fluxes are calculated from their defi nition according to the Pick’s law in equation 4.10:
Fir = - D t C T ~ ^ Vi = l , . . . , i V (4.10)
Where:
Fir = radial molar flux of component i in the m embrane [mol s“ ^ m"^]. Ct — to ta l concentration (from the ideal gas law P /R g a s /T ) [mol/m^].
D f = effective diffusivity of component i (see equation 3.8) [m^/s].
If equation 4.10 is used to calculate the molar fiuxes, the m em brane mass balance can be calculated w ith equation 4.11 in term s of molar fractions only:
Vi = l , . . . , i V (4.11) Where:
R = rate of reaction [mol k g ^ s“ ^]. p c = catalyst density [kg/m^]
i^i = stoichiometric coefficient of component i.
The boundary conditions can be calculated with equations 4.12 if interphase mass transfer lim itations are neglected, otherwise equations 4.13 should be used:
(4.12)
^ ~ ^ — kitCt {xit
f' — Rms — ^ir ~ ^isCs — Xis) (^-l^)
V% = 1, . . . , IV Where:
Xit 5 Xis = molar fractions of component i in tube and shell side respectively^. Ct,Cs — to tal concentration of tube and shell side respectively, [mol/m^] kitikis = mass transfer coefficients of tube and shell side respectively, [m/s]
The values xu and Xis vary along the axial co-ordinate and determ ine a different radial problem at each value of z. The values of kis and ku are assumed constant along the axial coordinate and are calculated as for fully developed lam inar flow. Saracco and Specchia (1995) suggested equation 4.14 and 4.15 for the calculation of mass transfer coefficients for annular flow:
kis = 2.63—— ^ — (4.14)
XLtjis
The value of ku was calculated with equation 4.15:
kit = 3.66^ ' ^ (4.15)
4rxmt
_
4. Mathematical modelling of membrane reactor__________________________________________ ^
M embrane heat balance is a second order differential equation w ith the term s of heat generation due to reaction and conductive dissipation:
l d T \
^ ~ ^ j + ■ ( - ^ % ) = 0 (4-16)
The radial heat flux is calculated according to Fourier law: d T
Qr - - k m - ^ (4.17)
Where:
A Hr = heat of reaction [J/mol]
km = effective therm al conductivity of the membrane [J m “^K s“ ^]
The boundary conditions can be w ritten as in equation 4.18 and are valid in case interphase heat transfer resistance is neglected:
r = Rms T = Ts r = Rmt T = Tt otherwise, equations 4.19 should be used:
^ — Rmt ^ k m ^ \ ^ _ R ^ ^ — Qr ~ {R ~ Tt)
^ —
R m s^
^771^1^—^
— Qr — { T ~ T s )(4.18)
(4.19)
hs and ht are calculated applying the Chilton-Colburn analogy to equations 4.14 and 4.15. It should be noted th a t the radial fluxes in the boundary conditions of the radial mass and heat balances coincide w ith the generation term s of the respective axial balances.
4.3.3 Membrane model equations using D G M 4.3.3.1 Mass and heat balances
As was mentioned in §4.3 the DGM should be used if high accuracy is desired despite the difficulty of the numerical solution. M ulticomponent diffusion in a porous medium can be precisely investigated with the DGM even in presence of convection or external forces. Since carbon dioxide and water are assumed to be the only products of th e reaction between m ethane and air, the to tal number of components in the system is five. Accordingly five independent molar balances can be w ritten in term s of molar fluxes as indicated by equation 4.20.
_ l d{rFir) _). Vi = l , . . . , AT (4.20)
r or
The symbols used in the above equation follow the same notation of the previous sec tion. The main difference with the simplified approach of the previous section is the way
the fluxes Fir are calculated. Equations 4.21 defines the radial molar fluxes of the N com ponents in a generic m ixture according to the DGM formulation.
X i F j r - X j F i r Fir _ 1 dxi X, / B o P \ g p
i t . . P D ti P D l , - d r P R , r . s T \ ^ i D l ^ J dr y i = 1, 2 , . . . , iV
(4.21)
The effective diffusivity in the DGM assumes a different meaning from the one expressed in equation 3.8. The effects of molecular effective diffusivity and Knudsen diffusivity are joined together in one single variable when the Pick’s law is used. If the DGM is used, all the contributions of various transp ort mechanisms are explicitly represented by different term s in the equations. Equation 4.22 shows how the presence of the porous medium affects the molecular diffusivity Dfj and generates a further diffusion resistance due to the collision of molecules of gas with the solid wall.
4 „ iSRgasT £
= = (4.22)
= Knudsen diffusivity [m^/s]
K q = Knudsen constant [m]
Bq = Perm eation constant [m^]
D f - = effective molar diffusivity [m^/s]
Dij = binary diffusivity of components i and j [m^/s] Mi = molecular weight of species i [g/mol]
Rgas — ideal gas constant [J mol~^ K “ ^] J = ratio porosity / tortuosity
H = gas viscosity [kg m~^ s~^]
The variables of equations 4.20 and 4.21 are the N molar fluxes, the N molar fractions and the pressure whose gradient gives rise to convective fluxes. If all the N molar fractions are considered independent, the system is well posed only if consistency equation 4.23 is considered:
N
^ ^ X i = 1 (4.23)
2 • N boundary conditions are needed for th e molar balances and th e DGM equations. Saracco et al. (1995b) suggested the use of equations 4.24 and 4.25 for the cases of negligible and relevant convective flow respectively.
4. M athematical modelling of membrane reactor T r — R r n t ~ R gasT t S r = Rms — h s R ^ Its ^ z = 1, 2, . . . , A/^ N \ / N ^ — Fmt ^ / TV \ — GXp E / j , ^ — -Rms ^ ^ — GXp 2 = 1 ,2 N RgasTt ^ J = 1 Ptkit / N \ (4.25) ' RgasTs
E
'
J = 1 PskiThe same notation defined in the previous equations was followed and the symbols x\ and
x f were introduced to identify the values of and The symbols T and
S on the left side of equations 4.24 and 4.25 stand for tub e and shell, as the equations of the respective lines refer to the boundaries of tube and shell side.
As can be noted, in both the expressions 4.24 and 4.25 pressure appears only in the calculation of the to tal concentration of the fluid phases^. Accordingly the pressure of the m embrane does not seem to have any explicit constraint. The boundary conditions used in this work were slightly different from the ones developed by Saracco et al. (1995b). Only N — 1 of the N 4.24 and 4.25, were used and two further conditions were imposed directly on the pressure according to equations 4.26
T r = Rmt => P = Ft (4.26)
S r = Rms P = Ps
W here it is assumed th a t pressure does not vary appreciably between the bulk of the fluid phases and the surface of the membrane.
Equations 4.24 and 4.25 are obtained using the film theory th a t is based on the as sum ption of perfect mixing of the gas in each radial section of the fluid phase except from one small film near the interface gas-solid (Bird at al. 1960). Mass and heat are exchanged only in such film where laminar regime is assumed. Integrating the constitutive equations of the film theory, a mass transfer coefficient can be defined th a t allows the calculation of