- Email: [email protected]
Modelling of Multicomponent Reactive Absorption using Continuum Mechanics
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
^25
Modelling of Multicomponent Reactive Absorption using Continuum Mechanics D. Roquet, P. Cezac and M. Roques Laboratoire de Genie des Precedes de Pau (L.G.P.P.) 5, rue Jules Ferry, 64000 Pau, France email : [email protected] A new model of multicomponent reactive absorption based on continuum mechanics is presented. It includes the equations of conservation of mass for species, momentum and energy. The diffusion is described by the Maxwell-Stefan model. The steady state resolution is performed in an equivalent two dimensional geometry. Simulation results are compared with experiments carried out on a laboratory scale column.
1. INTRODUCTION Wet scrubbing is a widely used technique to remove heavy metal and acid compounds from flue gases. Because of the limitations of atmosphere emissions, optimisation of such processes is essential. Thus, the model taking into account the description of all physical and chemical phenomena is the most accurate tool available to provide a maximum of information on the process. However the theoretical description is quite complex: the different phases involved are multicomponent systems and the pollutants may react in the aqueous phase. The competitive chemical reactions are instantaneous or kinetically controlled. Furthermore the description of transport phenomena should take into account the interactions between species in multicomponent diffusion and the non ideality of the fluids thanks to a specific model of thermodynamics. The aim of this article is to present a general model of multicomponent reactive absorption based on the considerations above using "empirical parameters. Firstly we will present a brief summary of the differents approaches that will lead us to the description of the model. Then we will apply it to the absorption in packed columns of acid gases in a aqueous solvent and we will compare the results with experiments carried out on a laboratory scale pilot
2. STATE OF THE ART Two kinds of model are presented in the literature. Firstly, the theoretical stage model, which describes the column as a succession of stages at thermodynamic equilibrium. Mass and heat transfers are driven by this equilibrium. It is very suitable in order to take into account each competitive chemical reaction. However it is an ideal case of mass transfer description and the simulated results could only be compared with experiments using an empirical coefficient called efficiency. Secondly, the transfer models which use empirical coefficients to predict the transfer of mass between phases. In packed columns, the two-film model (Whitman, 1923) is the most widely used. This model
326 allows us to define mass transfer coefficients ICL and kc for binary systems applying Pick's law of diffusion. In this case the coefficients could be obtained by experiments or by correlations (Lee and Tsui, 1999). Taylor and Krishna (1993) studied the case of multicomponent systems without reaction using the Maxwell-Stefan law of diffusion. When reactions occur, the mass transfer coefficient is modified using an enhancement factor. This factor is difficult to obtain in competitive chemical reactions (Versteeg and al., 1990). That's why recently, new models of non equilibrium have been proposed (Rascol and al., 1999 ; Schneider and al., 1999). Given an estimation of the diffusion film thickness, the mass balance relations are numerically solved in each film using an appropriate law of diffusion. This approach allows us to take directly into account the enhancement due to reaction without providing the enhancement factor. This is good progress in mass transfer description but in multicomponent mass transfer, the film thickness should be different for each component. Thus, we have developed a general model of multicomponent reactive absorption in a ionic system using continuum mechanics (Truesdell, 1969 ; Slattery, 1981) to describe local phenomena in each phase. This description does not require to use empirical mass transfer coefficient or film thickness.
3. GENERAL MODEL Let us consider two phases (cpi and 92) separated by an interface I and ^ the unit normal. Let us consider the phases as a multicomponent body in which all quantities are continuous and differentiable as many times as desired. On this system two kinds of relation can be written thanks to the continuum mechanics theory in multicomponent system (Truesdell, 1969 ; Slattery, 1981) : conservation of properties in each phase and boundary conditions at the interface. 3.1 Equations of conservation Using the transport theorem, the balances of each conservative quantity could be written as an equation of conservation applied to each point of the continua. In order to describe the evolution of the various components we require to solve the conservation of mass for species iG {l,...,n^,-l}. It takes into account the n^ kinetically controlled chemical reactions and the nre instantaneously balanced chemical reactions, as dissociation equilibrium in aqueous phase. The enhancements of these equilibria are obtained implicitly thanks to chemical equilibrium equations. |:(Pi)+V(p,v,)=Xv-jr,+Sv:,^, ,i6{l,...,n,-l}
(1)
Xvi,-^i=0, ke{l,...,nj
(2)
As we use only ric - 1 equations of continuity for species, we have to solve the equation for overall conservation of mass.
|:(p.)+v(p,v)=o
(3)
327 Equation (3) has been added to the system to prevent the total mass balance from numerical approximations. Then, in order to describe the movement of the various constituents, we need to solve equations of momentum conservation. But as we don't know any fluid behaviour that can express the stress tensor for species, we use an approximate equation based on the Maxwell-Stefan approach (Taylor and Krishna, 1993 ; Slattery, 1981).
c^RTd- =CiV^i +CiSi V T - — V P - p i
. .e{l,...,n,-l}
(4)
Pt
Neglecting the effect of heat transfer on the mass transfer (Soret effect), the driving force di is defined as follows ; ^__|._CjCj(v.-vJ
(5)
c?D^
Three main reasons have led us to this approach. Firstly, this description allows for the effect of the thermodynamic activity in ionic systems to be taken into account. Secondly, the effects of interactions between chemical species on rates of diffusion can be expressed. Finally, the external forces, such as the electrostatic ones occurring within an ionic system, are included. These forces are expressed thanks to the Nernst-Planck equation (Taylor and Krishna, 1993) and the electrical potential is implicitly obtained by solving the local electroneutrality. fi=-^F^-V(p,ie{l,...,nJ
(6)
A. = o £z,-^
(7)
M,
In order to describe the behaviour of the whole phase, we use the conservation of the overall momentum. The stress tensor in laminar flow is achieved from the Newtonian model. (8)
— (p,v)+V(p,v®v)-VT-pJ=0 dt
(-PP + XVvl[ XV.vJ++2riD
(9)
Finally, in order to describe the evolution of temperature, the conservation of energy is required. Thus, neglecting the viscous and the diffusive dissipations and the effect of mass transfer on heat transfer (Dufour effect), the energy balance is
o-MM.U' \^ +v
p V
V
u +—v 2
-V(KVT)-P,Q
(10)
Thus the global system is composed of the conservation of mass for species and of the overall mass, of Maxwell-Stefan's law of diffusion, of conservation of overall
328 momentum and of the conservation of energy. The thermodynamic properties in the aqueous phase, such as enthalpy, internal energy, density or activity coefficients, are calculated from thermodynamic models adapted for electrolytes. The gas phase is considered as a perfect gas. Assuming the summation equations for mass concentration and for mass flow, the system can be solved. However we should present the boundary conditions used at interface. 3.2 Boundary conditions As the interface is assumed to be immobile, the normal of the mass flows is conserved. It is applied to nc-1 species and to the overall mass to prevent numerical approximations.
p^^r-prvr 1-^=0, ie{i,...,n,-i}
(11)
prv
(12)
prv
i-i=o
As we assume that the thermodynamic equiUbrium is reached at the interface, the chemical potentials for species, the temperatures and the pressures are equal. ~» I
iir^'-fir''=0,ie{l,...,nj
(13)
p
(14)
rp(p,
(15)
I _rp(P| '
:0
4. NUMERICAL RESOLUTION An equivalent two dimensional geometry (figure 1) is generated from gas and liquid fractions, and from the interfacial area. These three characteristics could be obtained by experiments or by correlations (Lee & Tsui, 1999). The height of packing is conserved.
^
0(^
Figure I: Two dimensional geometiy, height of packing (z) by equivalent length (x)
329 The overall two dimensional system of differential and algebraic equations is solved in steady state using a finite volume method of discretisation. This method generates a non linear algebraic system solved by Newton-Raphson's method taking into account the sparse structure of the Jacobian matrix for minimising the storage space and the CPU time. Furthermore, the two dimensional grid uses a variable step to minimise the end effects. Thus, the equations are solved simultaneously and we obtain the profiles of concentration, temperature, pressure and velocity at each point of the geometry.
5. EXPERIMENT AND SIMULATION RESULTS We investigated the C02 absorption in soda inside a packed column at pH 10,1. The steady state simulations are validated by experiments performed on a laboratory scale packed column. The experiment measurements are reported in figure 2 and compared with the continuum mechanics model and with a rate based model taking into account the enhancement of the reactions (Versteeg and al. 1990). The phase's fractions and the wet interfacial area that permit us to define the two dimensional geometry are obtained with correlations (Lee and Tsui, 1999). The simulated pH profile with the continuum mechanics model is in accordance with the experiments, and is better than the ratedbased model. Figure 3 and 4 are concentration profiles obtained thanks to the continuum mechanics model. These concentration profiles are suitable and in agreement with the theory. 11 n
*
le+1
e^q^eiiment
10 - -.-Q . . .continuummechanics model A rate-based mo del
X
le+0
n
le-1
9 -
's ^^'^ |le-3
K 8-
^le-4 7-
P le-5
,-a**
6i i
5 -
0,0
le-6 A
A
A
-^^
le-7 arx—X
— — 1
1,0 height of packing (m)
Figure 2: comparison ofpH evolution
2,0
2,78
3,28 3,78 liquid fllm length (mm)
Figure 3: concentration profiles at the outlet
6. CONCLUSION A new model of multicomponent reactive absorption has been presented. The comparison of the experiment and its simulation allows us to validate the model. Furthermore, global information, such as concentration or pH profiles, could be obtained thanks to a local description of transfer phenomena. One of the most interesting aspects of this model is that it requires only two parameters i.e. the wet interfacial area and the phase's fractions.
330
0,0 Cco2 (mol.m"^)
Figure 4 : Evolution of CO2 concentration (molni^) in the two-dimensional geometry
NOTATIONS c d D D f H m P r R s S T
molar concentration, mol.m'^ driving force, m'^ Maxwell-Stefan diffusivity, m^.s'^ deformation tensor, s'^ external force, N height of packing, m mass flowrate, kg.s'^ pressure. Pa rate of chemical reaction, kg.m'^.s" gas constant, 8,314 J.mol'.K'^ entropy, J.kg•^K'^ area, m^ temperature, K
T u V K A, r\ \i V
stress tensor, N.m' mass internal energy, J.kg"^ velocity, m.s'^ thermal conductivity, W.m ^K'' bulk viscosity, Pa.s dynamic viscosity, Pa.s mass chemical potential, J.kg'^ stoechiometric coefficient of chemical reaction p mass concentration, kg.m'^ ^ enhancement of chemical equilibrium, kg.m'ls"^
REFERENCES Whitman, W.G., 1923, Chem. Met. Sci., 29, 147 Lee, S.-Y. and Y.P. Tsui, 1999, Chem, Eng. Prog., July, 23-49 Taylor, R. and R. Krishna, 1993, Multicomponent mass transfer, Wiley, New-York Versteeg, G.F., J.A.M. Kuipers, F.P.H. Van Beckum and W.P.M. Van Swaaij, 1990, Chem. Eng. Sci., 45, 183-197 Rascol, E., M. Meyer and M. Prevost, 1999, ECCE2, Montpellier Schneider, R., E.Y. Kenig, A. Gorak, 1999, Chem. Eng. Res. Des., 77, 633-638 Truesdell, C , 1969, Rational thermodynamics, McGraw-Hill, New-York Slattery, J.C, 1981, Momentum, energy, and mass transfer in continua, McGraw-Hill, 2"^ edition. New-York
^25
Modelling of Multicomponent Reactive Absorption using Continuum Mechanics D. Roquet, P. Cezac and M. Roques Laboratoire de Genie des Precedes de Pau (L.G.P.P.) 5, rue Jules Ferry, 64000 Pau, France email : [email protected] A new model of multicomponent reactive absorption based on continuum mechanics is presented. It includes the equations of conservation of mass for species, momentum and energy. The diffusion is described by the Maxwell-Stefan model. The steady state resolution is performed in an equivalent two dimensional geometry. Simulation results are compared with experiments carried out on a laboratory scale column.
1. INTRODUCTION Wet scrubbing is a widely used technique to remove heavy metal and acid compounds from flue gases. Because of the limitations of atmosphere emissions, optimisation of such processes is essential. Thus, the model taking into account the description of all physical and chemical phenomena is the most accurate tool available to provide a maximum of information on the process. However the theoretical description is quite complex: the different phases involved are multicomponent systems and the pollutants may react in the aqueous phase. The competitive chemical reactions are instantaneous or kinetically controlled. Furthermore the description of transport phenomena should take into account the interactions between species in multicomponent diffusion and the non ideality of the fluids thanks to a specific model of thermodynamics. The aim of this article is to present a general model of multicomponent reactive absorption based on the considerations above using "empirical parameters. Firstly we will present a brief summary of the differents approaches that will lead us to the description of the model. Then we will apply it to the absorption in packed columns of acid gases in a aqueous solvent and we will compare the results with experiments carried out on a laboratory scale pilot
2. STATE OF THE ART Two kinds of model are presented in the literature. Firstly, the theoretical stage model, which describes the column as a succession of stages at thermodynamic equilibrium. Mass and heat transfers are driven by this equilibrium. It is very suitable in order to take into account each competitive chemical reaction. However it is an ideal case of mass transfer description and the simulated results could only be compared with experiments using an empirical coefficient called efficiency. Secondly, the transfer models which use empirical coefficients to predict the transfer of mass between phases. In packed columns, the two-film model (Whitman, 1923) is the most widely used. This model
326 allows us to define mass transfer coefficients ICL and kc for binary systems applying Pick's law of diffusion. In this case the coefficients could be obtained by experiments or by correlations (Lee and Tsui, 1999). Taylor and Krishna (1993) studied the case of multicomponent systems without reaction using the Maxwell-Stefan law of diffusion. When reactions occur, the mass transfer coefficient is modified using an enhancement factor. This factor is difficult to obtain in competitive chemical reactions (Versteeg and al., 1990). That's why recently, new models of non equilibrium have been proposed (Rascol and al., 1999 ; Schneider and al., 1999). Given an estimation of the diffusion film thickness, the mass balance relations are numerically solved in each film using an appropriate law of diffusion. This approach allows us to take directly into account the enhancement due to reaction without providing the enhancement factor. This is good progress in mass transfer description but in multicomponent mass transfer, the film thickness should be different for each component. Thus, we have developed a general model of multicomponent reactive absorption in a ionic system using continuum mechanics (Truesdell, 1969 ; Slattery, 1981) to describe local phenomena in each phase. This description does not require to use empirical mass transfer coefficient or film thickness.
3. GENERAL MODEL Let us consider two phases (cpi and 92) separated by an interface I and ^ the unit normal. Let us consider the phases as a multicomponent body in which all quantities are continuous and differentiable as many times as desired. On this system two kinds of relation can be written thanks to the continuum mechanics theory in multicomponent system (Truesdell, 1969 ; Slattery, 1981) : conservation of properties in each phase and boundary conditions at the interface. 3.1 Equations of conservation Using the transport theorem, the balances of each conservative quantity could be written as an equation of conservation applied to each point of the continua. In order to describe the evolution of the various components we require to solve the conservation of mass for species iG {l,...,n^,-l}. It takes into account the n^ kinetically controlled chemical reactions and the nre instantaneously balanced chemical reactions, as dissociation equilibrium in aqueous phase. The enhancements of these equilibria are obtained implicitly thanks to chemical equilibrium equations. |:(Pi)+V(p,v,)=Xv-jr,+Sv:,^, ,i6{l,...,n,-l}
(1)
Xvi,-^i=0, ke{l,...,nj
(2)
As we use only ric - 1 equations of continuity for species, we have to solve the equation for overall conservation of mass.
|:(p.)+v(p,v)=o
(3)
327 Equation (3) has been added to the system to prevent the total mass balance from numerical approximations. Then, in order to describe the movement of the various constituents, we need to solve equations of momentum conservation. But as we don't know any fluid behaviour that can express the stress tensor for species, we use an approximate equation based on the Maxwell-Stefan approach (Taylor and Krishna, 1993 ; Slattery, 1981).
c^RTd- =CiV^i +CiSi V T - — V P - p i
. .e{l,...,n,-l}
(4)
Pt
Neglecting the effect of heat transfer on the mass transfer (Soret effect), the driving force di is defined as follows ; ^__|._CjCj(v.-vJ
(5)
c?D^
Three main reasons have led us to this approach. Firstly, this description allows for the effect of the thermodynamic activity in ionic systems to be taken into account. Secondly, the effects of interactions between chemical species on rates of diffusion can be expressed. Finally, the external forces, such as the electrostatic ones occurring within an ionic system, are included. These forces are expressed thanks to the Nernst-Planck equation (Taylor and Krishna, 1993) and the electrical potential is implicitly obtained by solving the local electroneutrality. fi=-^F^-V(p,ie{l,...,nJ
(6)
A. = o £z,-^
(7)
M,
In order to describe the behaviour of the whole phase, we use the conservation of the overall momentum. The stress tensor in laminar flow is achieved from the Newtonian model. (8)
— (p,v)+V(p,v®v)-VT-pJ=0 dt
(-PP + XVvl[ XV.vJ++2riD
(9)
Finally, in order to describe the evolution of temperature, the conservation of energy is required. Thus, neglecting the viscous and the diffusive dissipations and the effect of mass transfer on heat transfer (Dufour effect), the energy balance is
o-MM.U' \^ +v
p V
V
u +—v 2
-V(KVT)-P,Q
(10)
Thus the global system is composed of the conservation of mass for species and of the overall mass, of Maxwell-Stefan's law of diffusion, of conservation of overall
328 momentum and of the conservation of energy. The thermodynamic properties in the aqueous phase, such as enthalpy, internal energy, density or activity coefficients, are calculated from thermodynamic models adapted for electrolytes. The gas phase is considered as a perfect gas. Assuming the summation equations for mass concentration and for mass flow, the system can be solved. However we should present the boundary conditions used at interface. 3.2 Boundary conditions As the interface is assumed to be immobile, the normal of the mass flows is conserved. It is applied to nc-1 species and to the overall mass to prevent numerical approximations.
p^^r-prvr 1-^=0, ie{i,...,n,-i}
(11)
prv
(12)
prv
i-i=o
As we assume that the thermodynamic equiUbrium is reached at the interface, the chemical potentials for species, the temperatures and the pressures are equal. ~» I
iir^'-fir''=0,ie{l,...,nj
(13)
p
(14)
rp(p,
(15)
I _rp(P| '
:0
4. NUMERICAL RESOLUTION An equivalent two dimensional geometry (figure 1) is generated from gas and liquid fractions, and from the interfacial area. These three characteristics could be obtained by experiments or by correlations (Lee & Tsui, 1999). The height of packing is conserved.
^
0(^
Figure I: Two dimensional geometiy, height of packing (z) by equivalent length (x)
329 The overall two dimensional system of differential and algebraic equations is solved in steady state using a finite volume method of discretisation. This method generates a non linear algebraic system solved by Newton-Raphson's method taking into account the sparse structure of the Jacobian matrix for minimising the storage space and the CPU time. Furthermore, the two dimensional grid uses a variable step to minimise the end effects. Thus, the equations are solved simultaneously and we obtain the profiles of concentration, temperature, pressure and velocity at each point of the geometry.
5. EXPERIMENT AND SIMULATION RESULTS We investigated the C02 absorption in soda inside a packed column at pH 10,1. The steady state simulations are validated by experiments performed on a laboratory scale packed column. The experiment measurements are reported in figure 2 and compared with the continuum mechanics model and with a rate based model taking into account the enhancement of the reactions (Versteeg and al. 1990). The phase's fractions and the wet interfacial area that permit us to define the two dimensional geometry are obtained with correlations (Lee and Tsui, 1999). The simulated pH profile with the continuum mechanics model is in accordance with the experiments, and is better than the ratedbased model. Figure 3 and 4 are concentration profiles obtained thanks to the continuum mechanics model. These concentration profiles are suitable and in agreement with the theory. 11 n
*
le+1
e^q^eiiment
10 - -.-Q . . .continuummechanics model A rate-based mo del
X
le+0
n
le-1
9 -
's ^^'^ |le-3
K 8-
^le-4 7-
P le-5
,-a**
6i i
5 -
0,0
le-6 A
A
A
-^^
le-7 arx—X
— — 1
1,0 height of packing (m)
Figure 2: comparison ofpH evolution
2,0
2,78
3,28 3,78 liquid fllm length (mm)
Figure 3: concentration profiles at the outlet
6. CONCLUSION A new model of multicomponent reactive absorption has been presented. The comparison of the experiment and its simulation allows us to validate the model. Furthermore, global information, such as concentration or pH profiles, could be obtained thanks to a local description of transfer phenomena. One of the most interesting aspects of this model is that it requires only two parameters i.e. the wet interfacial area and the phase's fractions.
330
0,0 Cco2 (mol.m"^)
Figure 4 : Evolution of CO2 concentration (molni^) in the two-dimensional geometry
NOTATIONS c d D D f H m P r R s S T
molar concentration, mol.m'^ driving force, m'^ Maxwell-Stefan diffusivity, m^.s'^ deformation tensor, s'^ external force, N height of packing, m mass flowrate, kg.s'^ pressure. Pa rate of chemical reaction, kg.m'^.s" gas constant, 8,314 J.mol'.K'^ entropy, J.kg•^K'^ area, m^ temperature, K
T u V K A, r\ \i V
stress tensor, N.m' mass internal energy, J.kg"^ velocity, m.s'^ thermal conductivity, W.m ^K'' bulk viscosity, Pa.s dynamic viscosity, Pa.s mass chemical potential, J.kg'^ stoechiometric coefficient of chemical reaction p mass concentration, kg.m'^ ^ enhancement of chemical equilibrium, kg.m'ls"^
REFERENCES Whitman, W.G., 1923, Chem. Met. Sci., 29, 147 Lee, S.-Y. and Y.P. Tsui, 1999, Chem, Eng. Prog., July, 23-49 Taylor, R. and R. Krishna, 1993, Multicomponent mass transfer, Wiley, New-York Versteeg, G.F., J.A.M. Kuipers, F.P.H. Van Beckum and W.P.M. Van Swaaij, 1990, Chem. Eng. Sci., 45, 183-197 Rascol, E., M. Meyer and M. Prevost, 1999, ECCE2, Montpellier Schneider, R., E.Y. Kenig, A. Gorak, 1999, Chem. Eng. Res. Des., 77, 633-638 Truesdell, C , 1969, Rational thermodynamics, McGraw-Hill, New-York Slattery, J.C, 1981, Momentum, energy, and mass transfer in continua, McGraw-Hill, 2"^ edition. New-York