Modelling of a biomass fired furnace for production of lime

Chemical Engineering Science 64 (2009) 3417 -- 3426

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Modelling of a biomass fired furnace for production of lime F. Marias a,∗ , B. Bruyères b a b

Laboratoire de Thermique Energétique et Procédés (EA 1932), Université de Pau et des Pays de l'Adour, Rue Jules Ferry, BP 7511, 64075 Pau cedex, France Bruyères & Fils, Saint Front-sur-Lémance, 47500 Fumel, France

A R T I C L E

I N F O

Article history: Received 3 November 2008 Received in revised form 2 April 2009 Accepted 16 April 2009 Available online 24 April 2009 Keywords: Mathematical modelling Chemical reactor Biomass Limestone

A B S T R A C T

This paper deals with the mathematical modelling of a furnace devoted to the limestone processing. The novelty of this reactor is that the energy required by the endothermic reaction of calcination is provided by the combustion of biomass, making it compatible with sustainable development. The model is based on balance equations completed with phenomenological relations for the estimation of heat and mass transfer inside the reactor and for the estimation of the kinetics of the reaction. In a first section, the equations describing the behaviour of a single reacting particle of limestone are derived. The model relies on the assumption that the reacting particle obeys the shrinking core model. Then this model of a single particle is introduced in a more general one-dimensional model describing the whole furnace, where the different relevant properties depend upon the location within the reactor. Some industrial information is compared to numerical predictions, thus validating the model. Also the results provided by the numerical model allow for a better comprehension of the different phenomena occurring within the reactor. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction The aim of the paper is to develop a mathematical model for the industrial equipment depicted in Fig. 1. This device is designed to produce lime (CaO) by thermal decomposition of limestone (CaCO3 ). Because of the endothermicity of the calcination reaction CaCO3 → CaO + CO2 rH(298 K) = 182.1 kJ mol−1 lime manufacturing is an energy-intensive process. When heat is brought to the system using fossil fuels, mathematical models have been developed and validated for several kinds of kilns (Senegacnik et al., 2007a,b; Bes et al., 2007; Iliuta et al., 2002). Whereas some research is also carried on solar energy supply (Meier et al., 2006), the original feature of the installation presented here is that the thermal power required by the endothermic reaction of decarbonation is brought by combustion of biomass. This combustion is ensured by specific burners developed by the Bruyères & Fils company. Dealing with the design of new reactors of this kind, and once the load has been fixed, the main uncertainty lies in the position of the burners. Indeed, the height (zburner , see Fig. 1) and the depth at which they are inserted into the furnace (pburner ) are important parameters. In order to optimise this design, the choice has been made to develop a mathematical model able to represent the main

∗ Corresponding author. E-mail address: [email protected] (F. Marias). 0009-2509/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2009.04.022

physical and chemical processes occurring in such an installation, given a set of geometrical and operating parameters. The influence of the depth parameter is beyond the scope of the present study. It would have required a full description of the aerodynamics inside the reactor (which is a highly porous media), which depends upon the organisation of limestone blocks inside the furnace. Thus, in a first attempt, it has been chosen to focus on the first parameter, the height of the burner above the air distribution. More precisely, the model that has been written is a one-dimensional model assuming that the properties of the media do not depend upon their radial position inside the reactor. Of course, this is an important hypothesis. However, some results of the model show good agreement between numerical prediction and experimental results, which strengthens our assumptions. The first section of the paper is devoted to the formulation of the mathematical model. At first, the mathematical description of the fate of a single particle of limestone is given. Then this formulation is embedded in a more global reactor model. The second section of the paper presents numerical results of the model as well as validation aspects. Finally, a conclusion is drawn. 2. Mathematical modelling As mentioned in the introduction the model developed here is of the one-dimensional type. This means that the various properties of the media held within the reactor are supposed to be dependent solely upon the height above the bottom of the reactor. This

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CaCO 3

z

Air +CO2+HO2

.

m sol, H

d sol, H

Y CaCO3, H

T sol, H

reac reac

Y CaO, H

reac

reac

reac

ρ sol, H

reac

Hreac

H Biomass Air for biomass transport and combustion

p

burner

zburner

Dreac Solid system CaO

Air

Gaseous system CO 2

Fig. 1. Scheme of the system under investigation.

Energy

.

m air

assumption is a strong assumption. However, because of the radial and circumferential arrangement of the burners inside the kiln (which we are unable to describe further for confidentiality reasons), we expect that the biomass is uniformly distributed over cross sectional area of the kiln at the burners. Compared to the model developed by Senegacnik et al. (2007a,b) our model is slightly different because it includes the kinetics of the decarbonation reaction. On the other hand, compared to the work that was performed by Iliuta et al., 2002, our model does not compute the exact profiles of temperature and carbon dioxide concentration within the particle. This is of course a limitation, which has leaded us to assume a particular profile of temperature within the reacting particle. In a first paragraph, attention is given to the general concept of the model derived in the frame of this study. Then, physical and chemical processes occurring within the solid phase (limestone under decarbonation) are described and translated into mathematical formalism. Finally, the codification of the different processes occurring within the gaseous phase is presented.

Thus, the solid system is the place where both heterogeneous heat and mass transfer coupled with heterogeneous reaction occur.

2.1. General concept of the model

2.3. Assumptions

Because of the different natures of the materials present in the reactor (solid particles of limestone, gas) and because of the countercurrent mode of operation, the choice has been made to “separate” the two materials flowing in opposite directions. Hence, Fig. 2 sums up the way the overall model was built. Dealing with the energy yield brought by biomass combustion, it is hereafter supposed that this combustion is fully described by an instantaneous and complete one-step reaction. The air required by this combustion and the amount of H2 O and CO2 produced are computed according to the stoichiometry of the reaction.

The following assumptions are postulated in order to derive the mathematical model:

2.2. The solid system Afterwards, the terminology “solid system” will be used for the description of the solid particles taken as a whole. The composition, temperature and density of these particles change with respect to their position within the reactor from the top to the bottom. Once their thermal level is sufficient, the endothermic reaction of decarbonation takes place, leading to a release of gaseous carbon dioxide.

.

m bio

0

.

m gaz, 0

T gaz, 0

Y N 2, 0

Pgaz, 0

Y O 2, 0 Y CO

2,

YH

0

2O, 0

Fig. 2. Scheme representing the general concept of the model.

(A1) the solid particles entering the reactor are supposed to be spherical; (A2) the solid particles entering the reactor are supposed to be pure limestone CaCO3 ; (A3) within the reactor, particles are neither submitted to attrition nor agglomeration, hence sintering is neglected (industrial observation); (A4) within the reactor, the solid particles are not submitted to external shape modification. They leave the reactor with the same shape and size they had when they entered. In the frame of the mathematical model, this means that their external diameter is constant within the reactor; (A5) the reaction of decarbonation takes place as a shrinking core reaction (see below); (A6) within a single particle, heat transfer proceeds by conduction inside a solid;

F. Marias, B. Bruyères / Chemical Engineering Science 64 (2009) 3417 -- 3426

r0

3419

r0 rC

CaCO 3

CCO

Initial particle

CCO

2 ,s

CCO , gas 2

CaO Reacting particle

2, C

Fig. 3. Basic scheme for the description of the shrinking core model used in the study.

(A7) transport of particles inside the reactor is supposed to be of the plug flow type. The velocity of particles is computed according to the overall porosity of the reactor and the operating mass flow rate of the incoming limestone blocks. (A8) the diffusion coefficient of carbon dioxide in the mixture is supposed to be constant with the value DCO2 ,gas = 16 × 10−6 m2 s−1 .

Tgas Ts

TC

2.4. Single particle Before describing the behaviour of particles inside the reactor, the fate of a single particle is focussed here. Because of the low internal porosity of the incoming limestone blocks, it has been chosen to use the shrinking core model in order to describe the reactive particles (Garcia-Labiano et al., 2002; Cheng and Specht, 2006). This model is depicted in Fig. 3. Basically, it can be said that as the reaction proceeds towards the centre of the particle of un-reacted limestone, it leaves behind a layer of lime in which carbon dioxide produced by the reaction is drained out by diffusion. If the single particle is placed in a surrounding gas with a temperature Tgas and a molar concentration in carbon dioxide CCO2 ,gas , the following variables are introduced (see Fig. 4) • Ts : temperature at the external surface of the particle • TC : temperature at the surface of the reacting core • CCO2 ,s : molar concentration of CO2 at the external surface of the particle • CCO2 ,C : molar concentration of CO2 at the surface of the reacting core For such a particle, the mathematical expression describing the progress of the shrinking core ensues from the material balance on the reacting core of pure limestone: d dt



4 3 r  3 C CaCO3



decar = 4rC2 SCaCO 3

(1)

decar stands for the superficial reaction rate of limestone dewhere SCaCO 3 carbonation. According to literature (Hu and Scaroni, 1996; Stanmore and Gilot, 2005), this superficial reaction rate can be computed as: decar = −MCaCO3 kc f (PCO2 ,C ) SCaCO 3

(2)

where  kc = 6.078 × 107 exp

−205 000 RTC



(mol m−2 s−1 )

(3)

and

< 10−2 P

f (PCO2 ,C ) = 1

ifPCO2 ,C

f (PCO2 ,C ) =

if10−2 PCO2 ,equ. < PCO2 ,C < PCO2 ,equ

PCO2 ,equ −PCO2 ,C PCO2 ,equ

CO2 ,equ

r0

rC

0

Fig. 4. Internal profiles of molar carbon dioxide and temperature within a reacting particle.

The equilibrium partial pressure of CO2 at the reacting core is computed according to (Baker, 1962):   −19 680 (5) PCO2 ,equ = 1.886 × 1012 exp TC Thus, Eq. (1) can be rewritten as:

=

−MCaCO3 d (rC ) = k f (P ) CaCO3 c CO2 ,C dt

In order to complete the computation of the velocity at which the reacting core shrinks () the knowledge of the partial pressure of carbon dioxide at the core (PCO2 ,C ) as well as its temperature (T) are required. 2.4.1. CO2 The computation of the partial pressure of carbon dioxide at the core (PCO2 ,C ) as a function of the partial pressure of carbon dioxide in the external gas (PCO2 ,gas ) is computed as follows. (1) The molar rate of CO2 transferred by convection and diffusion inside the gaseous film surrounding the particle is evaluated as: ˙ film = 4r2 kg (CCO ,s − CCO ,gas ) N 2 2 0 CO2

(4)

(7)

where kg stands for the overall external mass transfer coefficient to a single particle. In the case of a fixed bed (because of the high residence time of solid particles, around 48 h, the reactor under investigation can be considered as a fixed bed) the relation of Stewart, (quoted by Bird et al., 2002), is used for the estimation of this coefficient: Sh = Sc1/3 Re(2.19 Re−2/3 + 0.78 Re−0.381 )



(6)

(8)

where Sh = 2kg r0 /DCO2 ,gas (1 − reac ) stands for the Sherwood's number of the particle; Re = 2r0 gas Ugas /(1 − reac )gas represents the

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Reynolds's number of the particle; Sc= gas / gas DCO2 ,gas stands for the Schmidt's number of the particle; DCO2 ,gas represents the molecular diffusion coefficient of CO2 inside the gas surrounding the particle; reac the overall porosity of the reactor; gas the density of the surrounding gas; Ugas the superficial velocity of gas inside the reactor; gas the dynamic viscosity of the surrounding gas. (2) The molar rate of CO2 transferred by diffusion inside the layer of lime produced “behind” the shrinking core is computed according to: ˙ CaO = N CO2

4r0 rC DCO2 ,eff r0 − rC

(CCO2 ,C − CCO2 ,S )

(9)

where DCO2 ,eff stands for the effective diffusion coefficient for CO2 inside the layer of lime. It is defined as: DCO2 ,eff =

CaO D CaO CO2 ,gas

(10)

where CaO represents the porosity of the lime layer produced “behind” the shrinking core; CaO stands for the tortuosity of this layer. (3) The molar rate of CO2 produced by the decarbonation reaction at the reacting core can be computed as: ˙ decar = 4r2 kc f (PCO ,C ) N 2 CO2 C

˙ CaO ˙ film = N N CO2 CO2

(12)

˙ CaO = N ˙ decar N CO2 CO2

(13)

and the use of the ideal gas law

where r H represents the heat of reaction ( r H(298 K) = 181.2 kJ mol−1 ). (4) The estimation of the total energy held within the shrinking core and within the lime layer requires a further assumption. Indeed, the temperature of the particle is known at its external surface and at the limit of the reacting core. Hence, it is hereafter supposed that the temperature of the core is uniform and equal to TC . Moreover, it is supposed that the temperature follows a linear profile from the external surface (Ts ) to the temperature of the core (TC ). As it was shown by Iliuta et al. (2002), the temperature profile inside the particle is slightly different from this kind of profile. However, the exact computation of this profile would have required describing the inner particle with a one dimensional model, in order to balance energy yield and consumption at each position inside the particle. This would have drastically increased the complexity of the system, and this is the reason why we have approximate the exact profile of temperature as described above. It is then possible to estimate the total enthalpy of both the lime layer and the limestone core: HCaCO3 =

4  r3 cp,CaCO3 (TC − Tref ) 3 CaCO3 C

HCaO = 4CaO cp,CaO     Ts − Tc Ts − Tc × rc − Tref (r03 −rc3 ) (r04 − rc4 )+ TC − 4(r0 − rc ) r0 −rc

(19)

(20)

(5) Finally, one is able to write the energy balance over the lime layer and over the limestone core: d(HCaCO3 ) ˙ decar ( r H + MCO hc ) + Q˙ CaO =N 2 CO2 CO2 dt

(21)

d(HCaO ) ˙ decar MCO (hs + hc ) − Q˙ CaO + Q˙ =N film 2 CO2 CO2 CO2 dt

(22)

(15)

In the last expressions, hcCO2 and hsCO2 stand, respectively, for the enthalpy to weight of gaseous CO2 leaving the reacting core and the enthalpy to weight of gaseous CO2 leaving the particle. These quantities are estimated according to: hcCO2 =



Tc Tref

cp,CO2 (T) dT

(23)

cp,CO2 (T) dT

(24)

where the emissivity of the particle as been imposed to a value of one. Also, as in the case of mass transfer, the external heat transfer coefficient is evaluated according to the relation of Stewart (quoted by Bird et al., 2002):

hsCO2 =

Nu = Pr1/3 Re(2.19 Re−2/3 + 0.78 Re−0.381 )

2.5. Reacting particles inside the reactor

(16)

where Nu = 2hT r0 / gas (1 − reac ) stands for Nusselt's number of the particle; Pr=gas cp,gas / gas stands for Prandtl's number of the particle; gas stands for the thermal conductivity of the surrounding gas; cp,gas stands for the heat capacity of the surrounding gas. (2) The heat transferred inside the lime layer (produced “behind” the shrinking core) is estimated as: 4 CaO rC r0 (TC − Ts ) Q˙ CaO = r0 − rC

(18)

(14)

2.4.2. Temperature The evaluation of the temperature of the core (TC ) as a function of the temperature of the surrounding gas (Tgas ) is performed in a similar manner. (1) The heat transferred from the external gas towards the particle is computed: 4 − Ts4 )] Q˙ film = 4r02 [hT (Tgas − Ts ) + (Tgas

Q˙ decar = 4rC2 kc r Hf (PCO2 ,C )

(11)

(4) Finally, the computation of the molar concentration of carbon dioxide at the shrinking core is computed assuming steady state for the particle, which results in equality of the above mentioned molar fluxes.

PCO2 ,C = RTC CCO2 ,C

(3) The heat required by the decarbonation reaction (endothermic) is computed according to:

(17)

where CaO represents the thermal conductivity of the lime layer.



Ts Tref

The previous paragraphs have shed light on the estimation of the rate of shrinkage of un-reacted limestone () (and thus the rate of conversion of a single particle). This rate requires the knowledge of internal data: the molar concentration of carbon dioxide at the shrinking core CCO2 ,C and its temperature (TC ). These data are estimated according to the external molar concentration of carbon dioxide in the surrounding gas (CCO2 ,gas ) and its temperature (Tgas ). However, because the properties of the gas change along the reactor, the immediate surrounding of each particle is also modified. Thus, it is necessary to compute the relevant properties as a function of the height above the bottom of the reactor.

F. Marias, B. Bruyères / Chemical Engineering Science 64 (2009) 3417 -- 3426

In order to fully describe the solid system, the knowledge of the following variables is required: n(z)

volumetric number of particles held within the reactor at a height z external radius of the particles held within the reactor at a height z radius of the core of the particles held within the reactor at a height z partial pressure of CO2 at the reacting core of the particles held within the reactor at a height z temperature of the reacting core of the particles held within the reactor at a height z mass fraction of limestone inside the particles held within the reactor at a height z mass fraction of lime inside the particles held within the reactor at a height z density of the particles held within the reactor at a height z

r0 (z) rc (z) PCO2 ,C (z) TC (z) YCaCO3 (z) YCaO (z)

sol (z)

The following paragraphs give insights into the equations that allow for the computation of these variables. Given assumption A3, the volumetric number of particles held within the reactor is not a function of the height above the bottom of the reactor. If the cross-sectional area of the reactor is supposed to be a constant (Sreac ), then: ˙ sol,Hreac 6m

n=

sol,Hreac d3sol,Hreac Usol Sreac

(25)

˙ sol,Hreac is the total mass flow rate of limestone at the input of where m the reactor (height Hreac ); sol,Hreac the density of the incoming limestone; dsol,Hreac the diameter of the incoming particles of limestone; Sreac is the cross sectional area of the reactor; and Usol is the overall velocity of the particles in the reactor. This last quantity is estimated as a function of the total height of the reactor (Hreac ) and the residence time of particles within the reactor (sol ) Usol =

Hreac

(26)

sol

Given the global porosity of the reactor (reac ) the residence time is computed according to:

sol =

sol,Hreac (1 − reac )Hreac Sreac ˙ sol,Hreac m

dsol,Hreac 2

d(rC ) = − dz

YCaCO3 =

(r03

rC3 CaCO3 3 − rC )CaO + rC3 CaCO3

YCaO = 1 − YCaCO3

(32)

(33)

Finally, the composition of the particle leads to the value of its density:

sol = YCaCO3 CaCO3 + YCaO CaO

(34)

2.6. The gaseous system This system is composed of the gaseous species (N2 , O2 , CO2 , H2 O) held within the reactor. Its composition, as well as its temperature, changes along the reactor because of • biomass combustion, • heat and mass transfer with reacting particles, • upward convective transport of the gas. 2.7. Assumptions The following hypotheses are used for the gaseous system. (H1) Because of the nearby atmospheric pressure prevailing inside the reactor, the ideal gas law is expected to be valid. (H2) The combustion of biomass is expected to be driven by a singlestep, complete and instantaneous reaction. The corresponding energy yield, air need and gaseous yield are computed according to the stoichiometry of the reaction. The biomass taken into account is a mixture of liquid water and C6 H9 O5 , a chemical component for which the lower heating value is supposed to be known. (H3) Axial diffusion across the reactor is neglected. 2.8. Modelling of the gaseous system

(28)

The change of the radius of the shrinking core along the reactor is computed according to Eq. (29): Usol

The derivation of the equations yielding the partial pressure and the temperature of CO2 at the core has been given in the previous paragraphs. As it is strictly the same for the particles inside the reactor, this formulation will not be remembered here. Given the radius of the reacting core, one is able to compute the composition of the particles:

(27)

Given assumption A4, the external radius of the particles is constant throughout the reactor: r0 =

3421

(29)

Energy balances for a single particle flowing inside the reactor yield: Usol

d(HCaCO3 ) ˙ decar (r H + MCO hc ) + Q˙ CaO ) = −(N 2 CO2 CO2 dz

(30)

Usol

d(HCaO ) ˙ decar MCO (hs + hc ) − Q˙ CaO + Q˙ ) = −(N film 2 CO2 CO2 CO2 dz

(31)

The complete characterisation of the gaseous system is performed according to the computation of the following data: gas

YN2 (z) gas YO2 (z) gas YCO2 (z) gas YH2 O (z) Ugas (z) gas (z) hgas (z) Tgas (z) Pgas (z)

mass fraction of nitrogen in the gas at a height z mass fraction of oxygen in the gas at a height z mass fraction of carbon dioxide in the gas at a height z mass fraction of gaseous water in the gas at a height z superficial velocity of the gas at a height z density of the gas at a height z enthalpy to weight of the gas at a height z temperature of the gas at a height z local pressure drop of the gas at a height z

Given assumption H3, the balance for chemical species is derived: gas

d(gas Ugas Yk ) dz

= Rbio + Rdecar + Rair k k k

(35)

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F. Marias, B. Bruyères / Chemical Engineering Science 64 (2009) 3417 -- 3426

In this equation, Rbio , Rdecar and Rair stand respectively for: k k k • The net chemical rate of production of species k by biomass combustion • The net chemical rate of production of species k by decarbonation • The source of species k brought by the air for biomass combustion

Before entering the paragraph devoted to the derivation of the energy balance, it is firstly required to give the relation between enthalpy to weight of the gas and its temperature. Given hypothesis H1 (the gaseous system obeys the ideal gas law), and given that the reference value of pure component at reference temperature is equal to zero, one can write: 4 



T

2.8.1. Chemical reaction rates associated with biomass combustion The chemical reaction considered for the description of biomass combustion is:

hgas =

C6 H9 O5 + 5.75O2 → 6CO2 + 4.5H2 O

where cp,k stands for the heat capacity of species k. The derivation of the energy balance leads to:

(36)

˙ bio is fed at Then, assuming that a total mass flow rate of biomass m a height z above the bottom of the reactor, the following chemical reaction rates are computed: ˙ bio MO2 5.75m 1 Rbio O2 = − Sreac z MC6 H9 O4 Rbio CO2 =

˙ bio MCO2 6m 1 Sreac z MC6 H9 O4

Rbio H2 O =

˙ bio MH2 O 4.5m 1 Sreac z MC6 H9 O4

2.8.2. Chemical reaction rates associated with decarbonation This reaction is a source for carbon dioxide:

2 Rdecar CO2 = 4rC nkc f (PCO2 )MCO2

+

1 ˙ air hair ) ˙ LHVbio + m (m Sreac z bio

−

4Hext (Tgas − Text ) Dreac

(39)

4 

Ykair

k=1



Tair Tref

cp,k (T) dT

(40)

where Tair stands for the value of the temperature of the incoming air at the burners. ext The term 4H Dreac (Tgas −Text ) represents the heat losses of the reactor. The density of the gas is computed according to ideal gas law: Pgas Mgas RTgas

(41)

gas

(42)

k=1

2.8.3. Source terms associated with the injection of air at the burners This quantification is performed according to the composition of the incoming air:

1 ˙ air YOair m 2 Sreac z

Finally, the local pressure drop is evaluated according to Ergun's Law: 2 −Pgas (1 − reac )2 gas Ugas (1 − reac ) gas Ugas = 150 + 1.75 3 2 3 (2r0 ) z reac reac (2r0 )

(43)

where reac stands for the overall porosity of the reactor.

Rair CO2 = 0

3. Solving the system

Rair H2 O = 0 1 ˙ air YNair m 2 Sreac z

The summation of the species balance leads to the overall mass balance: = Rdecar CO2 +

(38)

˙ decar MCO hs − Q˙ ) = n(N film 2 CO2 CO2

4  Yk 1 = Mgas Mk

Rdecar =0 N2

dz

cp,k (T) dT

Where the molar weight of the mixture is evaluated using:

=0

d(gas Ugas )

dz

gas =

=0 Rdecar O2

Rair N2 =

d(gas Ugas hgas )

hair =

where Sreac z represents the volume of biomass fed to the reactor by the burners

Rair O2 =

Tref

k=1

where LHVbio stands for the lower heating value of the wet biomass and hair stands for the enthalpy to weight of the incoming air:

Rbio N2 = 0

Rdecar H2 O

gas

Yk

1 ˙ air ) ˙ +m (m Sreac z bio

(37)

Table 1 sums up the overall set of state variables computed by the model, their associated equations as well as the nature of these equations (algebraic or ordinary differential). To sum up, it can be said that the model to be solved is composed of 13 algebraic equations and 9 ordinary differential ones. The discretization of the ordinary differential equations was performed using the finite volume method (Patankar, 1980) with “upwind” scheme for interpolation. Finally, the resulting set of equations is composed of 22 × NZ algebraic equations, where NZ stands for the number of volumes used for the discretization.

F. Marias, B. Bruyères / Chemical Engineering Science 64 (2009) 3417 -- 3426

3423

Table 1 Summary of variables (and associated equations) computed in the model. Variables

Equations

Name

Symbol

Name

Number

Type

Total enthalpy of the reacting core Total enthalpy of the lime layer Temperature of the external surface of the particle Temperature of the reacting core Molar concentration of carbon dioxide at the external surface of the particle Molar concentration of carbon dioxide at the reacting core Radius of the reacting core Molar flux of carbon dioxide in the film Molar flux of carbon dioxide in the lime layer Molar flux of carbon dioxide produced by decarbonation Heat flux inside the film Heat flux in the lime layer Heat flux produced by decarbonation Mass fraction of species k in the gas Superficial velocity of the gas Enthalpy to weight of the gas Temperature of the gas Density of the gas

HCaCO3 HCaO Ts TC CCO2 ,s CCO2 ,C rc ˙ film N CO2 ˙ CaO N CO2 decar ˙ NCO 2 Q˙ film Q˙ CaO Q˙ decar

Energy balance Energy balance Model for enthalpy inside the lime layer Model for enthalpy inside the limestone core Non accumulation of carbon dioxide at the surface Non accumulation of carbon dioxide at the reacting core Species balance External mass transfer flux Diffusion mass flux Reaction flux External heat flux Diffusion heat flux Reaction flux Species balance Overall mass balance Energy balance Model for enthalpy to weight of the gas Ideal gas law

(29) (30) (19) (18) (12) (13) (29) (7) (9) (11) (15) (17) (18) (35) (37) (39) (38) (41)

EDO EDO Alg. Alg. Alg. Alg. EDO Alg. Alg. Alg. Alg. Alg. Alg. EDO (Eq. 4) EDO EDO Alg. Alg.

Ergun's law

(43)

Alg.

Local pressure drop

gas

Yk Ugas hgas Tgas

gas

Pgas z

This set of algebraic (and non-linear) equations is solved using the Newton Raphson's method. Because the number of non-zero elements of the Jacobian matrix is small (108×NZ − 39) compared to its size ((22×NZ)2 ), the sparsity of this matrix was fully exploited. As an example, with a number of 150 volumes, (NZ = 150) the voidage of the Jacobian matrix is 99.85%. 4. Results This section is devoted to the analysis of the results yielded by the model. In a first paragraph, some insights are given into the existing installation on which the model has been validated. However, for confidentiality reasons, some of the geometrical parameters, as well as some operating parameters, are not provided. 4.1. Geometrical and operating parameters As mentioned in the introduction to this section, the precise values of the geometrical parameters of the furnace will not be provided here. However, in order to help the reader to “visualize” the industrial apparatus, some orders of magnitude are given. Height of the reactor (Hreac ) : 20 m Internal diameter of the reactor (Dreac ) : 3 m (working diameter, out of refractory lining) Height of the first burner stage (from the bottom of the furnace): 5m Height of the second burner stage (from the bottom of the furnace): 7 m The size distribution of the incoming limestone was given by the industrial partner. The corresponding `volume-surface mean' or `Sauter Mean' (Sevillen et al., 1997) is dsol,Hreac = 5.66 × 10−2 m. The incoming mass flow rate of limestone is approximately ˙ sol,Hreac =1.74 kg s−1 (150 t.d−1 ) for a total mass flow rate of biomass m ˙ bio = 0.22 kg s−1 (0.8 t.h−1 ). This biomass is 15% of approximately m wet (on a raw basis) for a lower heating value of 18 MJ kg−1 (on a dry basis). The total mass flow rate of biomass is distributed over two stages of burners, each of them being composed of several burners distributed over the circumference of the reactor. The vol-

umetric flow rate of incoming air at the bottom of the reactor is approximately 2 Nm3 s−1 at ambient temperature. 4.2. Validation Numerical results were obtained using 150 volumes for the discretization of the reactor (NZ =150). Global mass, energy and species balance over the whole reactor were satisfied up to a relative tolerance of 10−6 . A precise validation would have included a comparison of internal profiles of temperature, gas and solid composition. However, this information was not available on the industrial apparatus. Thus, the validation that was performed is relative to the different outputs of the reactor. Table 2 sums up these comparisons. The results show very good agreement with a maximal relative error of 12.1% in the case of the volumetric concentration of carbon dioxide in the outflow gas (20.3% for the numerical result and 18.1% for the experience) with the other relative errors being less than 5%. This relatively high value might be associated with the assumption used for the description of biomass (C6 H9 O5 ). Exact composition of this feed would lead to better estimation of he carbon dioxide level at the stack. The information brought by the analysis of Table 2 is global information. The main interest of the mathematical model developed in the frame of this study is that it gives insights into the different phenomena occurring within the reactor. Fig. 5 gives a representation of the evolution of the relevant temperatures along the reactor. The first remark that can be drawn is that there is no significant difference between the temperature of the gas and the temperature of the particle, either at his surface or at the reacting core. Given this first remark, one can analyse the temperature profiles from the top of the reactor to its bottom. This evolution is a succession of four steps. Firstly, there is a sharp increase in the temperature of the system. Because the gas temperature at the top of the reactor is slightly above the particle temperature, heat is transferred from the gas to the particle (see also Fig. 6). Then, once the difference between the temperature of the gas and the temperature of the particle has been eliminated, the heat flux from the gas to the particle (Q˙ film ) tends to zero (Fig. 6), which leads to a decrease in the slope of the particle temperature versus height curve (Fig. 5). Thus, from the top of the reactor to the second stage of burners, particles are heated by the countercurrent flowing gas which has

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Table 2 Comparison between experimental results and numerical prediction. Experimental results

Relative error (%)

16.4 454.7 9251.8 134.3 11.3 20.3 5.8 85.2

16.0 472 9340 130 11.9 18.1 5.9 84

2.5 3.7 0.9 3.3 5.0 12.1 1.7 1.4

Ts 1000

Molar flux (mol.s-1)

TC Tgas

Temp. (°C)

800 600

. decar. NCO r0 2 rC

3.0x10 -4

1.5x10 -4

0.018 st

1 stage burner

400

0 200

0

nd

2 stage burner

Height (m) 1ststage burner

2nd stage burner

0.024

Radius (m)

Residual mass fraction of CaCO3 (%) Global pressure drop of the reactor (mmCE) Volumetric output flow rate of gas (Nm 3/h) Temperature of the outflow gas (◦ C) Volumetric fraction of O2 (outflow gas, dry basis) (%) Volumetric fraction of CO2 (outflow gas, dry basis) (%) Volumetric fraction of H2 O (outflow gas, wet basis) (%) Mass flow rate of produced lime (t/d)

Numerical prediction

0.012 H reac

Fig. 7. Profile of molar flux of carbon dioxide from a single particle along the reactor. Evolution of the external and reacting core radii of a single particle along the reactor.

Hreac

Height (m) Fig. 5. Temperature profiles along the reactor.

80

nd

2 stage burner

Heat Flux (W)

60

. Q film . Q CaO . Q decar.

0.6 Mass fraction (-)

st

1 stage burner

40

N2 st

0.4

1 stage burner

nd

2 stage burner

O2 CO 2 HO 2

0.2

20 0

0 -20

Height (m)

Hreac

Fig. 8. Profiles of mass fraction of the gas along the reactor.

Height (m)

Hreac

Fig. 6. Profiles of heat fluxes to a single particle along the reactor.

been heated by biomass combustion between the two burner stages. At this location, the maximal value of the temperature of the gas is encountered, and once again, because there is a slight difference between the gas temperature and the particle surface temperature, heat is transferred towards the particle (Q˙ film > 0). This is also the location where the decarbonation reaction takes place. Indeed, both Fig. 7 and Fig. 8 indicate this phenomenon. More precisely, as Q˙ decar deviates from its zero value (see Fig. 6), the molar flux of carbon dioxide produced by decarbonation (or flowing across the lime layer, or leaving the particle) sharply increases, whereas the radius of the

reacting core decreases (Fig. 7). From the particle point of view, it can also be stated that their temperature begins to decrease before they leave the heated zone, which is imparted to the consumption of energy by the endothermic reaction of decarbonation. Then, whilst they flow towards the bottom of the reactor, the particles give away the heat to the countercurrent flowing gas. This results in a decrease in the temperature of the particle (Fig. 5) and a negative value of Q˙ film (Fig. 6). Fig. 8 shows the profiles of mass fraction of gaseous species inside the reactor. These profiles are strictly constant, except in the reacting zone, where both biomass combustion and decarbonation reaction contribute to the gas composition. Fig. 9 gives a representation of the profiles of superficial velocity and relative pressure along the reactor. The profile of velocity is strongly imparted by the

F. Marias, B. Bruyères / Chemical Engineering Science 64 (2009) 3417 -- 3426

2.0

-1000

1.6

-2000

1.2

-3000

-1

Velocity (m.s )

nd

2 stage burner

U gas Pgas

0.8

Relative pressure (Pa)

st

1 stage burner

-4000 -5000

0.4 Height (m)

H reac

Fig. 9. Profile of gas velocity and absolute pressure of the gas along the reactor.

temperature profile of the gas because of the dependence of the gas density with temperature according to ideal gas law, and also because of the supply of gaseous matter in the reacting zone (carbon dioxide and water from the combustion of biomass, carbon dioxide from the decarbonation). It can also be stated that a maximal value of 2.3 m s−1 is encountered for the gas in the reactor. The profile of relative pressure is linked to the profile of velocity because of Ergun's law. The modifications of the slopes in the curve representing its evolution versus height are imparted to the same processes as those responsible for the velocity profiles. From the analysis of these Figs. 5–9 it can be stated that the reactor mainly operates as a countercurrent heat exchanger. The reaction section occupies only approximately a tenth of the reactor. Another interesting result arising form this study is that it shows that biomass can be used as an alternative fuel. Indeed, important parameters in the decarbonation of limestone are the heat released by the combustion reaction and the associated production of carbon dioxide that might reduce the reaction rate of decarbonation. Burning 1 kg of methane leads to the release of 50 MJ and to the production of 2.75 kg of carbon dioxide. On the other hand, burning 1 kg of dry biomass leads to the release of 18 MJ and to the production of 1.64 kg of CO2 . This means than the ration of production of carbon dioxide per MJ released is higher in the case of biomass, than in the case of methane. This means that one could expect that this parameter would reduce the overall efficiency of the kiln. This study shows that it is not the case and that biomass can be used to fire the furnace.

where the surrounding conditions are affected by the height above the bottom of the reactor. The final mathematical model is composed of nine ordinary differential equations and 13 algebraic ones. The ordinary differential equations have been discretized according to finite volume method and the overall algebraic system has been solved using a Newton–Raphson's method. The assumptions that were formulated for the derivation of the model were validated by comparison of the numerical predictions of the model with industrial results. Of course, this comparison is a “macroscopic” one. Indeed, a comparison with, for example, internal profiles of temperatures would have been more relevant. However, because the industrial profiles were unavailable, this was the only way to validate the model. The analysis of the different results of the models, such as curves of state variable versus height above the bottom of the reactor, gives insights into the processing of limestone within the reactor. What can be concluded is that the reactor mainly operates as a countercurrent heat exchanger because the reacting zone is only approximately the tenth of the reactor. Finally, it can be said that this model is currently used for the design of a new industrial reactor based on the same concept that the one described in the paper.

Notation cp,gas cp,k CCO2 ,s CCO2 ,C CCO2 ,gas dsol,Hreac DCO2 ,gas DCO2 ,eff hgas hT Hreac kc kg

5. Conclusion In this paper, a mathematical model describing the processing of limestone furnace has been developed. The innovative nature of this reactor is that it is heated with biomass combustion. It is a vertical cylinder tube where fresh limestone is fed at the top while air is conveyed to its bottom. Because the endothermicity of the decarbonation reaction, and because this reaction only begins at high temperature (approximately 800 ◦ C for kinetics reasons), heat is required by the process. In our case, this heat is brought thanks to the combustion of biomass. The mathematical model that represents this installation relies on several assumptions and on the balance equations of mass, species and energy. The conversion of the limestone to lime in a single particle has been described using the shrinking core model. Energy balances have been formulated for both the reacting core and the lime layer surrounding the core. The carbon dioxide produced by the decarbonation reaction has been supposed to leave the particle at the same rate than the one at which it is produced. The model for a single particle has been included in a more general model of reactor

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˙ air m ˙ bio m ˙ sol,Hreac m Mk n ˙ film N CO2 ˙ CaO N CO2 ˙ decar N CO2 PCO2 ,C PCO2 ,equ.

heat capacity of the surrounding gas, J kg−1 K1 heat capacity of species k, J kg−1 K1 molar concentration in CO2 at the external surface of the particle, mol m−3 molar concentration in CO2 at the reacting core, mol m−3 molar concentration in CO2 in the gas surrounding a particle, mol m−3 external diameter of the incoming limestone blocks, m diffusion coefficient of CO2 in the gas, m2 s−1 effective diffusion coefficient of CO2 in the lime layer, m2 s−1 enthalpy to weight of the gas, J kg−1 external heat transfer coefficient to a single particle, W m−2 K−1 total height of the reactor, m kinetic constant for the decarbonation reaction, mol m−2 s−1 external mass transfer coefficient to a single particle, m s−1 total mass flow rate of incoming air at the burners, kg s−1 total mass flow rate of incoming biomass at the burners, kg s−1 total mass flow rate of incoming limestone at the top of the reactor, kg s−1 molecular weight of species k, kg mol−1 volumetric concentration of particles inside the reactor, m−3 molar flow rate of CO2 transferred in the gaseous film surrounding a single particle, mol s−1 molar flow rate of CO2 transferred by diffusion in the lime “behind” the reacting core in a single particle, mol s−1 molar flow rate of CO2 produced by the decarbonation reaction at the shrinking core, mol s−1 partial pressure of CO2 at the shrinking core, Pa equilibrium partial pressure of CO2 allowed for decarbonation, Pa

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Q˙ film Q˙ CaO Q˙ decar r0 rC R Rair k Rbio k Rdecar k Sreac decar SCaCO 3 Ts TC Tgas Ugas Usol gas Yk Ykair

F. Marias, B. Bruyères / Chemical Engineering Science 64 (2009) 3417 -- 3426

heat flow rate transferred from the surrounding gas to the surface of a single particle, W heat flow rate transferred inside the lime layer “behind” the reacting core in a single particle, W heat flow rate consumed by decarbonation reaction in a single particle, W initial radius of the incoming limestone blocks, m radius of the reacting core of a single particle, m ideal gas law constant, J mol−1 K−1 source term for species k associated with the supply of air at the burners, kg m−3 s−1 chemical reaction rate of species k associated with the combustion of biomass, kg m−3 s−1 chemical reaction rate of species k associated with decarbonation reaction, kg m−3 s−1 cross-sectional area of the reactor, m2 superficial chemical reaction rate of decarbonation, kg m−2 s−1 temperature of the external surface of a single particle, K temperature of the reacting core of a single particle, K temperature of the gas surrounding a single particle, K superficial velocity of the gas within the reactor, m s−1 velocity of the particles within the reactor, m s−1 mass fraction of species k in the gas, dimensionless mass fraction of species k in the air, dimensionless

Greek letters

r H Pgas z CaO gas

molar heat of decarbonation reaction, J mol−1 pressure drop in the gas, Pa height of a burner stage, m porosity of the lime layer, dimensionless thermal conductivity of the gas, W m−1 K−1

gas CaCO3 gas sol,Hreac CaO sol 

dynamic viscosity of the gas, Pa s density of limestone, kg m−3 density of the gas, kg m−3 density of the incoming blocks of limestone, kg m−3 tortuosity of the lime layer, residence time of particles within the reactor, s rate of shrinkage of the shrinking core, m s−1

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