6880
Ind. Eng. Chem. Res. 2003, 42, 6880-6892
Investigation of a Mineral Melting Cupola Furnace. Part II. Mathematical Modeling R. Leth-Miller,*,†,‡ A. D. Jensen,‡ P. Glarborg,‡ L. M. Jensen,† P. B. Hansen,† and S. B. Jørgensen‡ Rockwool International A/S, Hovedgaden 584, DK-2640 Hedehusene, Denmark, and Department of Chemical Engineering, Technical University of Denmark, Building 229, DK-2800 Lyngby, Denmark
A mathematical model of a mineral melting cupola furnace for stone wool production has been developed for improving cupola operation. The 1-D, first-engineering-principles model includes mass and heat balances for the gas phase, five solid phases, and four liquid phases. The gas and solid/liquid phases flow countercurrently. Seven chemical reactions account for the conversions of coke, iron oxide, limestone, and gaseous species. The heterogeneous reactions of coke conversion are limited by both kinetics and mass transport. Heat transfer between phases is modeled including both convection and radiation. The model predicts gas concentrations; mass flow rates; and temperature profiles of the solid, melt, and gas in the cupola, as well as heat loss to the water-cooled walls. Inputs to the model include the coke, rock, and blast air properties, the blast air amount, and the coke percentage in the charge. The unknown model parameters are estimated on the basis of input/output measurements. A comparison of the predicted and measured concentration and temperature profiles inside the cupola shows good agreement. Introduction The production of stone wool products involves melting rock materials and subsequent spinning of the melt to fibers (wool). The melting of the rocks is usually carried out in a cupola furnace. A detailed process description is the cupola furnace is given in Part I of this work,1 and a schematic of such a cupola furnace is shown in Figure 1. Cupolas are widely used in the foundry industry, and cupola models have previously been developed. However, the focus of previous cupola models has aimed at foundry-related problems, which are different from those of stone wool production. Stanek et al.2-11 developed a steady-state 1-D model that includes heat and mass balances. A primary aim of this model is to predict the composition of the melt leaving the cupola. Predicting the composition of the melt from a foundry cupola is much more complex than predicting the composition of the melt in a stone wool cupola, because fewer chemical reactions take place in the latter. On the other hand, stone wool production requires an accurate prediction of the melt temperature, and this is not possible with the model of Stanek et al.,2 which neglects radiation. A 3-D model of a foundry cupola has also been developed.12 Predictions with this model compare favorably with reported experimental data of gas composition and gas temperature. However, 3-D computational fluid dynamics (CFD) calculations are computationally expensive, and systematic calibration of the model might be virtually impossible. Also, it might be necessary to make several simplifications and assumptions not necessary in a 1-D model. Selection of the number of spatial dimensions is, of course, closely linked with the aim of * To whom correspondence should be addressed. Tel.: +45 4656 0300. Fax: +45 4655 5990. E-mail: rasmus.leth.miller@ rockwool.com. † Rockwool International A/S. ‡ Technical University of Denmark.
Figure 1. Cupola furnace.
the model. On the basis of a 3-D model, Viswanathan et al.12 concluded that the gas is in plug flow from 0.3 m above the tuyeres and upward. This conclusion is the basis for the 1-D assumption made for the model developed in this work. Models of blast furnaces, which are similar to cupolas, have also been developed.13-16 These models are focused
10.1021/ie030770u CCC: $25.00 © 2003 American Chemical Society Published on Web 11/22/2003
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6881 Table 1. Assumptions Made during the Development of the Mathematical Model General Assumptions A.1 A.2 A.3
steady state constant pressure ideal gas law applies
A.3 A.4 A.5 A.6 A.7 A.8 A.9 A.10 A.11
Geometric Assumptions radially homogeneous properties, i.e., 1-D assumption constant bulk porosity heterogeneous reactions only on the outer surface of the coke spherical coke particles no disintegration of the coke particles spherical raw material particles of each type no disintegration of the raw material particles all droplets spherical all droplets the same size
A.18 A.19 A.20 A.21 A.22
Mass-Related Assumptions negligible vertical heat conduction and mass diffusion negligible axial mass dispersion net mass flow away from the reacting coke does not influence mass and heat transfer ash does not interact with anything in the cupola and leaves as fly ash in the flue gas at the same temperature as it enters chemical reactions apart from those in Table 3 do not affect the overall mass and energy balances coke particles are initially the same size and shrink uniformly during conversion no ash layer present on the reacting coke surface raw materials melt on the outer surface only raw material particles of each type are initially the same size
A.23 A.24 A.25 A.26 A.27 A.28
Thermal Assumptions heat of gas-phase reactions released in the gas phase heat of heterogeneous reactions released in the coke negligible vertical heat conduction in the solid phase and heat dispersion thermally homogeneous coke and raw material particles radiation between nongaseous phases not damped by the gas negligible heat transfer between nongaseous phases through physical contact, i.e., conduction
A.14 A.15 A.16 A.17
on on-line control of different aspects of the blast furnace. Among the models are a simple distributed model,13 a state-space model,14 a model for predicting slag viscosity using neural networks,15 and an expert system for on-line diagnoses of blast furnace conditions.16 The aim of this work is to obtain new insights through mathematical modeling into the conditions inside the cupolas used for stone wool production and, thereby, to develop a tool that can be used for optimization of cupola operation; for trouble shooting; and for predicting the impact of operational changes, such as the introduction of new raw materials in the process. The thermal efficiency of stone wool cupolas is only approximately 50%, i.e., approximately 50% of the heat released from combustion of the coke is used for heating and melting the rocks, whereas the rest is lost to the surroundings, mainly to the cooling water and as CO in the flue gas. Thus, there is significant potential for improvement of the energy efficiency of mineral melting cupolas. To achieve the goals of this work, a mathematical model of a mineral melting cupola is developed. Subsequently, the model is calibrated to experimental input/ output data, and the model predictions of internal conditions are validated against the experimental data reported in Part I of this work.1
complex, heterogeneous chemical and physical nature of coal prohibits the development of a truly molecularscale model of fixed-bed combustion/gasification,17 which is similar to the present cupola process. The development of the model focuses on detailed descriptions of the chemistry and mass- and heatexchange mechanisms rather than on spatial dimensions. Three-dimensional geometry and detailed chemistry are not yet an option, but the development of CFD tools and computer hardware seems to make such a combination possible within the relatively near future. In the following sections, the different parts of the model are presented. First, the balance equations are described, followed by the constitutive equations and correlations. In the subsequent solution of the equations, the constitutive equations are substituted into the balance equations. The model can handle one coke type and four different types of raw materials (rocks) in each simulation, i.e., the model calculates mass and energy balances for four types of rock with individual physical properties. The fractions in the charge are defined as follows
mc fc ) mc + mri
∑i
Model The mathematical model of the cupola furnace developed here is a static 1-D model, i.e., the properties in any cross section are assumed to be constant, and the variations are limited to the vertical direction. The model is based on process knowledge, i.e., it is a firstengineering-principles model. It involves a number of assumptions, the most important of which are listed in Table 1. Such assumptions are necessary because the
frj,sol )
mrj,sol
∑i
(1)
(2)
mri,sol
Both the coke fraction and the raw material fractions are functions of position because the coke burns and the raw materials melt in the cupola. At the top of the
6882 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
cupola (z ) Hf), the fractions are given as the charge composition. Mass Balances. Using matrix and vector notation, the mole balance equation is written as
0)-
∂N + Afυ·r ∂z
(3)
(4)
The matrix of stoichiometric coefficients, υ, and the vector of chemical reactions, r, are defined in the chemical reaction section. The mass balance of the rocks can be calculated from the coke balance using algebraic equations only. The inlet boundary conditions for the gas species are given as input to the model in terms of the blast air flow rate and composition. The blast air is typically ordinary atmospheric, air but it might be enriched with oxygen. For the coke, the boundary condition is NC ) 0 at z ) 0. In the formulation of the mass balance, plug flow of the gas and particles is assumed, i.e., dispersion is neglected (assumption 15). The mass balances of the rocks are trivial given that the mass flow rate is constant through the cupola and the fractions of coke and of each of the rock types in the charge are inputs to the model. Energy Balances. In the energy balances, the enthalpies are calculated with reference to the temperature Tref ) 298.15 K. The heats of reactions in the model are for T ) Tref. The governing equation for the different phases is
0)-
∂
[Hj(Tj)·Nj] +
∂z
∑i -∆Hrj,irj,iaj + ∑i qi
(5)
where the subscript j denotes either gas, coke, or raw material. The terms in the equation represent the following: (i) -∂/∂z[Hj(Tj)‚Nj], energy transport by convection; (ii) b∑i - ∆Hrj,irj,i, energy released in homogeneous gas-phase reactions; and (iii) ∑iqi, heat transferred from other phases and the furnace wall by convection and radiation. For the gas phase, aj ) b. For the gas phase and the coke
Hj(Tj) ) Mw,j
∫TT cp,j dTj j
ref
equation
(6)
The enthalpy function for the raw materials, Hr(Tr), is described in the Raw Material Properties section. Gas Phase. The temperature of the blast air that is given as input to the model serves as a boundary condition, i.e., at z ) 0. The blast air temperature is typically 500-800 °C. It is assumed that the heat released in the gas-phase reactions is released in the gas phase (assumption A.23, see Table 1). Coke Particles. The temperature of the coke as it enters the cupola at the top (i.e., at z ) Hf) is given as input to the model and provides the required boundary condition. The coke is not preheated, so the temperature is the ambient temperature, i.e., between -10 and 30 °C, depending on the factory and the time of year. The storage facilities of the coke also result in varying moisture contents, typically 5% (w/w), but values of up to 17% (w/w) are also encountered.
ref
nA,gfc ) km,A(CA,g - CA,c)ac
km,A )
25
(1000T K)
1.75
atm m2 P s
60
(395T K)
1.75
atm m2 P s
60
atm m2 P s
60
DCO2 ) 1.6 × 10-5
DH2O ) 2.2 × 10-5
(273T K)
1.75
ShA ) jDScA1/3Rep
Fg )
19
ShADA 2rc
DO2 ) 2.2 × 10-4
where the molar flow vector, N, is given by
N ) [NCO2 NCO NO2 NH2O NH2 NC]T
Table 2. Model Equations Describing the Mass Transfer between the Gas Phase and the Particle Surface
mg Vg
P
)
RT
25
∑x M
1 0.357 jD ) b Re 0.359
i
i
61
i
25
p
Rocks. The energy balances for rock particles cover both solid and liquid rock. It is assumed that the melts from different rocks do not mix until they reach the melt bath, i.e., outside the control volume of the model. Katz18 has found evidence of this in foundry cupola probe measurements. The boundary condition is the temperature of the rocks at the inlet of the cupola. The inlet temperature of the rocks is the same as for the coke, given that they are stored in the same building. Particles. In the model, it is assumed that the particles have a uniform temperature, i.e., there is no temperature gradient in the particles (assumption A.26, Table 1). This assumption is reasonable because heating of the center of the particle to 90% of the surface temperature takes approximately 5 min (see ref 19) and the heating rate is only ∼0.2 K/s. For the coke, exothermic and endothermic reactions on the surface influence the heating rate. However, it is mostly in the combustion zone of the cupola that the gradient can be large, and in this zone, the reaction rate is controlled by mass transfer; therefore, the temperature gradient in the coke is of lesser importance. Mass Transfer. The mass transfer between the gas phase and the coke, NA,gfc, is modeled using the film model, i.e., Fick’s law in a gas film surrounding the particles (see, e.g., ref 19). The equations needed for the mass-transfer model are listed in Table 2. The masstransfer coefficient for species A, km,A, is calculated from the diffusion coefficient and the Sherwood number. In the evaluation of the Reynolds number, it is assumed that the ideal gas law applies given that the cupola runs at 1.0-1.1 bar. In the first equation in Table 2, ac is the surface area of the coke per unit volume of the cupola. The calculation of ac is described in the Coke Properties section. The formation of CO in the gasification reactions of coke with CO2 or H2O generates two gas molecules from only one gas molecule. Thus, there will be a net flow away from the particle, known as Stephan flow;17,20 however, this is neglected (assumption A.16). The surface area of the coke is calculated from the mass and density, assuming that the coke particles are spherical (assumption A.6) and that all of the coke particles have the same size (assumption A.19). To
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6883
simplify the model, it is assumed that there is no size distribution of the coke. The correlation for mass transfer to particles in fixed beds is a general correlation. However, the coke and raw material particles are unusually large compared to the particles in typical fixed beds, and for this reason, a correction factor will be estimated from experimental data on full-scale cupolas. The mass-transfer coefficient is thus km,A,eff ) km,Aδkm, where δkm ) 1 indicates no change to the original correlation. Heat Transfer by Convection. The cupola model contains five solid phases (coke and four types of raw material) and four liquid phases (melt from the four types of raw material) that interact with the gas. The gas also exchanges heat with the water-cooled wall. The convective transfer between gas and particles in a fixed bed is modeled as transfer through a thin gas film. The heat-transfer coefficient is calculated from the correlation
mg hgx ) 0.61Re-0.41γcg Pr2/3 Af
(7)
where the subscript x is either c for coke, r for raw material, or l for melted raw material. The equation is valid for Re > 50 based on df.19 The factor γ is a shape factor (for spheres, γ ) 1.0). The convective heat transfers from the gas phase to the liquid melt droplet and to the solid raw material are calculated from
qgl,i ) hgl,i(Tg - Tl,i)al,i
(8)
qgr,i ) hgr,i(Tg - Tr,i)ar,i
(9)
where al,i and ar,i are the surface areas of liquid and solid rock, respectively, of type i per unit volume of the cupola. The calculation of the surface areas is described in a following section. The convective heat transfer from gas to coke is modeled using the expression
qgc ) hgc(Tg - Tc)ac
hgwdf [6(1 - b)]0.185 0.815 ) 0.069 Re kg b Tw -0.73 0.5 Pr (12) Tg
( )
Heat Transfer by Radiation. In the cupola model, a simple model for radiation is used. The wavelength spectrum is not considered and the emission is calculated as
qrad ) σT 4 A
(13)
where σ is the Stephan-Boltzmann constant (σ ) 5.672 × 10-8 W‚m-2‚K-4) and is the emissivity. The values for the gas emissivity and absorption ratio depend on the contents of CO, CO2, and H2O (diathermal molecules). Values of the emissivity and adsorption ratio as a function of gas composition and temperature have been published.24 For simplicity, the emissivity was set equal to the absorption ratio and was given a constant value of g ) 0.01. The net radiation from the gas phase to a liquid or solid phase (raw materials, coke, or cupola wall) is calculated from19
qrad,gx ) σg(Tg4 - Tx4)ax
with x ) li, ri, c, w (14)
where ax is the surface area of the liquid or solid phase per unit volume of the cupola. In the following paragraphs, expressions for radiation between solid and liquid bodies are presented. For all of these expressions, the absorption of radiation in the gas phase is neglected, i.e., the radiation emitted from a solid/liquid body is also assumed to reach another solid/liquid body. For simplicity, the dampening of the radiation is not subtracted from the radiation emitted from one solid/liquid phase to another. The solid/liquid particles/droplets are assumed to have so little physical contact that the only mechanism for heat exchange is radiation (assumption A.28). The coke-melt radiation is modeled as
qcli ) σ(cTc4 - liTli4)ali
(15)
(10)
where ac is the surface area of the coke per unit volume of the cupola. The calculation of ac is described in a later section. The convective heat transfer to or from a reacting coke surface has been reported to be more difficult to model than indicated by eq 10 due to the Stephan flow.17,21 Correction models accounting for Stephan flow exist,22 but they are not used here because the mass flow away from the particles is low. The heat transfer from the gas phase to the watercooled wall is calculated using an expression similar to that used for the particles, namely
qgw ) hgw(Tg - Tw)aw
Nugw )
(11)
where the surface area of the wall per unit volume of the cupola is simply aw ) 4/df, given that the cupola is a cylinder. The heat-transfer coefficient, hgw, is calculated from a correlation valid for fixed beds23
The surface area of the liquid is used because it is assumed that the liquid hold-up is so small that the droplets are not visible to each other. The surface area of the coke is larger, but this only leads to coke-coke radiation, and because the coke particles all have the same temperature at a given axial position, this radiation need not be taken into account. It is also assumed that no solid raw materials are present in positions in the cupola where melt is present. That is not the case in the melting zone where solid and liquid raw materials coexist, but for simplicity, this is not considered. The coke is an almost perfect blackbody, so the emissivity is set to c ) 1. The liquid is known to have an emissivity around l ) 0.85 (unpublished work at Rockwool). For the radiation from the coke to the rocks, it is assumed that, in the region where rocks are present in solid form, coke particles are not visible to each other, and melt droplets are not present. Thus, the radiation is
qrad,cr ) σ(cTc4 - rTr,i4)acfs,rifri
(16)
6884 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
The coke surface area is moderated with the factors fs,ri and fri to account for the disappearance of the rock as it melts and for the fact that the radiation is divided among the four different rocks. The rock emissivity is the same as for the liquid rock, i.e., r ) 0.85. The coke is assumed not to be in contact with the wall, so the heat is exchanged only by radiation. The radiation is calculated as
qrad,cw ) σ[c(ξTc,wTc)4 - wTw4]ac
(17)
where Tw is the temperature of the inside of the wall. The factor ξTc,w is a correction factor accounting for the fact that the coke surface temperature is lower at the wall than the average value of Tc used in the cupola model. The parameter ξTc,w eventually has to be estimated from experimental data. The rock-rock radiation is neglected, because the temperature difference is small. Heat Transport in the Water-Cooled Wall. The wall temperature is calculated from ordinary heat conduction equations, taking heat transfer from the steel to the cooling water into account. The temperature of the wall can thus be determined from eq 25
qrad,gw + qrad,cw + qgw )
(
)
dw 1 + hwo λw
reaction C(s) + CO2(g) f 2CO(g)
I
(
kCO2 ) 4 × 108 exp -
( )
30 000 K /(m/s) T
(
)
kH2O ) 1.2 × 109 exp -
kH2OCH2O,sac/(mol‚s-1‚m-3)
rH2O ) (kH2O ) 3kCO2 assumed according to ref 33) ∆Hr,H2O ) 131 328 J/mol ref 62 III
C(s) + 1/2O2(g) f CO(g)
(
kO2 ) (600 T/K) exp -
18 000 K /(m/s) T
)
/(mol‚s-1‚m-3)
rO2 ) kO2CO2,sac ref 33 ∆Hr,O2 ) -110 490 J/mol ref 62
CO(g) + 1/2O2(g) f CO2(g)
IV
20 130 K -1 /[s ‚(mol/m3)-3/4] T
(
)
kCO ) 1.3 × 1010 exp -
rCO ) kCOCCOCH2O1/2CO21/4/(mol‚s-1‚m-3) ref 33 ∆Hr,CO ) -282 830 J/mol ref 60
(18)
where Ea ) 24-36 kJ/mol. Using this expression, the ratio is more than 25 at 1500 °C, so the CO2 production is neglected in the model. Coke also react with CO2 and H2O to form CO and, in the case of H2O, also H2. The H2 and CO react with oxygen in the gas phase to form H2O and CO2. The rate constants for the heterogeneous reactions might not be very important to know precisely because mass-transport limitations also affect the rates of reactions. Hobbs et al.17 concluded, on the basis of reviews of refs 2633, that the most important parameter with respect to the dominant reaction mechanism of large particle combustion and gasification is the diameter. For large particles (>5-20 mm), gas film resistance is often the dominant mechanism,17,34 at least at temperatures over 1200 K. Hence, especially the coke combustion reaction rate constant need not be determined with high accuracy. In the mineral phase, three reactions are included: calcination of limestone and reduction of iron oxides (FeO and Fe2O3) to metallic iron and CO in reaction with the coke surface. These are important chemical reactions related to the minerals, because they consume a considerable amount of energy.
)
C(s) + H2O(g) f H2(g) + CO(g)
II
(Tw - Twater)
(19)
30 000 K /(m/s) T
rCO2 ) kCO2CCO2,sac/(mol‚s-1‚m-3) ref 33 ∆Hr,CO2 ) 172 340 J/mol ref 62
-1
where Twater is the temperature of the boiling water in the cooling jacket and hwo is the heat-transfer coefficient between the cooling water and the outside of the inner cupola wall. The heat-transfer coefficient, hwo, is estimated from experimental data. Chemical Reactions. The chemical reactions in the cupola model are listed in Table 3. Coke reacts with oxygen to form CO. In the coke-O2 reaction, the ratio of CO to CO2 formation is very high at high temperatures. A reported correlation for this ratio is26
Ea CO ) 102.5 exp CO2 RT
Table 3. Chemical Reactions Included in the Modela
H2(g) + 1/2O2(g) f H2O(g)
V
3420 K / T s-1‚(mol/m3)-3/2
(
kH2 ) 1.631 × 109(T/K)-3/2 exp -
)
rH2 ) kH2CH23/2CO2/(mol‚s-1‚m-3) ref 63 ∆Hr,H2 ) -241 818 J/mol ref 60 VI
CaCO3(s) f CaO(s) + CO2(g) ∆Hr,CaCO3 ) 177 500 J/mol ref 35
VII
FeO(s) + C(s) f Fe(s) + CO(g) r ) kFeO,red(1 - z) for 0 < z < 1 r ) 0 for 1 < z ∆Hr,FeO ) 75 340 J/mol ref 62
VIII
Fe2O3(s) + 3C(s) f 2Fe(s) + 3CO(g) r ) kFe2O3,red(1 - z) for 0 < z < 1 r ) 0 for 1 < z ∆Hr,Fe2O3 ) 238 220 J/mol ref 62
a
∆Hr values given at 298.15 K.
No fundamental model of iron reduction is available in the literature. The iron reduction reaction expression used in the model is developed on the basis of measurements of melt composition at the tuyere level and higher in the cupola from Part I.1 The reaction rate constant is adjusted so that 30% of the iron oxides are reduced in the cupola. Additionally, the measurements showed that the reduction takes place in the region from the melt bath and 0.5 m up. The rate expression is related to the position in the cupola, and the same expression is used for all cupolas simply because of lack of knowledge about the mechanism. The calcination reaction (reaction VI in Table 3) is modeled as illustrated in Figure 2. The reaction rate function is zero below 750 °C and then increases to a
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6885
The heat capacity of coke is determined from eq 2637
R
(∑ )
cp,c ) 3
wi
i
Figure 2. Normalized rate of calcination reaction.
maximum at 800 °C, thereafter decreasing to zero again at 850 °C.35 The reaction rate function is scaled to satisfy the mass balance in the cupola model. The simple model is used to reduce the system of equations and can be justified by the fact that the calcination of the limestone represents less than 3% of the energy consumption and so the exact location is less important. The scaling factor is determined from an overall carbon balance of the cupola. The reaction rate vector used in the mass and enthalpy balances, r, is given by
r ) [rCO2 rH2O rO2 rCO rH2 rCaCO3 rFeO rFe2O3]T (20)
[ ][
0 1 0 -1 1 -1
0 1 -1/2 0 0 -1
1 -1 -1/2 0 0 0
0 0 -1/2 1 -1 0
1 0 0 0 0 0
0 1 0 0 0 -1
0 3 0 0 0 -3
]
NA P Ng RT
CCO2,s )
CH2O,s )
2km,O2 kO2 + 2km,O2
(21)
(22)
CO2,g
km,CO2
(23)
CCO2,g
(24)
C kH2O + km,H2O H2O,g
(25)
kCO2 + km,CO2 km,H2O
T (1200 K/T)
Coke Properties. The density of the coke is approximately 1000 kg/m3 (more precise data for some types of coke are reported in ref 36). The ash in the coke is assumed to leave the cupola as fly ash at Tref and not to interact with any of the phases (assumption A.17).
-1
Mw,i
)( 2
J
kg·K
)
(26)
where wi is the weight fraction of element i (e.g., C, H, or O). The coke surface area per unit volume of the cupola is needed for computing the rates of heterogeneous reactions. An expression for the surface area is derived as follows. A constant number flow of coke particles, Nc,p, at all vertical positions can be assumed given that all of the coke particles are initially identical (assumption A.19), i.e.
dNc,p ≡0 dz
(27)
From this assumption and the formulas for the volume of a sphere and the connection between mass, volume, and density, the expression for the coke radius can easily be derived as
x
and
CO2,s )
(
e
rc ) rc,in
The gas-phase concentrations at the coke surface needed to evaluate the rates of the heterogeneous reactions are found using the ideal gas law and setting up a mass balance over the gas film, i.e.
CA )
e(1200 K/T)
3
and the reaction coefficient matrix, υ, is given by
υCO2 -1 υCO 2 υO2 0 υ) υ ) 0 H2O υH2 0 -1 υC
-1
1200 K
mc ) rc,in mc,in
3
x
Nc Nc,in
(28)
where rc and Nc are the coke radius and the carbon molar flow rate at a given vertical position, respectively. The expression for the surface area of coke per unit volume of the cupola, ac, can easily be derived from eq 28; the definitions of fc, fr,i, and b; and the formulas for the volume and surface area of a sphere. The expression for ac is
ac )
1 - b
3 rc
1+
Fc 1 - fc fr,i Fr,i fc
(29)
∑i
Equations 28 and 29 represent the unreacted shrinking particle model (assumption A.5), which can be implemented as two algebraic equations. In the unreacted shrinking particle model, it is assumed that the coke reacts at the outer surface only. This assumption is reasonable because the Thiele modulus, φ, for the reaction with O2 is large, being of magnitude
kg 4 m2 m 2 10 103 10 (0.1 m)2 kF S r p p p s m kg 2 ) 1011 ≈ φ ) 2 De m 10-5 s (30) The Thiele modulus is the ratio of the rate of reaction to the rate of diffusion into the particle, and a large value indicates that the reaction is confined to a narrow zone close to the surface and that transport into the particle is negligible.38-40 Only an insignificant ash layer is assumed to be formed because of the small amount of ash in the coke
6886 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
and the rapid gas flow (assumption A.20). It is additionally assumed that the coke does not break in the cupola (assumption A.7). The only mechanisms to break the coke particles in the cupola are mechanical and thermal stress, and the coke particles are assumed to be sufficiently strong to withstand the stress in the bed, an assumption that is supported by the experiments described in Part I1 and observations when the cupola is emptied for maintenance. The reactivity with CO2 has been measured for a number of cokes by thermogravimetric analysis (TGA) (see ref 36). Raw Material Properties. The raw materials used in stone wool cupolas are rocks, briquettes, and limestone. The rocks consist of different crystalline phases with different chemical compositions, and they are typically of volcanic origin. Briquettes are composed of granulated wool waste from the production line, different powder mineral materials, and cement to bind the powder. Briquettes are useful for tailoring the chemical composition of the charge and for recycling waste wool. Limestone is especially used on lines without briquettes for adjusting the viscosity of the melt. The cupola model requires an enthalpy function for each of the four raw materials that are included in the model. The enthalpy function includes the integrated heat capacities and the heat of fusion and is given by
Hri(T) )
∫T
T ref
ref
i
i
rri ) rri,0
(31)
where fs,ri is the solid fraction as a function of temperature. Experimental studies of a number of raw materials41 show that the heat of fusion for the raw materials is very small and can be neglected. Leth-Miller et al.41 presented a model that predicts the composition of a rock in terms of crystalline phases, the heat capacities of the material in solid and liquid form, the heat of fusion, and the solid fraction function, on the basis of only the chemical composition of the material. It is an empirical model based on several submodels42-50 that each are based on extensive experimental data. The limestone melts at very high temperature, so the transformation from solid to liquid state in the cupola is not melting, but rather dissolution into the melt of the other raw materials. The cupola model assumes that there is no contact between the different raw materials, so in the model, the solution is described as resulting from melting with no heat of fusion, assuming a function for the fraction. The solid fraction curve used is described in ref 41. The decrease in size of the raw material particles in the cupola can be described in the same way as for the coke particles because the same assumptions are applied except that the raw materials shrink by melting (assumption A.21). The shrinking particle assumption is reasonable given that the raw materials are heated from the outside, even though the model assumes that there is no internal temperature gradient (assumption A.26). The raw materials consist of several different crystalline phases that melt at different temperatures, but it is assumed that the raw materials melt on the surface only. The size of the raw material particles can be evaluated from the expression
x
3
mri
mri,0
) rr,0
x
Nri
(32)
Nri,0
where r, m, and N are the radius, mass flow rate, and molar flow rate of the raw material, respectively. The expression for the surface area of rock per unit volume of the cupola, ar,j, can easily be derived from the definitions of fc, fr,i, and b and the formulas for the volume and surface area of a sphere. The expression is
3 ar,j ) rr,j Fr,j
1 - b fc
1
Fc 1 - fc fr,j
+
(33)
Fr,j fr,i
∑i F
r,i fr,j
The subscripts i and j indicate raw material type. The model handles four types of raw material that have independent physical properties such as melting point and heat capacity. The rocks only change size through melting on the surface and do not break or disintegrate in any way (assumption A.9). An empirical correlation is used for the liquid holdup, Hl,7
( )( ml Af µl
dc
Hl ) 21.2
cp,s,rifs,ri dT + ∆Hfus,ri(1 - fs,ri) +
∫TT cp,l,r (1 - fs,r ) dT
3
0.51
) ( )
dc3gFl µl
-0.44
Sc dc Vc
-0.60
(34)
where the hold-up is defined as
Hl ≡
Vl Vf
(35)
The correlation was developed for melted steel and iron in a coke bed, but it is assumed also to be valid for the raw materials used for stone wool production. The liquid surface area per unit volume of the furnace, al, is evaluated as
Ndrop 3 A ) Hl Vf drop rd
al )
(36)
where Adrop ) 4πrd2 is the surface area of one droplet. The droplet radius is unknown; however, the quenching experiment (Part I1) indicates a size range. In the model, the size is estimated by a fit to experimental data. Gas-Phase Heat Capacity. Most published fixedbed models assume that the gas heat capacity is independent of temperature and gas-phase composition.17 This is a poor approximation, and in this cupola model, the heat capacity is a function of both the temperature and the gas-phase composition. The heat capacity of a gas mixture is calculated as51
cp,g,mix )
∑i Xicp,i
(37)
where Xi is the mole fraction of species i. Expressions for heat capacities for the gas species N2, O2, CO2, CO, and H2O were taken from ref 51. The heat capacity of H2, cp,H2, which is almost independent of temperature, was assumed to be constant at cp,H2 ) 30 J/mol‚K.25 Numerical Solution. The model consists of eqs 3 and 5, together with the boundary conditions at z ) 0
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6887
Figure 3. Dimensionless mole flows of the gas species except N2.
Figure 4. Temperatures of the gas, coke, and four raw materials.
and z ) Hf, the constitutive equations concerning mass and heat transfer, and the rates of reactions substituted into the balance equations. The model is discretized using orthogonal collocation.52 The method of orthogonal collocation uses polynomial approximation of the differential equations, resulting in a set of algebraic equations. The set of algebraic equations is solved using a Levenberg-Marquardt (L-M) method53 that starts in the steepest descent direction and, as convergence is approached, switches gradually to the Newton-Raphson (N-R) direction.54 To further stabilize the solver, backtracking55 is used. Backtracking means that the full L-M or N- R step is used only if the residual function value is decreased. Otherwise, the step is decreased until the residual function decreases. To improve the numerical solution, the model equations were made dimensionless with the blast air flow rate. For the simulations presented here, 30 internal nodes and one node at each of the two boundaries were used. The solution was tested with 50 internal nodes, and the result was found to be virtually the same, indicating that 30 internal nodes gives a sufficiently accurate solution. The model was solved on a Pentium III, 700MHz computer in approximately 2 min if the initial guess was sufficiently good. If no initial guess is available, the model can be solved in a number of steps, where the energy balance is not solved initially. When the gas concentration equations have been solved assuming a fixed temperature profile, the energy balance is solved without changing the gas concentrations, rates of reaction, or solid fraction curves. Finally, the entire model is solved by using first the coke temperature; then the gas-phase temperature; and as the last step, the raw material temperatures. This safe method of solution takes approximately 15 min. The solution does not converge if too little coke is fed to the cupola. In practice, this means that the cupola is extinguished. Model Results. The developed cupola model can predict the capacity of the cupola (in kilograms of melt per hour), the temperatures of the melt and flue gas, and the composition of the flue gas. The model also predicts the gas, coke, and raw material temperatures and mass flow rates and the gas concentration profiles inside the cupola. Figure 3 shows a plot of the mole fractions of the gas components predicted by the model. The plot in Figure 3 shows that oxygen is consumed within the first 0.6 m. Its reaction with coke forms the net product CO2, because CO is rapidly oxidized in the gas phase when oxygen is present. When the oxygen is consumed, CO and H2 are formed, and CO2 and H2O are consumed until the temperature has decreased sufficiently that
the gasification reactions have virtually stopped. At that point, the CO2 concentration increases again when the limestone decomposes, and the H2O concentration increases as the coke and rocks are dried. The concentrations can decrease in some regions of the cupola even when the species is not reacting. This is due to dilution when the molar flow rate increases as CO is formed (because one O2 molecule creates two CO), as the lime decomposes, and as the raw materials and coke dry. Figure 4 shows the temperature profiles of the gas, coke, and rocks predicted by the model. At the tuyeres, the gas has a temperature of 500 °C, whereas the coke and raw materials enter at the top at ambient temperature. The inlet air is heated by the combustion of CO in the gas phase to a maximum of 2200 °C at 0.3 m above the tuyeres. Subsequently, heat from the hot gas is transferred to the countercurrently flowing raw materials and the coke. At some point, the raw materials start to melt, but this is not visible on the temperature profile because the heat of fusion is so low. This differs from foundry cupolas, where the heat of fusion of the scrap iron and pig iron is so large that the temperature is almost constant during the melting. At the bottom, the melt is cooled slightly by the colder blast air entering the cupola. Calibration The cupola model developed contains a number of parameters, some of which are better known than others. The most uncertain parameters will be adjusted to obtain good agreement with full-scale observations. This section describes how the model is calibrated to the experimental values reported in Part I.1 The data that will be used for the calibration of the model are the measured input and output data (Tm, Tflue, CCO,flue, qcw). The other experimental data from the probe measurements and the quenching experiment will be used for validating the model in the next section. Parameters. The model of the cupola contains a number of parameters that can be determined independently of the cupola, such as the properties of the raw materials41 and coke.36 Other model parameters are very difficult (or impossible) to measure, and these must be estimated using the full cupola model and experimental full-scale cupola data. The heat- and mass-transfer coefficients are modeled using general correlations for spherical particles in fixed beds. The coke and rocks charged into the cupola are not spherical; hence, multiplicative surface area correction factors are used in the correlations, e.g., h/gc ) hgcδh,gc, where h/gc is used in eq 10 instead of hgc. The
6888 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
parameter δh,gc is then estimated with a value of 1 as the initial guess. The cupola model is 1-D, and thus, the temperature is an average temperature over the cupola cross section. Because the temperature at the wall is, in reality, lower than this average, the predicted radiation from the coke to the wall will be too high. To correct for this effect, the radiation from the coke to the wall can be predicted using a lower coke temperature than the predicted average. A multiplicative factor for the coke temperature is estimated for the radiation evaluation. The factor can be interpreted as the ratio of the real temperature of the coke at the wall to the average temperature (see eq 17). The melt looses energy in the melt bath. The melt bath is not included in the model, but if the melt temperature is to be compared with measured values, the heat loss in the melt bath must be taken into account. This is simply done by transferring an amount of energy to the cooling water given by a heat-transfer coefficient in the melt bath to the cooling water and the temperature difference between the melt and the cooling water. The heat-transfer coefficient is unknown and has to be estimated. Only a little information is available on the melt droplet size. The quenched cupola indicates that drops of 10-mm diameter are present, but much larger droplets are also found caught on the coke. The average droplet size in the model determines how large a surface area is available for heat exchange. The droplet size must therefore be estimated in the cupola model. The corrections to the mass- and heat-transfer correlations, the coke temperature ratio, the melt droplet size, and the heat-transfer coefficient from the melt bath to the cooling water will be estimated as described in the following section. Residual Function. The residual function that is minimized with a Levenberg-Marquardt method is given by N
Fresid )
{[kT (Tm,model,i - Tm,exp,i)]2 + ∑ i)1 m
[kTflue(Tflue,model,i - Tflue,exp,i)]2 + [kCOflue(COflue,model,i COflue,exp,i)]2 + [kQcw(Qcw,model,i - Qcw,exp,i)]2} + NPAR
∑j
(
wreg,j
)
Θj - Θj,0 Θj,0
2
(38)
where Tm is the temperature of the melt, Tflue is the temperature of the flue gas, COflue is the CO concentration in the flue gas, and Qcw is the cooling water loss. The constants, kx, are chosen such that all four terms have units of megawatts, i.e., the temperature deviation of the model compared to experimental data is converted to the difference in megawatts that the modeled and experimental values each represent in the cupola operation. Subscripts model and exp denote predicted and experimentally determined values, respectively. Subscript i denotes the index number of the data set and corresponding model solution, and N is the total number of data sets. The second summation is a regularization contribution, where wreg,j represents weights for each parameter Θj and Θj,0 represents the initial values of the parameters. Regularization is a method used to
balance the parameter variance and model bias in the minimization. The parameter variance is typically high when the model parameters are highly correlated. Here, the selected term is a Ridge regression. The regularization has earlier been used by Lei and Jørgensen56 for a nonlinear model and is described in general terms by, e.g., Hansen57 and Hastie et al.58 The four quantities in the residual function were chosen because they represent the major energy streams from the cupola. Procedure. The tuning of the model to experimental data is divided into four steps: (1) initialization of the parameter values, (2) sensitivity analysis, (3) parameter estimation through minimization of Fresid, and (4) validation. The first three steps are described below, and validation is described in the next section. Initialization of the Parameter Values. The initial values for the correction factors to the heat- and masstransfer correlations are unity, assuming that the correlations directly apply. The Tc,w/Tc ratio, ξTc,w, is set to a factor between 0 and 1 that gives reasonably good results (a manual tuning shows that ξTc,w ) 0.7 gives good results), and the heat-transfer coefficient in the melt bath is also initially manually tuned. The droplet size can be partly estimated from the quenched cupola where some droplets are found in the visible cross section. However, this estimation is highly uncertain because not all of the droplets could be caught during the quenching, and those that were caught might have been of a size that was not representative for normal cupola operation. Sensitivity Analysis. The sensitivity to each parameter was calculated as
∆j )
∂yi
∑i |ki ∂Θ ∆Θj|
(39)
j
where yi represents the output parameters of the model, ki represents weights to scale each term to the same units (megawatts in this case), and ∆Θj is a reference change or uncertainty for parameter j. The calculated sensitivity is thus (roughly) the resulting influence on the energy balance from a change in the model parameter. The sensitivity analysis was performed to determine the necessity of estimating each of the parameters. Parameter Estimation. The model parameters were estimated simultaneously using the Levenberg-Marquardt method.53,55 The required Jacobian was obtained numerically. The solutions to the parameters were found iteratively where the step for each iteration was calculated according to
(JTJ + λI)∆Θk+1 ) -JTFresid(Θk)
(40)
where Θk+1 ) Θk + ∆Θk+1 and Fresid is defined in eq 38. J is the Jacobian, and λ is set to a suitable value and reduced as the solution converges. For λ ) 0, the method is identical to the Newton-Raphson method. Evaluation of Fresid requires that the cupola model be solved for each data set, so the numerical evaluation of the Jacobian is expensive in terms of computing time. The problem can be formulated more efficiently by including the solution of the model and the minimization of the residual function in the same LevenbergMarquardt iteration, rather than using the nested structure with one iteration algorithm inside the other as is done here.59
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6889 Table 4. Measured and Modeled Melt Temperature (°C), Flue Gas Temperature (°C), CO Content in the Flue Gas (%) and Cooling Water Loss (MW) for Five Different Sets of Cupola Operation Conditions (Measured/Modeled) operating conditions coke (%) cal val (MJ/kg) Vblast/Vblast,max
1 13.5 28.465 0.802
2 16.1 31.194 0.802
3 17 31.194 0.813
4 15 30.721 1
5 14 30.721 0.879
std deva
Tm (°C) Tflue (°C) COflue (%) CW (MW)
1501/1485 165/176 6.4/5.4 1.64/1.63
1487/1510 158/154 9.6/6.4 1.34/1.59
1481/1517 176/154 10.9/6.9 1.52/1.56
1501/1541 195/193 6.5/7.9 1.78/1.70
1485/1525 175/160 6.6/6.8 1.61/1.50
23.7 12.7 2.27 0.14
a
Standard deviations of the five simulations from the measured data.
Calibration Results. The parameter estimation is based on five sets of data from the same cupola. The estimation procedure reduces the error of each of the data sets to less than 5% of the total energy consumption, which is approximately 8-10 MW. Some of the remaining error is due to the fact that the measured input and output energy flows did not match, i.e., the experimentally determined energy balance could not be closed. The correction factors were found to δh,gc ) 1.52, δh,gr ) 1.24, and δh,gc ) 1.13, which is reasonable because the deviation from a spherical shape will increase the surface area and thus the heat and mass transfer. The coke temperature ratio was found to ξTc,w ) 0.67, indicating that the coke temperature near the cupola wall is lower than the average, as expected. The melt droplet size was found to be rd ) 0.05 m, and the overall heat-transfer coefficient in the melt bath was estimated to be Umelt bath ) 387 W/K. The model can predict the output of the cupola using the new parameters with the precision indicated in Table 4, i.e., the melt temperature to within 45 °C, the gas temperature to within 25 °C, the CO content in the flue gas to within 4.5% (percentage points), and the cooling water loss to within 300 kW, in a 95 confidence interval, i.e., approximately two standard deviations. Considering the uncertainties in the measurements, this precision is fairly good. The precision was tested on the data sets that were used for the parameter estimation, which is not the optimal way to do it, but too few data sets were available for any to be left out of the parameter estimation and used only as validation data. Validation The calibration described in the previous section only compares the model with output data for operating cupolas. Extrapolation from the experimental data used in the calibration must be done with caution. The model is based on a detailed description of the processes inside the cupola, and in this section, the concentration profiles, temperature profiles, and mass flow profiles predicted by the model are compared to measurements for validation of the model. The validation data were obtained through the experiments described in Part I1 for two types of probes and the quenching. The data obtained from the two types of probe experiments (see Part I1) and the quenching experiment are used for the validation. Wall Probes. In this section, model simulations are compared to concentration and temperature profiles obtained with the wall probes (see Part I1). Figure 5 shows the measured CO concentration as a function of the vertical position at the wall, halfway from the center, and at the center, along with a CO concentration profile obtained through simulation with the cupola
Figure 5. CO profiles predicted and measured with the probes. (Vertical bars indicate the standard deviations of the measurements.)
Figure 6. Temperature profiles predicted and measured with the probes. (Vertical bars indicate the standard deviations of the measurements.)
model. Because the model is 1-D, the predicted curve can only be compared qualitatively to the experimental data. Figure 5 shows that the predicted CO concentration is very close to zero until 0.3 m above the tuyeres, and then the concentration increases to approximately 10% at 0.9 m above the tuyeres. The modeled behavior is very similar to the measured profiles, in terms of both the position in the cupola where the CO is formed and the final level of the CO. Figure 6 shows the measured temperature profiles and the predicted profiles of the gas, coke, and rock temperatures. The IR pyrometer was designed to measure the temperature of solid/liquid surfaces, so the temperature profile measured with the wall probes should be compared to the temperature of either the raw materials or the coke. Figure 6 shows that, at 0.45 m above the tuyeres, the model fits the measured data well. Higher in the cupola, the model predicts lower raw material and coke temperatures than the measured values. There is no apparent reason for this discrepancy. The conclusion is that there is qualitatively good agreement between the model-predicted and measured
6890 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
Figure 7. Dimensionless mass flows through the cupola of two raw materials in solid and/or liquid form.
the difference between the two measurements is not clear, but the simulation apparently agrees well with the measured gas temperatures. Discussion of the Model Validity. From the validation made in the above sections, it can be concluded that the model generally performs reasonably well. This indicates that the model includes the most important phenomena in a sufficiently detailed way, and that the assumptions made are reasonable for the model to describe gas temperature, CO concentration, and melt zone location. The good agreement with the melting zone location indicates that the temperature of the raw materials is also well predicted. The discrepancy between the measured data and the model results can, at least partly, be attributed to the lack of understanding of the cupola operation. There are some phenomena that have been omitted from the model but that have influence on the cupola. Indeed, the discrepancy is also a result of the fact that the phenomena included in the model does not encompass all of the important phenomena, i.e., the behavior that the model cannot explain must be explained in terms of phenomena that were excluded from the model. Discussion
Figure 8. Model-predicted gas temperature profile in the cupola and two gas temperature profiles measured with the top probes.
CO concentration profiles. The agreement for the temperatures in the cupola is less good. Quenching a Cupola. The results of quenching of the cupola (see Part I1) are compared to the model in qualitative terms using the obtained information about the position of the melting zone. Figure 7 shows simulated mass flow profiles in the cupola. In the simulation, two raw materials were included. At the top of the cupola, both raw materials are solid. The raw materials start to melt approximately 0.8 m above the tuyeres, leading to a decreasing solid mass flow rate and an increasing liquid mass flow rate. Approximately 0.4 m above the tuyeres, the raw materials are all melted, and the solid mass flow rate is zero, whereas the liquid mass flow rate is the same as the solid mass flow was at the top of the cupola. Coke is present at any vertical position. In the quenched cupola, there was evidence that the melting zone was located from 0.4 to 0.75 m above the tuyeres. The simulated results thus agree qualitatively with the experimental quenched cupola. In the wall probe experiments, it was possible to collect melt in the melt sample probe 0.9 m above the tuyeres, which indicates that the melting zone can extend to over 0.9 m above the tuyeres. However, these measurements were made on a different cupola than the one that was quenched. Top Probes. The predicted gas temperature profile is compared to the top probe measurements (see Part I1) in Figure 8. The radial position of the top probes was 0.2-0.3 m from the wall. The results show a very good prediction of the measurements of the gas profile marked 20.4, whereas the measurements made the day before, marked 19.4, are slightly colder at the top and slightly hotter 1.5 m above the tuyeres. The reason for
The model developed in this work has some limitations. Because the model is 1-D, it does not capture the effects indicated by the probe measurements that show significant differences between the CO concentration in the center and at the wall of the cupola. Extending the model to 2-D will probably be a very beneficial improvement. The chemical reactions can be described in more detail with, e.g., reaction mechanisms accounting for gas-phase radicals and with several reaction steps instead of the lumped expressions. The mechanisms for the reduction of iron and the calcination of limestone are very simple. This might not matter much for the limestone, but the model is not able to provide any reliable predictions of the amount of iron reduced in the cupola. Thus, the model cannot be used to optimize the cupola operation to minimize iron reduction. The model was calibrated to experimental data that are known to be of limited accuracy, and consequently, the precision of the model is also reduced. The deviations of the predictions from measured data are thus caused not only by inaccuracies in the model assumptions, but also by inaccuracies in the data. It is a weakness of the calibration that the experimentally determinated energy balances cannot be closed. Data reconciliation would improve this deficiency. Conclusion A mathematical model of a mineral melting cupola furnace has been developed. The model is a static 1-D model in which the one spatial dimension is the vertical position in the cupola. The model development is based on chemical engineering principles. The model was calibrated against experimental data and is able predict the melt temperature with a precision of 45 °C, the gas temperature with a precision of 25 °C, the CO content in the flue gas with a precision of 4.5% (absolute), and the cooling water loss with a precision of 3%. Considering the uncertainties in the measurements, this precision is acceptable.
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6891
A comparison of predictions with the probe measurements and results from a quenched cupola shows that the model describes the conditions inside of the cupola well, indicating that the relevant phenomena are included in the model and described in sufficient detail. Acknowledgment This work is part of the research program of Rockwool International A/S carried out in cooperation with CAPEC (Computer Aided Process Engineering Centre) and CHEC (Combustion and Harmful Emission Control) at the Department of Chemical Engineering, Technical University of Denmark. The project is funded by Rockwool and Erhvervsfremmestyrelsen (Danish Ministry of Business and Industry), and the industrial Ph.D. program is administrated by the Academy of Technical Sciences. List of Symbols a ) specific surface area, m2/m3 A ) area, m2 c ) specific heat capacity, J‚kg-1‚K-1 C ) concentration, mol/m3 d ) diameter, m D ) diffusion coefficient, m2/s Ea ) activation energy, J/mol f ) weight fraction Fresidual ) residual function g ) acceleration of gravity, m/s2 h ) heat-transfer coefficient, W‚m-2‚K-1 Hf ) height of cupola, m H ) enthalpy function, J/kg ∆H ) enthalpy of reaction or phase change, J/kg jD ) Chilton-Colburn factor J ) Jacobian k ) reaction rate constant, moln‚m-3n‚s-1 km ) mass-transfer coefficient, mol‚m-2‚s-1 l ) length, m L ) characteristic length, m (for particles, the characteristic length is the diameter dp; for the furnace, the characteristic length is the diameter df) m ) mass flow rate, kg/s M ) molcular weight, kg/mol N ) molar flow rate, mol/s P ) pressure, Pa q ) heat transfer, J/m3 r ) radius, m r ) rate of reaction, mol‚m-3‚s-1 R ) universal gas constant, J‚mol-1‚K-1 T ) temperature, K U ) overall heat-transfer coefficient, W/K v ) velocity, m/s V ) volume, m3 wreg ) weight for regularization X ) mole fraction z ) axial position, m ) porosity δ ) parameter correction factor ) emissivity λ ) heat conductivity µ ) viscosity, kg‚m-1‚s-1 F ) density, kg/m3 υ ) stoichiometric coefficient Θ ) adjustable parameter in model tuning ξTc,w ) Tc,w/Tc ratio Subscripts A ) species A
b ) bulk c ) coke f ) furnace flue ) flue gas g ) gas i ) component i j ) component j l ) liquid raw material m ) melt leaving the cupola melt bath ) melt bath p ) particle r ) rocks rad ) radiation ref ) reference s ) surface sol ) solid w ) wall Dimensionless Numbers Nu ) hL/k ) Nusselt number, heat transfer Pr ) cµ/k ) Prandtl number, heat capacity Re ) FνL/µ ) Reynolds number, velocity Sc ) µ/FD ) Schmidt number, diffusivity Sh ) kmL/D ) Sherwood number, mass transport
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Received for review April 20, 2003 Revised manuscript received June 4, 2003 Accepted June 20, 2003 IE030770U
