Dynamic Analysis of the Temperature and the Concentration Profiles of an Industrial Rotary Kiln Used in Clinker Production

ABSTRACT

Cement is one of the most used building materials in the world. The process of cement production involves numerous and complex reactions that occur under different temperatures. Thus, there is great interest in the optimization of cement manufacturing. Clinker production is one of the main steps of cement production and it occurs inside the kiln. In this paper, the dry process of clinker production is analysed in a rotary kiln that operates in counter flow. The main phenomena involved in clinker production is as follows: free residual water evaporation of raw material, decomposition of magnesium carbonate, decarbonation, formation of C₃A and C₄AF, formation of dicalcium silicate, and formation of tricalcium silicate. The main objective of this study was to propose a mathematical model that realistically describes the temperature profile and the concentration of clinker components in a real rotary kiln. In addition, the influence of different speeds of inlet gas and solids in the system was analysed. The mathematical model is composed of partial differential equations. The model was implemented in Mathcad (available at CCA/UFES) and solved using industrial input data. The proposal model is satisfactory to describe the temperature and concentration profiles of a real rotary kiln.

Key words:
Clinker; dynamic analysis; mathematical modelling; rotary kiln

INTRODUCTION

Some of the challenges facing the cement industries are the high energy demand of production, the continuous increase in fuel prices, process complexity and environmental impact (Atmaca and Yumruta 2014, Tsamatsoulis 2014, Kaddatz et al. 2013, Gartner and Macphee 2011, Schneider et al. 2011, Mujumdara et al. 2007). To address these challenges, there is great interest in optimising the cement production process (Copertaro et al. 2015, Utlu et al. 2006).

The best-known type of cement is called Portland cement, which is defined as a hydraulic cluster that is basically obtained by grinding a mixture of clinker and gypsum (Copertaro et al. 2015). Thus, one of the main steps of the process for obtaining cement is the synthesis of the clinker (Atsonios et al. 2015, Saidur et al. 2011). Synthesis occurs inside the rotary kiln and involves complex physical (phase changes) and chemical (endothermic and exothermic reactions) processes (Lourenço et al. 2013, Saidur 2011, Silva 2007, Boateng and Barr 1996). The main clinker components are C₃S (3CaO.SiO₂), C₂S (2CaO.SiO₂), C₃A (3CaO.AlO₃) and C₄AF (4CaO.Al₂O₃.Fe₂O₃). The intermediate reactions for obtaining the clinker and their respective heats of reaction are shown in Table I, and the temperature ranges of the formation of chemicals are shown in Table II.

TABLE I
Chemical reactions of the clinker production process ( Paula 2009 , Smith 2007 ).

Clinkerisation step	Chemical Reaction	Reaction heat (kJ/kg)
Evaporation of residual free water and water as a result of the clay combination	H2O(l) →H2O(v)	+2443 (25°C)
Calcination	CaCO3(s) ↔ CaO(s) + CO2(g)	+1766 (20°C)
Magnesium carbonate decomposition	MgCO3 (s)→ MgO(s) + CO2(g)	+1188 (20°C)
Formation of the liquid phase	3CaO(s) +Al2O3(s) → 3CaO.Al2O3(s) (C3A) 4CaO(s) + Al2O3(s) + Fe2O3(s) → 4CaO.Al2O3. Fe2O3(s) (C4AF)	-15 (20°C) -84 (20°C)
Formation of silicate dicalcium	2CaO(s) + SiO2(s) → 2CaO. SiO2(s) (C2S)	-717 (20°C)
Formation of tricalcium silicate	3CaO(s) + SiO2 (s) → 3CaO. SiO2(s) (C3S)	-528 (20°C)

TABLE II
Temperatures of the chemical reactions of clinker production ( Paula 2009 ).

Temperature	Chemical Reaction
Above 800°C	Start of the CaO manufacturing process
Between 800°C and 1200°C	Formation of C2S
Between 1095°C and 1205°C	Formation of C3A and C4AF
Between 1260°C and 1455°C	Formation of C3S from C2S
Between 1455°C and 1300°C	Crystallisation of C3A and the C4AF liquid phase C2S and C3S silicate practically remain unchanged both in form and composition

Clinker production can be performed in a dry or wet process (Paula 2009). On one hand, in the dry process, the mixture of agglomerates and aggregates are completely dried and ground to feed the kiln in powder form. On the other hand, in the wet process, the mixture is a mud that is fed into the kiln with approximately 30-40% of moisture (Saidur et al. 2011, Del Coz Díaz et al. 2002). This paper considers the dry clinker production, which is the most used process in Brazilian production of cement (Kihara and Visedo 2014).

A rotary kiln in a dry cement production can be divided into five zones (Stadler et al. 2011): heating zone, calcining zone, transition zone, firing zone and cooling zone. The position of the zones along the kiln depends on the temperature and the chemical reactions being performed in the solid (Spang 1972). The temperature profile along the rotary kiln is generally not directly measured due to the scarcity of sensors that supports its internal operating conditions. Thus, estimating the temperature profile along the kiln is an important object of study.

In a previous work (Spang 1972), a dynamic model was developed. The model was composed of partial differential equations describing the mass balance and the system power (Spang 1972). The model is capable of predicting the concentration and temperature profiles along a rotary kiln operating in counterflow (Spang 1972). A flame model was also developed to quantify the amount of energy supplied to the system (Spang 1972). The equations that comprise this model (Equations 1 to 29) are presented in the Appendix. The results qualitatively describe the behavior of kilns, but the steady state was not reached.

In the present paper, an adaptation of the previous model is implemented to make it more realistic (Spang 1972). Using realistic values of the operational properties provided by a Brazilian company, the concentration and temperature profiles of a real rotary kiln used in the industrial production of the clinker is obtained.

CALCULATION PROCEDURES

Initially, a modification of the original model (Spang 1972) was performed to obtain a more realistic description of the temperature and concentration profiles of the clinker components along the rotary kiln.

In the originally proposed model (Spang 1972) the variation of gas temperature over time was not considered. To improve the description of the process, an adjustment was made in the gas energy balance (Equation 13 of the Appendix), which consisted of adding a term of gas temperature variation into the equation. In this way, Equation 13 was substituted by Equation 30.

${\text{A}}_{\text{g}}{\text{C}}_{\text{pg}}{\text{ρ}}_{\text{g}}{\text{v}}_{\text{g}}\left(\frac{\partial \text{Tg}}{\partial \text{z}}\right)={\text{β}}_1\left({\text{T}}_{\text{w}}-{\text{T}}_{\text{g}}\right)+{\text{β}}_2\left({\text{T}}_{\text{s}}-{\text{T}}_{\text{g}}\right)-{\text{C}}_{\text{pg}}{\text{ρ}}_{\text{g}}{\text{A}}_{\text{g}}\left(\frac{\partial \text{Tg}}{\partial \text{t}}\right)+{\text{q}}_{\text{f}}$ (30)

${\text{T}}_{\text{g}}\left(\text{L},\text{t}\right)={\text{T}}_{\text{gi}}$

The flame model originally described (Spang 1972) (Equations 17, 18 and 19 of the Appendix) was replaced by an amount of energy supplied to the kiln by the fuel combustion. These changes enable the industrial professional to use the model more directly. In this paper, the quantity of energy proposed for the fuel combustion is constant and equal to 326.7 kW/m (3.9×105 BTU/ft.h). This amount, adjusted by trial and error, approaches the values commonly used in industry. Thus, the new proposed model is composed of Equations 1-12, 14-16 and 20-30 as listed in the Appendix.

After adaptations, realistic values of the operational properties of the clinker production as reported by a Brazilian industrial plant were employed in the model. The composition of the raw material used in the industrial production of the clinker is shown in Table III. The values were normalized in relation to CaO (kg/kg). Other industrial operating properties employed in the model are shown in Table IV. The inner radius of the kiln was estimated to be 1.84 m (6.05 ft), the initial solid temperature, 562 ºR (312 K), and the initial temperature of the kiln wall, 662 ºR (367 K). These estimates were performed based on previous data (Spang 1972) and industrial realities.

TABLE III
Input composition of the raw material in the rotary kiln reported by industry.

Chemical Species	Normalized Composition in relation to CaO (kg/kg)
CaCO3	1.6666
SiO2	0.4849
Al2O3	0.1133
Fe2O3	0.0651
H2O	0.0117

TABLE IV
Rotary kiln data reported by industry.

Variables	American Units	International System
Inlet gas temperature	1391.67 ºR	773.15 K
Radio of the kiln (r)	7.00 ft	2.15 m
Length of the kiln	295.28 ft	90.00 m
Outside radius (r2)	2.148 m	7.05 ft

The model was solved for different inlet speeds of gas and solid until the final composition of the clinker obtained in the simulation was similar to the actual values obtained in the cement industry ( V). The best values for the gas and the solid inlet speeds were 274.32 m/s (-900 ft/hr) and 5.4864 m/s (18 ft/hr), respectively. Next, the temperature and concentration profiles obtained for the rotary kiln were analyzed considering these values.

For further analysis, the established inlet speed of gas and solid were varied as -20%, -10%, +10% and +20% to characterize the differences between the systems profiles. Subsequently, the obtained profiles were analyzed.

The model was solved using the discretization method for finite differences. Forty points of discretization were defined along the length of the rotary kiln. Discretization of the points was implemented according to the incoming stream of gas and solids into the kiln (Figure 1). For the discretization of the equations related to the solids, backward differentiation was used because they are fed into the beginning of the kiln (z = 0), and forward differentiation was used in the equations related to the gas because it is fed into the end of the kiln (z = L). The ordinary differential equations, functions of time, resulted from the discretization that were resolved by the numerical integrating method of Runge-Kutta with a variable step for error control, with an established tolerance of 10^-7. The model was implemented in Mathcad.

Figure 1
Volume control used in the mathematical modelling of the rotary kiln (adapted from Spang 1972).

RESULTS AND DISCUSSION

Successive integrations were made in the model equations until the variations between the results were no longer observed, which indicates achievement of the steady state of the rotary kiln. The temperature profiles of the gas, solid, and wall along the kiln’s length in continuous operation are presented in Figure 2a. As expected, the kiln’s temperature increases along its length to a maximum and then decays.

Figure 2
Temperature profiles of the gas, the solid and the industrial rotary kiln’s wall (a); concentration profiles of CaCO3 (b); concentration profiles of water (c) and concentration profiles of C3S, C2S, C3AF and CaO (d) in the industrial rotary kiln at steady state.

The CaCO₃ concentration profile along the length of the kiln is shown in Figure 2b. The profile follows the general pattern of a previous simulation (Spang 1972). Initially, there is a high concentration of CaCO₃ that decreases along the length of the kiln until it reaches zero.

The water concentration profile along the length of the kiln in steady state is shown in Figure 2c. The concentration of molecular water contained in the beginning of the feeding of solids decreases along the kiln until it becomes zero, as expected.

The concentration profiles of C₃S, C₂S, C₃A, C₄AF, and CaO along the length of the kiln in continuous operation are shown in Figure 2d. A decreasing behavior in the concentration of reactants followed by the formation of the clinker components is observed, which is in agreement with the expected.

The clinker composition simulated by the model and the industrial clinker composition are presented in Table V. The final concentrations of the clinker components obtained by the model are relatively close to the actual concentrations of the cement industry. The CaO.Al₂O₃ and 12CaO.7Al₂O₃ elements were not taken into consideration during modelling because they were not considered as part of the main elements of the clinker formation. The authors believe that this consideration caused the difference between the real (industrial composition) and the simulated composition of 3CaO.Al₂O₃ presented in Table V.

TABLE V
Comparison of the results obtained by the model and the industrial composition of the clinker.

Chemical Species	Industrial Composition of the Clinker (mol/mol)	Simulated Composition of the Clinker (mol/mol)
3CaO.SiO2	0.5600	0.4630
2CaO.SiO2	0.2215	0.3074
CaO.Al2O3	0.0848	-
12CaO.7Al2O3	0.0087	-
3CaO.Al2O3	0.0816	0.1706
4CaO.Al2O3.Fe2O3	0.0433	0.0589
CaO	0.0000	0.0000
TOTAL	0.9999	0.9999

Note that the final composition of the clinker widely varies from one industry to another and may vary even between operations of a same kiln because the composition of raw material is not constant, i.e., it may depend on the natural sources. In addition, changes in the operating conditions of a kiln produces variances in the final clinker concentration. An important aspect to highlight is that during the simulation of the actual system, it was not trivial to set values of inlet gas flow and solid flow that together satisfied the required values for the profiles of temperature and clinker concentration.

The profiles of temperature and clinker concentration obtained by varying the inlet gas speed by -20%, -10%, +10% and +20% are shown in Figures 3 and 4, respectively. The temperature profiles in Figure 3 are very similar, with only slight differences in the shapes of the temperature peaks noticed. In Figure 4, it is observed that C₃A, C₄AF and CaO have similar profiles in all of the proposed systems, whereas C₃S and C₂S have different profiles in each one of them. In general, the increase of the inlet gas speed from -20% to +10% increases the final concentration of C₃S and decreases the final concentration of C₂S. However, with +20% increase of inlet gas speed there is a decrease of the final concentration of C₃S and an increase of C₂S concentration.

Figure 3
Temperature profiles of the gas (Tg), the solid (Ts), and the wall (Tw) of the industrial rotary kiln at the steady state. (a) -20% of inlet gas speed; (b) -10% of inlet gas speed; (c) +10% of inlet gas speed; (d) +20% of inlet gas speed.

Figure 4
Concentration profiles of C₃S, C₂S, C₃A, C₄AF and CaO in the industrial rotary kiln at steady state. (a) -20% of inlet solid speed; (b) -10% of inlet solid speed; (c) +10% of inlet solid speed; (d) +20% of inlet solid speed.

Figure 5
Temperature profiles of the gas, the solid, and the wall of the industrial rotary kiln at steady state. (a) -20% of inlet solid speed; (b)-10% of inlet solid speed; (c) +10% of inlet solid speed; (d) +20% of inlet solid speed

As the quantity of energy proposed for the fuel combustion is constant, the increase of the inlet gas speed increases the energy transfer from the solid that is near the output of the kiln to the solid inside the equipment. The energy transfer is responsible for the increase of temperature inside the kiln, which is necessary for the chemical reactions to occur (Table II). The increase of the inlet gas speed from -20% to +10% increases the consumption of C₂S and the production of C₃S while CaO is available (Figure 4a, b and c). A +20% increase of the inlet gas speed strongly promotes the formation of C₂S, produced in lower temperatures than C₃S (Table II), and consequently reduces the availability of CaO in the system. Therefore, even with an increase of temperature, the production of C₃S is limited (Figure 4d).

The temperature and clinker concentration profiles that were obtained by varying the inlet solid speed by -20%, -10%, +10% and +20% are shown in Figures 5 and 6, respectively. In Figure 5, it is observed that the temperature of the burning zones, which correspond to the highest peak, decrease with the increase of the inlet solid speed. There is also an enlargement of the transition zone that corresponds to the second highest peak. In Figure 6, the concentrations of C₃A, C₄AF and CaO along the kiln have relatively similar profiles in all of the proposed systems, whereas C₃S and C₂S have different profiles in each one of them. The increase of the inlet solid speed promotes a later formation of C₃S through the length of the kiln and decrease in the C₃S final concentration. There is also an increase in the final concentration of C₂S.

Figure 6
Concentration profiles of C₃S, C₂S, C₃A, C₄AF and CaO in the industrial rotary kiln at steady state. (a) -20% of inlet gas speed; (b) -10% of inlet gas speed; (c) +10% of inlet gas speed; (d) +20% of inlet gas speed.

As the formation of C₃A and C₄AF occurs before 1205^oC (Table II), a reduction of temperature in the system (Figure 5) does not significantly affect the production of these species, which is not true for C₃S that are formed at higher temperatures (Table II). An increase of the inlet solid speed promotes a decrease of the temperature profile of the kiln and consequently reduces the formation of C₃S from C₂S, which occurs at temperature above 1260 ^oC (Figure 2). For this reason the profiles of C₃S and C₂S are modified.

CONCLUSIONS

A mathematical model able to describe the temperature and concentration profiles of the clinker components along a real rotary kiln, in continuous operation, was developed. The mathematical model proposed in this paper does not require advanced computing to be solved, and it is easily adaptable to new industrial realities.

Variations of the gas and solid inlet speeds by

-20%, -10%, +10% and +20% were individuality analysed according to the resulting temperature and concentration profiles of the clinker. These results were obtained via the model in the simulation of the real process. These parameters of operation were chosen because they have a significant influence in the process of cement production, and they can be changed without much modification in the kiln. In addition, these parameters can be used to find new operating conditions for the equipment.

NOMENCLATURE

Notation Value Unit f2 Coefficient of conduction - solid to gas 4 (Btu/(h(ft)2°F) f3 Coefficient of conduction - wall to solid 4 (Btu/(h(ft)2°F) f4 Coefficient of conduction - wall to outside air 0.7 (Btu/(h(ft)2°F) F #Fe2O3/ #CaO Fi Initial value of #Fe2O3/#CaO 0.0469 #/#CaO GF Amount of fuel per hour 53000 #/h h0 Fraction of radiation 0.0758 K Thermal conductivity of the wall 0.9 (Btu/(h(ft)2°F) Kf Radiation rate - fuel 1/h k1 Radiation rate - CaCO3 1/h kα Radiation rate - C3S 1/h kβ Radiation rate - C2S 1/h K γ Radiation rate - C3A 1/h K δ Radiation rate - C4AF 1/h K ω Radiation rate - water 1/h L Total length of the kiln 400 ft M Depending on the subscript molecular weight of the chemicals M C1 Molecular weight of Carbon (C) lb P Pressure Btu/ft2 P Angle subtended by the surface of solid 3π/2 Radians Qc Heat generated by the chemical reactions (Btu/(ft3 h)) qF Heat generated by the fuel (Btu/(ft h)) R Ideal Gas constant 1.987 Btu lbmol−1R−1 Q Heat generated or moving into a region rF Particle size of the fuel 10-2 ft r1 Inside radius of the kiln 5 ft r2 Outside radius of the kiln 6 ft r3 Ratio of heat transfer in the chain section 5 ft R ω Reaction rate of water [h-1] S #SiO2/#CAO Si Initial value of #SiO2/#CAO 0.3398 # /#CAO Ta Temperature outside the kiln 561.7 °R Tg Temperature of the solid °R Tgi Input temperature of the gas 1700 °R Ts Temperature of the solid °R Tsi Initial temperature of the solid 562 °R Tw Temperature of the wall °R vg Velocity of the gas 40000 ft/h vs Velocity of the solid -150 ft/h Z Distance along the kiln ft Α #C3S/#CAO Β #C2S/#CAO β1, β2, β3, β4 Heat transfer coefficient (Btu/(h°R)) γ #C3A/#CaO δ #C4AF/#CaO ΔΗξ Heat of reaction - CaCO3 1275 Btu/#CaCO3 ΔΗF Heat of reaction - fuel -14000 Btu/#CaCO3 ΔΗα Heat of reaction - C3S 11 Btu/#C3S ΔΗβ Heat of reaction - C2S -381 Btu/#C2S ΔΗω Heat of reaction - water 970 Btu/#water ɛg Radiation coefficient - gas 0.273 ɛs Radiation coefficient - solid 0.500 ɛw Radiation coefficient - wall 0.751 ξ #CaCO3/#CaO ξi Initial value of #CaCO3/#CaO at the input 1.784 #/#CaO ρF Density of the fuel #/#ft3 ρg Density of the gas 0.05 #/#ft3 ρs Density of the solid 56 #/#ft3 ρw Density of the wall 112 #/#ft3 Ψ #CO2/#CaO ω #water/#CaO ωi Initial value of #water/#CaO 0.0649 #/#CaO

	Notation	Value	Unit
f2	Coefficient of conduction - solid to gas	4	(Btu/(h(ft)2°F)
f3	Coefficient of conduction - wall to solid	4	(Btu/(h(ft)2°F)
f4	Coefficient of conduction - wall to outside air	0.7	(Btu/(h(ft)2°F)
F	#Fe2O3/ #CaO
Fi	Initial value of #Fe2O3/#CaO	0.0469	#/#CaO
GF	Amount of fuel per hour	53000	#/h
h0	Fraction of radiation	0.0758
K	Thermal conductivity of the wall	0.9	(Btu/(h(ft)2°F)
Kf	Radiation rate - fuel		1/h
k1	Radiation rate - CaCO3		1/h
kα	Radiation rate - C3S		1/h
kβ	Radiation rate - C2S		1/h
K γ	Radiation rate - C3A		1/h
K δ	Radiation rate - C4AF		1/h
K ω	Radiation rate - water		1/h
L	Total length of the kiln	400	ft
M	Depending on the subscript molecular weight of the chemicals
M C1	Molecular weight of Carbon (C)		lb
P	Pressure		Btu/ft2
P	Angle subtended by the surface of solid	3π/2	Radians
Qc	Heat generated by the chemical reactions		(Btu/(ft3 h))
qF	Heat generated by the fuel		(Btu/(ft h))
R	Ideal Gas constant	1.987	Btu lbmol−1R−1
Q	Heat generated or moving into a region
rF	Particle size of the fuel	10-2	ft
r1	Inside radius of the kiln	5	ft
r2	Outside radius of the kiln	6	ft
r3	Ratio of heat transfer in the chain section	5	ft
R ω	Reaction rate of water		[h-1]
S	#SiO2/#CAO
Si	Initial value of #SiO2/#CAO	0.3398	# /#CAO
Ta	Temperature outside the kiln	561.7	°R
Tg	Temperature of the solid		°R
Tgi	Input temperature of the gas	1700	°R
Ts	Temperature of the solid		°R
Tsi	Initial temperature of the solid	562	°R
Tw	Temperature of the wall		°R
vg	Velocity of the gas	40000	ft/h
vs	Velocity of the solid	-150	ft/h
Z	Distance along the kiln		ft
Α	#C3S/#CAO
Β	#C2S/#CAO
β1, β2, β3, β4	Heat transfer coefficient		(Btu/(h°R))
γ	#C3A/#CaO
δ	#C4AF/#CaO
ΔΗξ	Heat of reaction - CaCO3	1275	Btu/#CaCO3
ΔΗF	Heat of reaction - fuel	-14000	Btu/#CaCO3
ΔΗα	Heat of reaction - C3S	11	Btu/#C3S
ΔΗβ	Heat of reaction - C2S	-381	Btu/#C2S
ΔΗω	Heat of reaction - water	970	Btu/#water
ɛg	Radiation coefficient - gas	0.273
ɛs	Radiation coefficient - solid	0.500
ɛw	Radiation coefficient - wall	0.751
ξ	#CaCO3/#CaO
ξi	Initial value of #CaCO3/#CaO at the input	1.784	#/#CaO
ρF	Density of the fuel		#/#ft3
ρg	Density of the gas	0.05	#/#ft3
ρs	Density of the solid	56	#/#ft3
ρw	Density of the wall	112	#/#ft3
Ψ	#CO2/#CaO
ω	#water/#CaO
ωi	Initial value of #water/#CaO	0.0649	#/#CaO

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APPENDIX

Material balance equations:

Water:

$\frac{\partial \text{ω}}{\partial \text{t}}=-{\text{R}}_{\text{ω}}-{\text{v}}_{\text{s}}\frac{\partial \text{ω}}{\partial \text{z}},R_{\omega }=K_{\omega }\omega \omega \le 0.1=\omega \left(\mathrm{0,}t\right)={\omega }_i{K}_{\omega }\omega >0.1$ (1)

$C{O}_2:\frac{\partial \text{Ψ}}{\partial \text{t}}=\frac{{\text{A}}_{\text{s}}}{{\text{A}}_{\text{g}}}\frac{{\text{ρ}}_{\text{s}}}{{\text{ρ}}_{\text{g}}}\frac{{\text{M}}_{\text{Ψ}}}{{\text{M}}_{\text{C}}}{\text{k}}_1\text{ξ}-{\text{v}}_{\text{g}}\frac{\partial \text{Ψ}}{\partial \text{z}},\text{Ψ}\left(\text{L},\text{t}\right)=0$ (2)

$CaC{O}_3:\frac{\partial \text{ξ}}{\partial \text{t}}=-{\text{k}}_1\text{ξ}\frac{\text{Mξ}}{{\text{M}}_{\text{C}}}-{\text{v}}_{\text{s}}\frac{\partial \text{ξ}}{\partial \text{z}},\text{ξ}\left(\mathrm{0,}\text{t}\right)={\text{ξ}}_1$ (3)

$C_{3}S:\frac{\partial \text{α}}{\partial \text{t}}=\frac{\text{Mα}}{{\text{M}}_{\text{C}}}{\text{k}}_{\text{α}}\left(\text{C}\right)\text{β}-{\text{v}}_{\text{s}}\frac{\partial \text{α}}{\partial \text{z}},\text{α}\left(\mathrm{0,}\text{t}\right)=0$ (4)

$C_{2}S:\frac{\partial \text{β}}{\partial \text{t}}=\frac{\text{Mβ}}{2{\text{M}}_{\text{C}}}{\text{k}}_{\text{β}}\left({\text{C}}^2\right)\text{S}-\frac{\text{Mβ}}{{\text{M}}_{\text{C}}}{\text{k}}_{\text{α}}\left(\text{C}\right)\text{β}-{\text{v}}_{\text{s}}\frac{\partial \text{β}}{\partial \text{z}},\text{β}\left(\mathrm{0,}\text{t}\right)=0$ (5)

$C_{3}A:\frac{\partial \text{γ}}{\partial \text{t}}=\frac{\text{Mγ}}{3{\text{M}}_{\text{C}}}{\text{k}}_{\text{γ}}\left({\text{C}}^3\right)\text{A}-{\text{v}}_{\text{s}}\frac{\partial \text{γ}}{\partial \text{z}},\text{γ}\left(\mathrm{0,}\text{t}\right)=0$ (6)

$C_{4}AF:\frac{\partial \text{δ}}{\partial \text{t}}=\frac{\text{Mδ}}{4{\text{M}}_{\text{C}}}{\text{k}}_{\text{δ}}\left({\text{C}}^4\right)\text{AF}-{\text{v}}_{\text{s}}\frac{\partial \text{δ}}{\partial \text{z}},\text{δ}\left(\mathrm{0,}\text{t}\right)=0$ (7)

$F{e}_2{O}_3:\frac{\partial \text{F}}{\partial \text{t}}=-\frac{{\text{M}}_{\text{F}}}{4{\text{M}}_{\text{C}}}{\text{k}}_{\text{δ}}\left({\text{C}}^4\right)\text{AF}-{\text{v}}_{\text{s}}\frac{\partial \text{F}}{\partial \text{z}},\text{F}\left(\mathrm{0,}\text{t}\right)={\text{F}}_{\text{i}}$ (8)

$A{l}_2{O}_3:\frac{\partial \text{A}}{\partial \text{t}}=-\frac{{\text{M}}_{\text{A}}}{4{\text{M}}_{\text{C}}}{\text{k}}_{\text{δ}}\left({\text{C}}^4\right)\text{AF}-\frac{{\text{M}}_{\text{A}}}{4{\text{M}}_{\text{C}}}{\text{k}}_{\text{γ}}\left({\text{C}}^3\right)\text{A}-{\text{v}}_{\text{s}}\frac{\partial \text{A}}{\partial \text{z}},\text{A}\left(\mathrm{0,}\text{t}\right)={\text{A}}_{\text{i}}$ (9)

$Si{O}_2:\frac{\partial \text{S}}{\partial \text{t}}=-\frac{{\text{M}}_{\text{S}}}{2{\text{M}}_{\text{C}}}{\text{k}}_{\text{β}}\text{S}\left({\text{C}}^2\right)-{\text{v}}_{\text{s}}\frac{\partial \text{S}}{\partial \text{z}},\text{S}\left(\mathrm{0,}\text{t}\right)=S_i$ (10)

$CaO:\frac{\partial \text{C}}{\partial \text{t}}=-{\text{k}}_1\text{ξ}-{\text{k}}_{\text{α}}\text{Cβ}-{\text{k}}_{\text{β}}\text{S}\left({\text{C}}^2\right)-{\text{k}}_{\text{γ}}{\text{C}}^3\text{A}-{\text{k}}_{\text{δ}}{\text{C}}^4\text{AF}-{\text{v}}_{\text{s}}\frac{\partial \text{C}}{\partial \text{z}},\text{C}\left(\mathrm{0,}\text{t}\right)=0$ (11)

$Fuel\left(coal/oil\right):\frac{\partial {\text{C}}_{\text{F}}}{\partial \text{z}}=\frac{1}{{\text{ρ}}_{\text{g}}{\text{v}}_{\text{g}}}\left[\frac{{\text{Ψ}}^M{\text{C}}_1{\left({\text{PM}}_{\text{a}}\right)}^2}{{\text{ρ}}_{\text{F}}{\text{M}}_{{\text{O}}_2}{\left({\text{RT}}_{\text{g}}\right)}^2}\right]{\text{k}}_{\text{F}}{\text{d}}_0{\text{C}}_{\text{F}},{\text{C}}_{\text{F}}\left(\text{L},\text{t}\right)={\text{C}}_{{\text{F}}_{\text{i}}}$ (12)

Thermodynamic energy balance equations:

$Gas:A_g{C}_{pg}{\rho }_g{v}_g\left(\frac{\partial Tg}{\partial z}\right)={\beta }_1\left(T_w-T_g\right)+{\beta }_2\left(T_s-T_g\right)+q_f$ (13)

$T_g\left(L,t\right)=T_{gi}$

$Solid:{\text{A}}_{\text{s}}{\text{C}}_{\text{ps}}{\text{ρ}}_{\text{s}}{\text{v}}_{\text{s}}\left(\frac{\partial \text{Ts}}{\partial \text{z}}\right)={\text{β}}_2\left({\text{T}}_{\text{g}}-{\text{T}}_{\text{S}}\right)+{\text{β}}_3\left({\text{T}}_{\text{w}}-{\text{T}}_{\text{s}}\right)-{\text{C}}_{\text{ps}}{\text{ρ}}_{\text{s}}{\text{A}}_{\text{s}}\left(\frac{\partial \text{Ts}}{\partial \text{t}}\right)+{\text{A}}_{\text{S}}{q}_c$ (14)

$T_s\left(\mathrm{0,}T\right)=T_{si}$

$Kilnwall:{\text{A}}_{\text{w}}{\text{C}}_{\text{pw}}{\text{ρ}}_{\text{w}}\left(\frac{\partial \text{Tw}}{\partial \text{t}}\right)={\text{β}}_1\left({\text{T}}_{\text{g}}-{\text{T}}_{\text{W}}\right)+{\text{β}}_3\left({\text{T}}_{\text{S}}-{\text{T}}_{\text{W}}\right)+{\text{β}}_4\left({\text{T}}_{\text{a}}-{\text{T}}_{\text{W}}\right)$ (15)

Heat of reaction:

${\text{q}}_c=\frac{{\text{ρ}}_{\text{s}}}{\left(1+{\text{A}}_{\text{i}}+{\text{F}}_{\text{i}}+{\text{S}}_{\text{i}}\right)}\left[-\Delta {\text{H}}_{\xi }{k}_1\xi -\Delta {\text{H}}_{\omega }{R}_{\omega }-\Delta {\text{H}}_{\beta }{k}_{\beta }S\left(C^2\right)-\Delta {\text{H}}_{\alpha }{k}_{\alpha }C\beta \right]$ (16)

Flame model:

${\text{q}}_f=\frac{{\text{G}}_{\text{f}}\left(-\Delta {\text{H}}_{\text{F}}\right)}{\left({\text{p}}_{\text{g}}{\text{v}}_{\text{g}}\right)}\left[\frac{{\text{Ψ}}^{\text{M}}{\text{C}}_1{\left({\text{PM}}_{\text{a}}\right)}^2}{{\text{ρ}}_{\text{s}}{\text{M}}_{{\text{O}}_2}{\left({\text{RT}}_{\text{g}}\right)}^2}\right]{\text{k}}_{\text{F}}{\text{d}}_0{\text{C}}_{\text{F}}$ (17)

${\text{d}}_0=1-\frac{{\text{k}}_{\text{F}}}{\left(\frac{{\text{D}}_{0}3}{{\text{r}}_{\text{f}}{}^2}\right)+{\text{k}}_{\text{F}}}$ (18)

${\text{D}}_0={\text{a}}_0{\text{T}}_{\text{g}}{}^{\frac{3}{2}}$ (19)

Reaction rate coefficient:

$k_{\text{i}}=A_i\text{exp}\left(\frac{-E_i}{R{T}_i}\right)$ (20)

$i=\mathrm{1,}\alpha ,\beta ,\gamma ,\delta ,\omega$

$k_{\text{F}}=\frac{3}{r_F}{A}_F\text{exp}\left(\frac{-E_F}{R{T}_g}\right)$ (21)

Coefficient of heat transfer:

${\text{β}}_1=r_1\text{p}\left[{\text{f}}_1+1.73\text{x}{10}^{-9}\left(1-{\text{h}}_0\right){\text{ε}}_{\text{g}}{\text{ε}}_{\text{w}}\left({\text{T}}_{\text{g}}{}^2+{\text{T}}_{\text{w}}{}^2\right)\left({\text{T}}_{\text{g}}+{\text{T}}_{\text{w}}\right)\right]$ (22)

${\text{β}}_2=2r_1\text{sin}\left(\frac{\text{p}}{2}\right)\left[{\text{f}}_2+1.73\text{x}{10}^{-9}{\text{ε}}_{\text{g}}{\text{ε}}_{\text{s}}\left({\text{T}}_{\text{g}}{}^2+{\text{T}}_{\text{s}}{}^2\right)\left({\text{T}}_{\text{g}}+{\text{T}}_{\text{s}}\right)\right]$ (23)

${\text{β}}_3=r_1\left(2\text{Π}-\text{p}\right)\left[{\text{f}}_3+1.73\text{x}{10}^{-9}{\text{h}}_0{\text{ε}}_{\text{w}}{\text{ε}}_{\text{s}}\left({\text{T}}_{\text{w}}{}^2+{\text{T}}_{\text{s}}{}^2\right)\left({\text{T}}_{\text{w}}+{\text{T}}_{\text{s}}\right)\right]$ (24)

${\text{β}}_4=2\text{Π}{f}_4{\text{r}}_2$ (25)

$\text{h}=1+\frac{2h_0\text{sin}\left(\frac{\text{p}}{2}\right)}{2\text{Π}-\text{p}}$ (26)

Area coefficient:

${\text{A}}_{\text{g}}=\frac{{\text{r}}_1{}^2}{2}\left(\text{p}-\sin \text{p}\right)$ (27)

${\text{A}}_{\text{s}}=\frac{r_1{}^2}{2}\left(2\text{Π}-p+\sin p\right)$ (28)

${\text{A}}_{\text{w}}=2\text{Π}\left(r_2{}^2-r_1{}^2\right)$ (29)

Publication Dates

• Publication in this collection
  Oct-Dec 2017

History

• Received
  30 Sept 2016
• Accepted
  24 Aug 2017