Numerical analysis of the thermal profile inside the wall of a rotary cement kiln

Silva, P. H. G. da; Moreira, J.; Costa, A. O. S.; Costa Jr., E. F.

doi:10.1590/0366-69132020663802902

P. H. G. da Silva ^* * pedro.h.guimaraes.silva@gmail.com

Federal University of Minas Gerais, Av. Pres. Antônio Carlos 6627, 31270-901, Belo Horizonte, MG, Brazil

https://orcid.org/0000-0002-5220-5804

J. Moreira

Federal University of Minas Gerais, Av. Pres. Antônio Carlos 6627, 31270-901, Belo Horizonte, MG, Brazil

A. O. S. Costa

Federal University of Minas Gerais, Av. Pres. Antônio Carlos 6627, 31270-901, Belo Horizonte, MG, Brazil

E. F. Costa Jr.

Federal University of Minas Gerais, Av. Pres. Antônio Carlos 6627, 31270-901, Belo Horizonte, MG, Brazil

* pedro.h.guimaraes.silva@gmail.com

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Figure 1
Schematic of the cross-section of clinker kiln, showing the layers that make up the wall and their dimensions.

Figura 1
Esquema da seção transversal do forno de clínquer, mostrando as camadas que compõem a parede e suas dimensões.

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Figure 2
Schematic drawing of the cylindrical kiln, showing the longitudinal section plane (yellow rectangle). The rectangular section indicated by the red arrow, which corresponded to the intersection between the section plane and the wall, was the section used to construct the mesh.

Figura 2
Desenho esquemático do forno cilíndrico, exibindo o plano de corte meridional (retângulo amarelo). O trecho retangular indicado pela seta vermelha, correspondente à intersecção entre o plano de corte e a parede, foi a seção utilizada para a construção da malha.

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Figure 3
Illustration of the domain, showing the layers in which it divides, and their dimensions.

Figura 3
Ilustração do domínio, explicitando as camadas em que se divide e suas dimensões.

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Figure 4
Representation of the mesh obtained by discretization of the domain, detailing the steps of the mesh, and the 19 groups of nodes with equations and boundary conditions in common. The number of nodes in the figure is merely illustrative and does not correspond to the actual amount used for resolution.

Figura 4
Representação da malha de pontos obtida por discretização do domínio, detalhando os passos da malha e os 19 grupos de pontos com equações e condições de contorno em comum. A quantidade de pontos na figura é apenas representativa, não correspondendo à quantidade real utilizada para resolução.

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Figure 5
Representation of the ghost node T_f on the steel surface. The black dots are real nodes, and the white dot is a ghost node.

Figura 5
Representação do ponto fictício T_f na superfície do aço. Os pontos pretos são reais e o branco é fictício.

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Figure 6
Representation of the ghost nodes used for the interfaces between two materials in the cases in which all nodes are considered to belong to material Y (a) and material X (b). The black dots are real nodes, and the white dots are ghost nodes.

Figura 6
Representações dos pontos fictícios utilizados para os nós das interfaces entre dois materiais nos casos em que se consideram todos os pontos como pertencentes ao material Y (a) e material X (b). Os pontos pretos são reais e os brancos são fictícios.

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Figure 7
Representations of the ghost node T_f on: a) left surface; and b) right surface. The black dots are real nodes and the white dots are ghost nodes.

Figura 7
Representação do ponto fictício T_f na: a) superfície esquerda; e b) superfície direita. Os pontos pretos são reais e os brancos são fictícios.

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Figure 8
Graph of temperatures used as initial values for the algorithm as a function of kiln length, adapted from an available kiln internal temperature profile ³⁰30 K.E. Peray, The rotary cement kiln, Chem. Publ., New York (1986)..

Figura 8
Gráfico das temperaturas utilizadas como valores iniciais para o algoritmo em função do comprimento do forno, adaptado de perfil disponível de temperatura interna do forno ³⁰30 K.E. Peray, The rotary cement kiln, Chem. Publ., New York (1986)..

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Figure 9
Temperature map obtained as a result. The white dashed lines that divide the map horizontally mark the division between layers of different materials.

Figura 9
Mapa de temperatura obtido como resposta do algoritmo. As linhas tracejadas brancas que dividem o mapa horizontalmente marcam a divisão entre as camadas de diferentes materiais.

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Figure 10
Comparisons between kiln surface temperatures measured experimentally and predicted by the models for: a) this study; and b) reference study ²⁰20 E. Mastorakos, A. Massias, C.D. Tsakiroglou, D.A. Goussis, V.N. Burganos, A.C. Payatakes, Appl. Math. Model. 23 (1999) 55..

Figura 10
Comparações entre temperaturas da superfície do forno medidas experimentalmente e preditas pelos modelos para: a) este estudo; e b) estudo de referência ²⁰20 E. Mastorakos, A. Massias, C.D. Tsakiroglou, D.A. Goussis, V.N. Burganos, A.C. Payatakes, Appl. Math. Model. 23 (1999) 55..

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Table I
Atmospheric air properties at film temperature ²⁷27 Y.A. Cengel, J.M. Cimbala, Fluid mechanics: fundamentals and applications, McGraw-Hill, New York (2013)..

Tabela I
Propriedades do ar atmosférico na temperatura de filme ²⁷27 Y.A. Cengel, J.M. Cimbala, Fluid mechanics: fundamentals and applications, McGraw-Hill, New York (2013)..

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Table II
Properties and thermophysical constants used in the algorithm.

Tabela II
Propriedades e constantes termofísicas utilizadas no algoritmo.

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Table III
Parameter values chosen by the user.

Tabela III
Valores de parâmetros arbitrados pelo usuário.

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NOMENCLATURE

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Table A-I
Chemical composition of Magkor A1 refractory brick (wt%) ⁽³⁶⁾, and output stream of an industrial clinker kiln (wt% calculated from molar fraction data in ⁽³⁷⁾).

Tabela A-I
Composição química de tijolo refratário Magkor A1 (% mássica) ⁽³⁶⁾ e corrente de saída de forno de clínquer industrial (% mássica calculada a partir de dados de fração molar em ⁽³⁷⁾).

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Table A-II
Mesh node groups with equations and boundary conditions in common.

Tabela A-II
Grupos de nós da malha com equações e condições de contorno em comum.

(A)

(B)

(C)

(D)

(E)

(F)

(G)

(H)

(I)

(J)

(K)

(L)

(M)

(N)

(O)

(P)

(Q)

(R)

(S)

(T)

(U)

(V)

(W)

(X)

(Y)

(Z)

(AA)

(AB)

(AC)

(AD)

Table I Atmospheric air properties at film temperature 27.

Property	Value
Kinematic viscosity (ν)	2.745x10^-5 m²/s
Thermal conductivity (k)	0.03374 W/(m.K)
Prandtl number (Pr)	0.7041

Table II Properties and thermophysical constants used in the algorithm.

Property	Value	Source
k_s	13.4 W/(m.K)	³3 K.S. Mujumdar, K.V. Ganesh, S.B. Kulkarni, V.V. Ranade, Chem. Eng. Sci. 62 (2007) 2590.
k_r	4.0 W/(m.K)	²²22 K.S. Mujumdar, V.V. Ranade, Chem. Eng. Res. Des. 84, 3 (2006) 165.
k_c	0.5 W/(m.K)	²²22 K.S. Mujumdar, V.V. Ranade, Chem. Eng. Res. Des. 84, 3 (2006) 165.
ρ_s	8238 kg/m³	²³23 F.P. Incropera, A.S. Lavine, T.L. Bergman, D.P. DeWitt, Fundamentals of heat and mass transfer, John Wiley Sons, Hoboken (2007).
ρ_r	3100 kg/m³	³³33 Magnesita Refratários, “MAGKOR-B-RA-LP-JP”, Techn. data sheet (2010).
ρ_c	1046 kg/m³	²²22 K.S. Mujumdar, V.V. Ranade, Chem. Eng. Res. Des. 84, 3 (2006) 165.
c_p,s	468 J/(kg.K)	²³23 F.P. Incropera, A.S. Lavine, T.L. Bergman, D.P. DeWitt, Fundamentals of heat and mass transfer, John Wiley Sons, Hoboken (2007).
c_p,r	1130 J/(kg.K)	³³33 Magnesita Refratários, “MAGKOR-B-RA-LP-JP”, Techn. data sheet (2010).
c_p,c	800 J/(kg.K)	²²22 K.S. Mujumdar, V.V. Ranade, Chem. Eng. Res. Des. 84, 3 (2006) 165.
T_ext	298.15 K	*
σ	5.67x10^-8 W/(m².K⁴)	²³23 F.P. Incropera, A.S. Lavine, T.L. Bergman, D.P. DeWitt, Fundamentals of heat and mass transfer, John Wiley Sons, Hoboken (2007).
ε	0.54	³⁴34 Engineering ToolBox, “Emissivity coefficients materials”, 34 Engineering ToolBox, “Emissivity coefficients materials”, www.engineeringtoolbox.com , ac. 09/02/2019. www.engineeringtoolbox.com...

Table III Parameter values chosen by the user.

Parameter	Value
npRs	5
npRr	15
npRc	5
npZ	181
nt	20000
t_final	400000

c_p,s	Specific heat capacity for steel at constant pressure	[J/(kg.K)]
c_p,c	Specific heat capacity for coating at constant pressure	[J/(kg.K)]
c_p,r	Specific heat capacity for refractory brick at constant pressure	[J/(kg.K)]
h	Modified Euler method time step	[s]
h_ext	Convective heat transfer coefficient on kiln surface	[W/(m².K)]
k	Thermal conductivity	[W/(m.K)]
k_s	Thermal conductivity for steel	[W/(m.K)]
k_c	Thermal conductivity for coating	[W/(m.K)]
k_r	Thermal conductivity for refractory brick	[W/(m.K)]
L	Kiln length	[m]
npRs	Number of nodes that steel layer thickness is divided	-
npRc	Number of nodes that coating layer thickness is divided	-
npRr	Number of nodes that refractory layer thickness is divided	-
npZ	Number of nodes that kiln length is divided	-
nt	Number of steps that total time interval is divided	-
Nu_D	Nusselt number	-
Pr	Prandtl number	-
$\dot{q}$	Heat generation rate	[W/m³]
r	Radial direction of cylindrical coordinates	-
Re_D	Reynolds number	-
R₁	Kiln radius from central axis to inner surface of coating layer	[m]
R₂	Kiln radius from central axis to inner surface of refractory layer	[m]
R₃	Kiln radius from central axis to inner surface of steel layer	[m]
R₄	Kiln radius from central axis to outer surface	[m]
T	Temperature	[K]
t	Time	[s]
T_ext	Temperature of surrounding	[K]
t_final	Final time	[s]
T_Xf	Ghost temperature for interface composed of X only	[K]
T_Yf	Ghost temperature for interface composed of Y only	[K]
z	Axial direction of cylindrical coordinates	-
α	Thermal diffusivity	[m²/s]
Δr_s	Step in r direction for steel layer	[m]
Δr_c	Step in r direction for coating layer	[m]
Δr_r	Step in r direction for refractory layer	[m]
Δz	Step in z-direction	[m]
ε	Emissivity of surface material	-
θ	Angular direction of cylindrical coordinates	-
ν	Kinematic viscosity	[m²/s]
ρ_s	Steel specific mass	[kg/m³]
ρ_c	Coating specific mass	[kg/m³]
ρ_r	Refractory brick specific mass	[kg/m³]
σ	Stefan-Boltzmann constant	[W/(m².K⁴)]

\frac{1}{r} = (\frac{T_{i + 1, j} - T_{i - 1, j}}{2 Δ r_{m}}) + \frac{T_{i + 1, j} - 2 T_{i, j} + T_{i - 1, j}}{2 Δ r_{m}^{2}} + \frac{T_{i, j + 1} - 2 T_{i, j} + T_{i, j - 1}}{2 Δ z^{2}} = \frac{1}{α} \frac{\partial T}{\partial t}

Table A-I Chemical composition of Magkor A1 refractory brick (wt%) (36), and output stream of an industrial clinker kiln (wt% calculated from molar fraction data in (37)).

Compound	Magkor A1	Output stream
MgO	89.5	-
Al₂O₃	9.0	11.8
Fe₂O₃	0.4	3.0
CaO	0.8	65.3
SiO₂	0.3	20.0

Table A-II Mesh node groups with equations and boundary conditions in common.

Group	Mesh location	Boundary conditions
1	Inner surface of the kiln (r=R₁)	Constant temperature over time due to continuous heat supply from the flame
2	Lateral surface close to the flame (z=0) in the section composed only by coating (R₁<r<R₂)	Adiabatic surface (no heat flow perpendicular to the surface)
3	Inside the coating layer (0<z<L; R₁<r<R₂)	No boundary condition
4	Lateral surface away from the flame (z=L) in the section composed only by coating (R₁<r<R₂)	Adiabatic surface (no heat flow perpendicular to the surface)
5	Lateral surface close to the flame (z=0) at the interface between the coating and refractory (r=R₂)	Adiabatic surface in the z-direction and heat received from the innermost layer is transferred by conduction to the outermost layer
6	Inside the interface between the coating and refractory (0<z<L; r=R₂)	Heat received from the innermost layer by conduction in the r direction is transferred by conduction to the outermost layer
7	Lateral surface away from the flame (z=L) at the interface between the coating and refractory (r=R₂)	Adiabatic surface in the z-direction and heat received from the innermost layer is transferred by conduction to the outermost layer
8	Lateral surface close to the flame (z=0) in the section composed only by refractory bricks (R₂<r<R₃)	Adiabatic surface (no heat flow perpendicular to the surface)
9	Inside the refractory layer (0<z<L; R₂<r<R₃)	No boundary condition
10	Lateral surface away from the flame (z=L) in the section composed only by refractory bricks (R₂<r<R₃)	Adiabatic surface (no heat flow perpendicular to the surface)
11	Lateral surface close to the flame (z=0) at the interface between refractory and steel (r=R₃)	Adiabatic surface in the z-direction and heat received from the innermost layer is transferred by conduction to the outermost layer
12	Inside the interface between refractory and steel (0<z<L; r=R₃)	Heat received from the innermost layer by conduction in the r direction is transferred by conduction to the outermost layer
13	Lateral surface away from the flame (z=L) at the interface between refractory and steel (r=R₃)	Adiabatic surface in the z-direction and heat received from the innermost layer is transferred by conduction to the outermost layer
14	Lateral surface close to the flame (z=0) in the section composed only by steel (R₃<r<R₄)	Adiabatic surface (no heat flow perpendicular to the surface)
15	Inside the steel layer (0<z<L; R₃<r<R₄)	No boundary condition
16	Lateral surface away from the flame (z=L) in the section composed only by refractory bricks (R₂<r<R₃)	Adiabatic surface (no heat flow perpendicular to the surface)
17	Crossing between the lateral surface close to the flame (z=0) and the outer surface (r=R₄)	Adiabatic surface in the z-direction; the heat received by conduction is dissipated by convection and radiation in the r direction
18	Outer surface of the kiln (0<z<L; r=R₄)	The heat received by conduction is dissipated to the environment by convection and radiation
19	Crossing between the lateral surface close to the flame (z=L) and the outer surface (r=R₄)	Adiabatic surface in the z-direction; the heat received by conduction is dissipated by convection and radiation in the r direction

[1] * pedro.h.guimaraes.silva@gmail.com

Brasil

Brasil

Numerical analysis of the thermal profile inside the wall of a rotary cement kiln

Análise numérica do perfil térmico na parede de um forno rotativo de cimento

Abstract