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RADIATION-CONDUCTION AND CONVECTION IN A HIGH TEMPERATURE PACKED BED
Abdelhamid Belghit
To cite this version:
Abdelhamid Belghit. HEAT TRANSFER BY SIMULTANEOUS RADIATION-CONDUCTION AND CONVECTION IN A HIGH TEMPERATURE PACKED BED. Eurotherm Seminar N°81 Reactive Heat Transfer in Porous Media, Jun 2007, Albi, France. pp.ET81- 1. �hal-00422188�
HEAT TRANSFER BY SIMULTANEOUS RADIATION-CONDUCTION AND CONVECTION IN A HIGH TEMPERATURE PACKED BED
Prof. A. Belghit
Department of Chemical Engineering, University of La Rochelle, Avenue Michel CREPEAU- 17000 La Rochelle – France
E-mail : [email protected]
Abstract
A numerical model of a packed bed reactor for gasifying coal in mixed control using concentrated solar radiation is proposed. Case’s normal-mode expansion technique is used to obtain solutions to the radiative transfer problem for the packed bed. The comparison between the radiative heat transfer and the exchanges by conduction and forced convection is analysed. The model permits the determination of temperature profiles for both the gas and the solid phases and the evolutions of thermal flux densities in the reactor.
Keywords: Radiative, Transfer, packed bed, Case.
1. Introduction
The gasification of coal is a very important economical operation. It is performed at relatively high temperatures, between 700°C and 1300°C. The produced gas, consisting primarily of CO and H2, can be used as a feedstock for many chemical processes (Gregg et al. [1], Aiman et al. [2], Taylor et al. [3], Belghit et al. [4]).
A moving bed reactor, for gasifying coconut charcoal with CO2 was experimentally studied (Taylor et al. [3], Belghit et al. [5]). Experiments were carried out on a vertical solar furnace ( Fig. 1).
Any serious technical evaluation of this process schould be based on a rigorous and precise functioning model of the gasification reactor. Belghit et al. [4], presented a numerical model for the moving bed pertinent to the case where the total rate is determined by mass transfer control, Rosseland’s approximation was utilised for analysis the radiative transfer.
In the present paper, we propose a rigorous model for the behaviour of the reactor in mixed control, which explicitly takes in to account the radiative exchange in the porous medium. Case’s normal-mode expansion technique [6] is used to obtain solutions to the radiative transfer problem for the packed bed. The comparison between the radiative heat transfer and the exchanges by conduction and forced convection is analysed.
2. Analysis
The heat and mass transfer equations for porous medium are written on a macroscale. Therefore, its geometry is caracterized by parameters such as porosity ε and contact surface ratio: A=6(1-ε)/d for solid (carbon) particles supposed spherical and pure. The gasification reaction is: C + CO2 → 2CO
To simplify the heat and mass transfer equations (Luikov [7], Szekely et al. [8]), it is assumed that the flow rate is fully established, the side effect is negligible in the reactor considered as a straight cylinder with a constant section, the reactor is considered adiabatic and composed of a stack of identical and spherical non- porous carbon particles. In addition the flow is assumed one-dimensional, this effect being confirmed for (Do/d)>8. The gas is taken as ideal. Moreover, the viscous friction can be neglected compared with other exchange modes (conduction, radiation and forced convection).
Fig. 1. Packed bed reactor for solar gasification.
The heat transfer equations are given by [9]:
- for the gas:
σθ
λ θ
ρ θ +
∂
∂
∂
= ∂
∂
∂ ( )
X X v X
cp (1)
- for the solid: ( * ) ( ) + = 0
∂ Τ
− ∂
∂
− ∂
∂ Τ
∂
∂
∂
σΤ
ρ
λ c v X
X X Q X
X c pc s
r
s (2) In these equations, the terms
σ
θ, represent heat production (convection) and σT represent heat (convection) andmasse production (diffusion and chemical reaction), in the control volume. They may be expressed as:
) ) (
1 ( ) 6
( θ
ε θ ε
σθ ε − Τ−
=
− Τ
= h
h d
A , ( )
) 1
( ) 6
6 (
) ( 2
s CO
C x C
H kg h d
d s
+ −
∆
−
− Τ
−
Τ = θ
σ
The energy equation for solid includes the net radiative heat flux density Qr(τ) which is related to the intensity L(τ, µ) by: Τ =
∫
−11 ( , )
2 )
(τ π L τ µ dµ
Q (3) where L(τ, µ) satisfies the equation of radiation transfer.
2.1. Equation of radiative transfer
The equation of radiative transfer for one-dimensional, emitting, absorbing, isotropically scattering, gray media can be written in the form:
∫
−Τ +
−
=
∂ +
∂ 1
1 4
2
' ) ' , 2 (
) ( ) '
1 ( ) , ) ( ,
( ω τ µ µ
π τ ω σ
µ τ τ
µ
µ τ n L d
L L (4)
With the boundary conditions:
L(o,µ)=F0′=φi µ>0 for τ=0 (5)
L(∞,−µ)= Lp(∞,−µ) µ>0 for τ→∞ (6) The solution of equation (4) can be written in the form [10]:
∫
++
= −
− 1
0 0
0) ( , ) ( ) ( , ) ( , )
( ) ,
(τ µ φ µ 0 φ µ τ µ
τ τ
p n n
L dn e n n A e
n n A
L (7)
where A(n0) and A(n), n∈[0,1] are the expansion coefficients, Lp(τ, µ) is a particular solution of equation (4), and φ(-n0,µ) and φ(-n,µ) are the normal modes [11]:
[
µ]
µµ φ
µ n F L o d
n W n N
A ( ) ( , ) p( , )
) ( ) 1
( 1 0 0
0 0
0 =
∫
− ′−{
µ}
µµ φ
µ n F L o d
n W N n w n n
A ( ) ( , ) p( , )
) ( ) ) (
( 1 0
0 ′−
=
∫
The expansion coefficients A(n0) and A(n) can be determined by constaining the solution equation (7) to meet the boundary conditions equations (5) and (6), and by utilizing the orthogonality property of the normal modes and various normalisation integrals as described in reference [10] provided that a particular solution Lp(τ, µ) is available. To determine a particular solution it is assumed that the inhomogeneous term is represented as a polynomial in t in the form [12]:
∑
=
+
−
= Τ
N
m
m me
0
) (
4( ) π
τ
α
τ and that the coefficients αm are determined by the method of Gauss-Seidel. Noting a that a particular solution Lp(τ, µ) for an inhomogeneous term of the form T4(τ) is given as [13]:
∑
=
+
−
− +
′ +
= −
M
m
m m
p m D m
e m L n
0
) 2 (
) ( ) (
) ( )
1 ) ( ,
( π µ
π α
π σ µ ω
τ
π τ
with: D(m) = 1-ω(m+π)th-1(1/(m+π)). Then we have:
− ′ + +
−
=
∑
= ( ) ( ) ( )
) 1 ) (
( 2 ) ( ) 1 (
) 2 ( )
1 (
0 2 '
0 0 0
0 D m DP m DS m
F n n n n N
A m m
M
m
m α γ α γ
α π
σ ω ω
+
− +
−
− ′ + + − +
−
=
∑
=
2 0 ) 1 ( ) 1 ( 0 ) 2 (
0 )
1 ( 0
2 '
0
)( (
) )(
( ) ( )
1 ) (
( 2 ) ( ) ) (
(
n n n n
m n DS
n m n
DP m D F n
n n N n W n n A
m
m M
m m
γ γ
α γ
α γ α
π σ ω ω
with: DS(m) = (1+m)D(m) et DP(m) = (1+m)DS(m). Then, we obtain:
- The intensity of radiation
dn e n n A e
n n A
L n
∫
−n−
+
= 1
0 0
0) ( , ) ( ) ( , )
( ) ,
( 0
τ τ
µ φ µ
φ µ
τ
(1 ) 1 ( ) ) (( ) )
0
2 2
m D
e m
m
n M m m
m
π τ
α π µ π
µ π
σ
ω +
−
=
∑
+ + + +
− ′
+ (8)
- The net radiation flux:
∫
−−
+
−
−
−
= 1
0 0 0 0
0 ( ) ( ) ( )
) 1 ( 2 )
( n A n e 0 n A n en0 nA n e dn
Qr n n
π τ τ
ω π τ
∫ ∑
=
+
−
+ ′
−
−
m
m
m n m
m DP n e
dn e n nA
0
) 1 (
0
2
) ( 3
) 2 (
π τ
τ α
π
σ (9)
Then dQr(τ)/dτ can be obtained by differentiating expression for Qr(τ) with respect to τ :
+ ′ +
−
−
=
∫ ∑
=
+
−
− − M
m
m n m
r n
m DS
e dn n
e n A e
n d A
dQ
0
) 2 (
1 0 0
) ( 3
) 2 ( )
( ) 1 ( ) 2
( 0 π
τ τ τ
α π ω σ
τ π τ
where τ is : τ =
∫
0X dx′, and X is the axial coordinate.The set of equations (1) and (2) may be written as:
) ) (
1 ( ) 6
( θ
ε ε λ θ
ρ θ − Τ −
∂ +
∂
∂
= ∂
∂ h
d X
X v dX
cp (10) )
)( 1
( ) 6 6 (
) (
) ( 2
s co
g s
pc c
s C C
x d h Hk
d v X
X c
X s
+ − + ∆
− Τ
∂ + Τ
= ∂
∂ Τ
∂
∂
∂ ∗
θ ρ
λ
+ ′ +
−
−
∫ ∑
=
+
−
− − M
m
m X n m
X n
X
m DS n e
dn e n A e
n A
0
) ( 1 2
0 0
) ( 3
) 2 ( )
( ) 1 (
2 0
α π
π ω σ
π (11)
-Boundary conditions [9]:
- At: X = 0: therefore an energy balance on the front (X=0) gives
0 4
0 4
0 )
( X
S g s
i g i
g ∂
Τ
− ∂
− Τ
=
=
Φ φ ε σ θ λ∗
α
α (12) and θ =θ0;C = C0;C1 = 0;v = v0;vs = 0 (13)
-At: X = L :
P = P
atm; d = d
L (14)1 = 0
∂
= ∂
∂
= ∂
∂
= ∂
∂
= ∂
∂ Τ
∂
X v X C X C X X
θ (15) The value of the total emissivity εg is determined by the correlation for (Borodulya et al [14]) : εg = εP0.485 which is valid when the porosity is about 0.4; εp is the emissivity of the material in the solid state.
With the hypothesis that a thermodynamic equilibrium exists, it becomes : αg = εg.
2.2. Parameters:
Heat and mass transfer coefficients
Heat and mass transfer coefficients h and kg are evaluated by empirical correlation’s to Nusselt and Sherwood numbers (Gunn [15]), for 0.35≤ε≤1 and Re≤105 :
Nu = (7 - 10ε + 5ε2)(1 + 0.7 Re0.2Pr1/3) + (1.33 – 2.4ε + 1.2ε2)Re0.7Pr1/3 Sh = (7 - 10ε + 5ε2)(1 + 0.7 Re0.2c1/3) + (1.33 – 2.4ε + 1.2ε2)Re0.7Sc1/3
Effective thermal conductivity of the packed bed
To calculate the effective thermal conductivity of the packed bed, the correlation is used [16]:
) 1 (
) 1
( 1
1
*
β φ λ
λ β φ
λ ε λ λ ε
λ
− +
− +
=
s
s
s
Where
φ β
φ ε εβ
−
= −
1
β and φ are parameters which depend on the geometric characteristics of porous medium; β a is function of the particles arrangement and varies between 0.9 and 1; 0.9 corresponds to a close arrangement and 1 to a lose one.
φ is given as : φ = 6.486 10-1 – 0.417 10-1 (λs/λ) + 1.585 10-3 (λs/λ)2 – 0.285 10-4 (λs/λ)3 + 0.238 10-6 (λs/λ)4 – 0.752 10-9 (λs/λ)5
3. Numerical methods
The system of coupled and non linear differential equations (10-11) with the imposed boundary conditions (12-15), are solved by using primarily the control volume method of (Patankar [17]), for the energy equations. The system of algebraic equations obtained after discretization of the different model equations, all with a tridiagonal matrix, are solved by the Thomas’s algorithm (Patankar [17]) , using and under-relaxation process for the prime model variables. A stability study leads to ∆X+ = ∆X/L =0.005 corresponding to a space- step of 1mm (L = 200mm) and to under-relaxation coefficients as follow: the temperature of the solid: 0.22.
The precision of the calculations is equal to 10-4.
4. Results
The proposed theoretical model allows the determination of gas and solid characteristics along the reactor, both of which depend on the physical properties of the fluid and the radiative and physical properties of the solid.
4.1. Temperature profiles
Fig.2 shows the dimensionless temperature profiles in the gas and solid (T+=T/θ0; θ+=θ/θ0) as a function of the dimensionless distance X+=X/L from the warm front of the reactor with the following parameters:
φi = 1000W, θ0=300K, dL = 0.001m, v0 = 0.08m/s, ε = 0.45, L = 0.2m, K = 1820 m-1, L = 0.2 m.
Fig. 2. Temperature distributions for the gas and the solid along the packed bed, T+ = T/θ0; θ+ = θ/θ0
It is noted that there are two temperature zones: the first corresponds to the heating of gas by the porous medium exposed to concentrated external incident radiation, and the second to an equilibrium thermal zone between the gas and solid phases.
In the solid, the temperature gradient is very high because the effective thermal conductivity of the porous bed is low (λs*=0.3 W/m.K) and also due to the radiative transfer influence.
4.2. Analysis of heat transfer
a) Evolution of dimensionless numbers
To illustrate the different heat and mass exchanges, we represent in Fig.3 the evolution of dimensionless numbers: Re, Pr, Sc, Nu et Sh. We note that prandtl number remains nearly constant and varies in the interval of 0.788 to 0.636. the Nusselt and Sherwood numbers persuit the same evolution between 4.6 and 6, but the Reynolds number varies from a value near 0 to 0.9.
Fig 3. Dimensionless variables profiles (Nu, Re, Sh, pr and Sch).
b) Interaction radiation-conduction
To know the contribution of all heat transfer modes taking place in the reactor, in the domain of high temperatures, we have tried to determine then by calculating the flux densities of radiation Qr, conduction Qod and convection Qcv. If convective transfer is characterized by the coefficient h and conduction transfer in the packed bed by an effective thermal conductivity λs*, radiative transfer may also be characterized by a radiative conductivity:
r 3K
16 Τ3
= σ
λ where K (m-1) is the extinction coefficient per unit volume of the packed bed.
The interaction between these modes of transfer is evaluated by defining a parameter which is a function of the two conductivities, known as the conduction-to-radiation parameter and expressed as:
3
*
'
4 Τ
= n
N sK σ
λ
A value of N = 0.79 is found when the average control parameters are: φi = 1000W, v0 = 0.08 m/s, dL = 0.001 m, K = 1820 m-1,n’ = 1, L = 0.2 m, θ0 = 300K.
Doornink and herring [18], showed that for N≤5, radiative heat his implied the coupling of these two energy exchange modes in current model.
c) Evolution of heat flux densities
- The net radiative heat flux Qr : is given by expression (9);
- The conductive heat flux : is given by Fourier’s law :
Qcd s X
∂ Τ
− ∂
= λ*
- The convective heat flux : Qcv = P.cp.v.T
The figure 4 shows the evolution of these parameters. We note the importance of radiative transfer in comparison with the other modes of heat transfer. The flux density Qr decreases slowly from 1.033 106 to 8.145 105 W .m-2. For the conductive flux density, we note especially a drop of its value besides the hot surface (at X = 0). This fact explains the greet value of the temperature gradient observed near this one in Fig. 2. We note a significant increase of the convective flux density from 3.64 104 to 4.798 104 W.m-2. This phenomenon is explained by the creation CO during the reaction whose overall degree of conversion of carbon is relatively high.
Fig 4. Fig.5
Fig 4. Heat flux density (Qr, Qcd and Qcv) profiles along the packed bed Fig 5. Ratios of heat flux density profiles: Qr/Qcd et (2) : Qr/Qcv
To appreciate the importance of radiative transfer in comparison with the other transfer modes, we have represented in Fig. 5 the ratios Qr/Qcd/ and Qr/Qcv.
Nomenclature
C molar concentration of CO2 in the gas, mol/m3 C0 initial molar concentration of CO2 at the entry
of reactor, mol/m3
C1 molar concentration of CO in the gas, mol/m3 Cp specific heat of the gas, J/kg.K
cpc specific heat of the solid, J/kg.K d particle diameter, m
D binary diffusion coefficient (CO2-CO), m2/s D0 diameter of the cylinder containing the porous
medium, m
h convective heat transfer coefficient, J/m2.K H(µ) Chandrasekhar’s H-function
K extinction coefficient, k = α+δ, m-1
kg mass transfer coefficient, m/s L length of the packed bed, m L(τ,µ) intensity of radiation, W m molar mass, kg/mol n continue eigenvalue n0 discrete eigenvalue n` refraction index
N molar flux density, mol/m2.s P pressure inside the reactor, Pa Patm pressure at the ambient conditions
S area of the cylinder containing the porous medium, m2
T temperature of the solid, K
v0 velocity of gas CO2 at the entry of reactor, m/s v fluid velocity, (v0=εv), m/s
vs solid velocity, m/s
X axial coordinate (positive in the flow direction), m
xi molar fraction of specie in the gas X(-µ) Case’s X-function
Greek letters αp absorptivity δ scattering coefficient
∆H molar enthalpy of reaction, J/mol ε bed void fraction (porosity) εp emissivity of the solid θ gas temperature, K
λ thermal conductivity of the gas, W/m.K λs thermal conductivity of the particles, W/m.K λs* effective thermal conductivity of the packed
bed, W/m.K
µg viscosity of the gas, N.s/m2
µ direction cosine between the directed intensity and the positive τ axis
ρ density of the gas, kg/m3 ρc density of the solid, kg/m3
σ Stefan-Boltzmann constant, Wm2.K4 τ optical thickness, m
φi incident radiative flux (solar energy); W Φi radiative flux density, Φi=φi/S, W/m2 φ(N,µ) continue normal mode
φ(n0,µ) discrete normal mode
ω albedo, the ratio of the scattering to the extinction coefficient, ε = δ/k
Dimensionless numbers Nu Nusselt number, hd/λ Pr Prandtl number, µg/cp/λ Re Reynolds number, v0dp/µg
Sc Schmidt number, µg/ρD Sh Sherwood number, kgd/D Subscripts
C carbon L the exit of gas S solid
0 reference value at the entry of reactor
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