8926
Ind. Eng. Chem. Res. 2009, 48, 8926–8933
Reaction-Diffusion Model for Irregularly Shaped Ammonia Synthesis Catalyst and Its Verification under High Pressure Tao Li,* Mao-sheng Xu, Bing-chen Zhu, Ding-ye Fang, and Wei-yong Ying State Key Laboratory of Chemical Engineering, East China UniVersity of Science and Technology, Shanghai 200237, P. R. China
A one-dimensional isothermal multicomponent reaction-diffusion model was established for irregularly shaped ammonia synthesis catalyst A301. The intrinsic kinetics equation was derived from the experimental data obtained under high pressure ranging from 7.5 to 10.5 MPa. A feasible method was developed to measure the shape factor of the irregularly shaped ammonia synthesis catalyst. The dynamics single pellet string reactor method as well as parameter estimation was applied to get the tortuosity factor of irregularly shaped ammonia synthesis catalyst. The orthogonal collocation method combined with the Broyden method was employed to solve the model equations, and the internal-diffusion efficiency factor and the concentration distributions of individual components in the catalyst were obtained. The multicomponent reaction-diffusion model was verified by the global kinetics data obtained in a gradientless reactor. The calculation data agreed well with experimental data, so the model can be used to describe the processes of multicomponent reaction and diffusion in the irregularly shaped catalyst. The integrative methodology of catalyst engineering presented in this paper can be extended to guide catalyst preparation and reactor design. 1. Introduction Gas-solid reactors are popular in the chemical process industry. The design, scale-up, and operation of these reactors require detailed knowledge of the complicated mass transfer, heat transfer, and reactions in a gas-solid catalyst. The shape and structure of catalysts are key parameters in a gas-solid reaction process. The method for catalyst preparation and optimization of catalyst size are challenges to chemical engineering, especially for irregularly shaped gas-solid catalysts. It is necessary to develop a method for studying of the reaction-diffusion process in irregularly shaped catalysts for optimization of catalyst size and reactor design. Through the reaction-diffusion model, one can get the information about temperature and concentration gradients and judge if there is a dead zone in the catalyst particle. The internal effective factor can also be obtained simultaneously, hence the efficiency of the catalyst. The knowledge resulting from the model can direct catalyst manufacturers to chose proper shape, size, and pore distribution etc. to increase catalyst efficiency. A traditional reaction, ammonia synthesis, and a traditional gas-solid catalyst, irregular ammonia synthesis catalyst, are selected to study on the complex coupled mass-transfer and reaction process in the catalyst. Irregularly shaped ammonia synthesis catalyst (A301), like all the other industrial ammonia catalysts, is a kind of iron catalyst prepared by fusion and used intensively in ammonia synthesis plants in China. It has higher activity under lower temperature until 320 °C and pressure until 11.0 MPa and is qualified for application in large-scale plants.1-3 Industrial ammonia synthesis is a multicomponent reversible single reaction. Reaction and diffusion of reactants and products in catalysts take place simultaneously. The mathematical model of the internal effectiveness factor should be developed according to the property of multicomponent reaction-diffusion in the catalyst particle. Zhu4,5 proposed a mathematical model of internal effectiveness factor of a regular (cylindrical) shift * To whom correspondence should be addressed. Tel.: +86 021 64252874. Fax: +86 021 64251002. E-mail address:
[email protected].
reaction catalyst. The catalyst was sphericized according to the equivalent specific outer surface area, and a multicomponent reaction-diffusion model was set up. For irregularly shaped ammonia synthesis catalyst, Zhu6 advanced a model of multicomponent reaction-diffusion and obtained numeric solution of internal effectiveness factor for ammonia synthesis catalyst. But none of the previous work from the literature is concerned with verification of the model of the internal effectiveness factor of irregularly shaped ammonia catalysts at high pressure by experiments. The analytical methods to estimate the parameters of the model for irregularly shaped ammonia catalysts, such as equivalent diameter, shape factor, and tortuosity factor, have not yet been investigated. A one-dimension isothermal multicomponent reactiondiffusion mathematical model is developed for industrial irregularly shaped ammonia synthesis catalyst A301 in this paper to describe the coupled reaction and diffusion process in the catalyst. The engineering parameterssshape factor and tortuosity factorswere measured to calculate the effective diffusivity and then combined with the intrinsic kinetics; the numeric solution of internal effectiveness factor of irregularly shaped catalyst A301 is gained based on the model. The model is verified by the global reaction rate obtained in a gradientless reactor under the conditions of high temperature and high pressure with satisfactory results. The integrative methodology developed in this paper can be extended equally to other irregular catalyst to optimize catalyst structure and reactor design. 2. Model Description For reactions in porous catalyst, reactions take place only on the catalytic surface of the pores. Since the pores in the pellets are not straight and cylindrical; rather, they are a series of tortuous, interconnecting paths of pore bodies and pore throats with varying cross-sectional areas.7 The effective diffusivity is defined to describe the average diffusion aroused contemporarily with reactions in the pellets. Considering the effect of the pore distribution on diffusion, Knudsen diffusion should be consid-
10.1021/ie9001266 CCC: $40.75 2009 American Chemical Society Published on Web 08/25/2009
Ind. Eng. Chem. Res., Vol. 48, No. 19, 2009
8927
ered in calculating diffusivity. So the effective diffusivity of component i of gas mixture in the catalyst pores is8 Deff,i )
θ δ
[ ( n
∫ ∑ D1 ∞
0
j*i
ij
yj - y i
)
Jj 1 + Ji Dk,i
]
-1
f(r) dr
(1)
where DK,i ) 9.7 × 103r√T/Mi For a spherical porous catalyst in a multicomponent reaction system, a steady-state material balance on a reaction component i as it enters, leaves, and reacts in a spherical shell of inner radius R and outer radius R + dR of the pellet is performed.7 Combined with the intrinsic kinetics equation, the one-dimensional isothermal reaction-diffusion model, actually, a material balance equation used to describe the process of reaction and diffusion of all reaction species in catalyst can be developed and the model equation is shown in eq 2.
[ ( ) ]( ) ( )(
)
(2)
Substituting specific radius x ) R/Rp into eq 2 gives eq 3, and we have d2yi dx2
+
[ ( ) ]( ) ( )(
)
of the effective diffusivity with the varying compositions is negligible.9 So, the model equation is d2yi dx2
dDeff,i dyi ZRgT Rp2 2 + /D ) Fr + eff,i R dR dR P Deff,i p i dR2
d2yi
Figure 1. Schematic diagram of intrinsic kinetics experiment: (1) Gas cylinder. (2) Purifier. (3 and 6) Pressure regulator. (4) Mass flowmeter. (5) Reactor. (7) Control valve. (8 and 11) Three-way valve. (9) 0.75% H2SO4 solution. (10) Drier. (12) Soap bubble flowmeter. (13) Chromatograph.
dDeff,i dyi ZRgT Rp2 2 + /Deff,i ) Fr x dx dx P Deff,i p i (3)
Equation 3 is a second-order nonlinear ordinary differential equation indicating the mole fractions of all reaction species in an isothermal catalyst pellet changing with the specific radius x of the catalyst pellet. The boundary conditions are (yi)x)1 ) (yi)s at R ) Rp(x ) 1) (mole fractions at the external surface of the catalyst) (dyi /dx)x)0 ) 0 at R ) 0(x ) 0) When the intradiffusion affection is intensive, reaction species approach equilibrium composition at the radius Rd. That is to say, no reaction takes place inside pellet with the radius smaller than Rd. This part of pellet where no reaction takes place is called the dead zone, and the concentration gradients of all reaction species at R ) Rd is zero. When the dead zone occurs in the catalyst, the boundary conditions are R ) Rp(x ) 1), (yi)x)1)(yi)s (mole fraction at the surface of the catalyst) R ) Rd(x ) xd), (yi)x)xd)(yi)e (equilibrium composition), (dyi /dx)x)xd ) 0 On the basis of the above approach of model development, as to the irregularly shaped ammonia synthesis catalyst A301, it is assumed that: (i) the model is one-dimensional; (ii) the internal effectiveness factor is calculated in terms of a sphere whose specific surface area is equal to that of the irregularly shaped A301 ammonia synthesis catalyst; (iii) the catalyst is isothermal; (iv) a dead zone may occur in the catalyst; (v) the reaction is a multicomponent one (H2, N2, NH3); (vi) the change
+
( )(
)
ZRgT Rp2 2 dyi ) F r (i ) H2, N2, NH3) x dx P Deff,i p i (4)
The boundary conditions of the model equation are (yH2)x)1 ) (yH2)s,
(yN2)x)1 ) (yN2)s, (yNH3)x)1 ) (yNH3)s at x ) r/Rp ) 1
(dyH2 /dx)x)xd ) 0, (dyN2 /dx)x)xd ) 0, (dyNH3 /dx)x)xd ) 0 at x ) xd 3. Experiments and Results about the Parameters in the Model 3.1. Intrinsic Kinetics of Ammonia Synthesis Catalyst. Intrinsic kinetics is a key term in developing the reaction-diffusion model. Intrinsic kinetics experiment is carried out in an isothermal integral reactor at high pressure. 3.1.1. Experimental Section. A scheme of the experimental setup is shown in Figure 1. The feed gas made by 99.999% H2, N2, and 99.99% methane is deoxidized by DG-Pd catalyst and dehydrated by 5A molecule sieve. The material gas is admitted into the isothermal integral tubular reactor by operating a forward pressure regulator. The reacted gas is discharged by a pressure regulator down to the atmospheric pressure and measured by a soap film flowmeter. The reacted gas without is analyzed by chromatography. Ammonia concentration in the exit gas is tested by chemical analysis method. The size of the reactor is Φ 24 mm × 6 mm × 1200 mm. The catalyst used is 0.090-0.076 mm in size. The amount of catalyst is 2.4168 g (Fp ) 2682.3 kg/m3). The operating conditions are the following: temperature ranging from 593.15 to 753.15 K, pressure ranging from 7.5 to 10.5 MPa, and space velocity from 5000 to 30000 h-1 based on the exit gas. The molar ratio of H2 to N2 in the feed gas is 2.7-3.0, and the mole fraction of methane yCH4 is less than 0.05. The internal and external diffusion effects are eliminated at the experimental conditions. The catalyst bed is in the isothermal zone. 3.1.2. Results. The intrinsic kinetic data shown in Table 1 are obtained under the above experimental conditions. 3.1.3. Intrinsic Kinetics Equation. The intrinsic kinetics equation shown in eq 5 takes the form of the Temkin equation. Because the reaction system under high pressure is a nonideal
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Ind. Eng. Chem. Res., Vol. 48, No. 19, 2009
Table 1. Experimental Data of Intrinsic Kinetics of A301 Catalyst yin
yout
no.
T/K
P/MPa
Vsp2/h-1
H2
N2
CH4
H2
N2
CH4
NH3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
594.87 595.13 615.10 615.22 634.45 635.75 654.92 654.13 675.52 676.49 695.13 698.06 713.44 714.90 733.91 734.07 752.79 753.34 677.77 675.81 676.18 675.94 676.06 676.45
7.60 7.60 7.65 7.60 7.65 7.65 7.60 7.60 7.55 7.60 7.75 7.65 7.60 7.60 7.70 7.65 7.60 7.60 10.5 9.15 5.85 7.55 7.50 7.50
6358.52 10276.03 5982.83 10257.30 6333.54 10086.35 5995.98 9582.76 6113.54 10286.80 5960.45 9864.13 5923.13 9330.27 5782.32 8792.05 6299.44 9958.89 10156.54 6962.013 9503.46 9282.68 19262.61 27699.06
0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276 0.7265 0.7265 0.7265 0.7265 0.7265 0.7265
0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2399 0.2493 0.2493 0.2493 0.2493 0.2493 0.2493
0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0325 0.0242 0.0242 0.0242 0.0242 0.0242 0.0242
0.6810 0.6862 0.6685 0.6830 0.6591 0.6671 0.6497 0.6565 0.6403 0.6501 0.6276 0.6405 0.6328 0.6483 0.6402 0.6501 0.6586 0.6606 0.6298 0.6238 0.6604 0.6493 0.6718 0.6776
0.2219 0.2260 0.2207 0.2246 0.2162 0.2178 0.2063 0.2109 0.2055 0.2086 0.2051 0.2094 0.2078 0.2101 0.2114 0.2124 0.2149 0.2154 0.2106 0.2083 0.2256 0.2185 0.2264 0.2313
0.0321 0.0311 0.0322 0.0375 0.0340 0.0330 0.0372 0.0362 0.0342 0.0374 0.0348 0.0351 0.0342 0.0341 0.0343 0.0336 0.0333 0.0333 0.0254 0.0241 0.0251 0.0257 0.0246 0.0258
0.0650 0.0567 0.0785 0.0614 0.0907 0.0821 0.1068 0.0959 0.1200 0.1039 0.1325 0.1150 0.1252 0.1075 0.1141 0.1039 0.0932 0.0907 0.1341 0.1431 0.0889 0.1066 0.0772 0.0653
system, the compositions of all reaction species and reaction equilibrium constants are indicated by fugacity.7 rNH3 )
dNNH3 dw
where ln Kf ) ln
(
) k1fN2
fNH3 fH21.5fN20.5
fH21.5 fNH3
)
- k2
fNH3 fH21.5
(
) kT Kf2fN2
fH21.5 fNH3
-
fNH3
)
fH21.5 (5)
) -2.691122 ln T + 4608.8543/T -
1.270855 × 10-4T + 10-4T + 4.257164 × 10-7T2 + 6.1937237 (6) The fugacity is expressed by fi ) piφi, and the fugacity coefficient φi is computed by the Beattie-Bridgeman equation of state. Table 2. Results Derived from Experimental Data no.
P/MPa
T/K
kT
ln kT
yNH3,exp
yNH3,cal
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
7.60 7.60 7.65 7.60 7.65 7.65 7.60 7.60 7.55 7.60 7.75 7.65 7.60 7.60 7.70 7.65 7.60 7.60 10.5 9.15 5.85 7.55 7.5 7.5
594.87 595.13 615.10 615.22 634.45 635.75 654.92 654.13 675.52 734.07 695.13 698.06 713.43 733.91 752.79 714.90 676.49 753.34 677.77 675.81 676.18 675.94 676.06 676.45
0.0684 0.0663 0.2087 0.2086 0.5556 0.5616 1.4799 1.3545 3.6907 37.3562 8.3453 9.3006 17.3827 38.8862 73.1589 18.6868 3.7902 75.2648 3.9462 3.6761 3.7165 3.7131 3.7369 3.8125
-2.6827 -2.7134 -1.5669 -1.5673 -0.5877 -0.577 0.392 0.3034 1.3058 3.6205 2.1217 2.2301 2.8555 3.6606 4.2926 2.9278 1.3324 4.321 1.3728 1.3019 1.3128 1.3119 1.3183 1.3383
6.50 5.67 7.85 6.14 9.07 8.21 10.68 9.59 12.00 10.39 13.25 11.50 12.52 10.75 11.41 10.39 9.32 9.07 13.41 14.31 8.89 10.66 7.72 6.53
6.58 5.24 8.23 6.38 9.48 7.72 11.23 9.09 12.38 10.17 13.34 11.32 12.66 11.55 11.18 10.84 9.32 8.91 13.38 13.89 8.78 10.73 7.79 6.63
According to the material balance, we have kT )
Vsp2(1 + yNH3) 3600
∫
dyNH3
yNH3,2
yNH3,1
(1 + yNH3)2FA
(7)
where FA ) [{Kf2fH21.5fN2/fNH3} - {fNH3/fH21.5}] and yNH3,1 and yNH3,2 are ammonia mole fractions at the inlet and outlet of the integral reactor, respectively. Romberg integration method is applied to integrate eq 7, and the results obtained from the experimental data are shown in Table 2. In terms of the Arrhenius equation kT ) k0e-Ea/(RgT), we obtain kT ) 2.0257 × 1013 exp(-164833/Rg/T) and the correlation coefficient is larger than 0.99. 3.2. Shape Factor. The shape factor of a particle, a dimensionless quantity, is defined as the ratio of the outer surface area Ss of a sphere to the outer surface area Sp of a nonspherical particle whose volume Vp is equal to the volume of that sphere. The shape factor is also equal to the ratio of diameter of a sphere ds with the same specific outer surface area of actual nonspherical particle to the diameter of a sphere dp with the same volume of actual nonspherical particle.10 Hence ψs ) Ss /Sp ) ds /dp
(8)
where ds ) 6Vp/Sp. The shape factor indicates the different level between a sphere and a particle. ψs ) 1 for a sphere, and ψs < 1 for a nonsphere. The shape factor of a regular particle can be readily calculated through the definition, but for an irregularly shaped one, the shape factor cannot be obtained directly for an unknown outer surface area. In this paper, a method of determining the shape factor of an irregularly shaped particle is developed through measuring the pressure drop of a fixed-bed packed by these irregularly shaped particles, so the equivalent diameter ds of specific outer surface area can be solved through the pressure drop equation and the shape factor can be obtained. Many empirical expressions can be applied to calculate pressure drop ∆P along a fixed bed when a single-phase fluid passes through the bed. In this paper, Ergun correlation is used to calculate the pressure drop. When Re < 10, the pressure drop
Ind. Eng. Chem. Res., Vol. 48, No. 19, 2009
of a fixed bed is
11,12
8929
15
∆P ) 150
(1 - ε)2 µu0 H ε3 ds2
(9)
Considering the wall effect, if the ratio of the packed bed diameter dt to the particle diameter ds is between 7 and 91, the modified Ergun correlation is
( )( )( )
150µ(1 - ε)M ∆P ds ε3 1 ) + 1.75 2 Gds FfG L 1 - ε M
(10)
where M ) 1 + 2/3(ds/dt)(1/(1 - ε)). If the pressure drop and void fraction ε of the bed are known, ds can be solved from eq 10. Then the shape factor ψs can be computed through dp according to eq 8. 3.2.1. Experimental Section. Experiments are performed to obtain the pressure drops under varying flow rates in a fixed column. Purified dry gas is pressurized into a buffer to maintain constant flow rate and then flows down through the packed bed consisting of 3.3-4.7 mm irregularly shaped A301 catalyst at various flow rates measured by a mass flow meter. A micropressure difference meter is connected to two pressure measuring taps of the column, 500 mm apart. The bed is a Φ 48 mm × 3.5 mm tube. Two, 650 and 500 mm high, flow zones are at two ends of the column for developing the flow. Two pressure measuring taps are connected to two pressure measuring loops at their cross sections of the column. 3.2.2. Results. Experimental data measured, V0 and ∆P, are given in Table 3. The voidage ε, which is 0.466, is measured using the displacement method. As to every set of experimental data, eq 10 can be solved and ds is obtained. The results also give an indication in Table 3. From Table 3, the average ds of the A301 catalyst is 0.192 cm. Wen13 proposed that the equivalent diameter dp may be calculated from the geometric mean of two consecutive sieve openings, so for 3.3-4.7 mm irregularly shaped ammonia synthesis catalyst, dp ) (0.33 × 0.47)1/2 ) 0.394 cm. Finally, the shape factor ψs of 3.3-4.7 mm irregularly shaped A301 ammonia synthesis catalyst is 0.487. This method is verified to be valid by testing a regular pellet.14 3.3. Tortuosity Factor. The tortuosity factor is an important parameter of the solid catalyst, and it has an effect on the global reaction rate of gas-solid catalytic reaction and the efficiency of the catalyst. The steady-state technique and the dynamic technique are widely applied to determine the tortuosity factor. The steady-state technique, which is only applicable to cylindrical or hollow cylindrical pellets, is operated readily, but the effect of the dead pores on the diffusion can not be reflected. Now, the dynamic method such as the single pellet string reactor method (SPSRM) is widely used to measure the tortuosity factor. Table 3. Calculation results of 3.3-4.7 mm irregularly shaped ammonia synthesis catalyst V0/L min-1
∆P/Pa
Re
0.40 0.60 0.80 1.00 1.20 1.60 2.00 2.40
18.75 30.46 39.67 53.65 64.09 87.80 112.76 136.86
1.560 2.257 3.059 3.689 4.456 5.909 7.287 8.830
ds/cm 0.200 0.193 0.196 0.190 0.191 0.190 0.188 0.189
Scott claimed that this method is suitable for all different shapes of catalysts. The pulse-response curves reflected the adsorption, desorption, and diffusion occurring in catalysts can be truly indicated mass-transfer properties. So SPSRM is used to obtain the effective diffusivity of the irregularly shaped ammonia synthesis catalyst A301 and, hence, the tortuosity factor. The effort can supply the development of the reactiondiffusion mathematical model with fundamental data. 3.3.1. Theory. According to the SPSRM, more than 50 catalyst pellets, the ratio of its diameter to the chromatography column diameter is 1.1:1-1.4:1 to avoid channeling, are in series packed in a chromatography column. The material balance equations were solved by Kucera,16 and nth moment expressions containing mass-transfer parameters were obtained. The equations are very complicated when n is larger than 3. In general, the first absolute and the second central moments, indicated in eq 11 and eq 12, containing enough masstransfer information are in common use. µ1′ ) µ2 )
t0,A L (1 + ζ0) + + µ1d ′ V 2
[
(11)
]
EA t0,A2 2L (1 + ζ0)2 /V2 + + µ2d ζ1 + V ε 12
(12)
where
(
FpKA 1-ε θ 1+ ε θ
ζ0 )
)
ζ1 ) ζa + ζi + ζe ζa ) ζi ) ζe )
(
(
2 1 - ε FpKA θ ε θKada
)
( ) (
) )
KA 1 1 - ε R2θ θ 1 + Fp ε 15 θ Deff,A
(
)
KA 5 1 - ε R2θ θ 1 + Fp ε 15 θ KmR
According to the above model, many parameters including effective diffusivity Deff,A can be obtained. From eq 13, we can determine the tortuosity factor of a catalyst. Deff,A )
θ D δ E,A
(13)
Considering the effect of the pore size distribution on the Knusen diffusion, the general diffusivity is written as following DE,A )
∫
∞
0
(
1 1 + DAB DK,A(r)
)
-1
f(r) dr
(14)
where DAB )
0.001T1.75(1/MA + 1/MB)0.5 0.101325p[(
∑ V)
1/3
A
+(
∑ V)
1/3 2 B
]
3.3.2. Experimental Section. The reduced catalyst pellets are packed in a chromatography column. The trace gas is injected into the column at different velocities of the carrier gas and different column temperatures. The adsorption and desorption occurring in the catalyst pellets are recorded by an online computer sampling system and the pulse-response curves
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Ind. Eng. Chem. Res., Vol. 48, No. 19, 2009
are autoanalyzed by this system.17 So, the data of the first absolute and the second central moments are collected. The experimental conditions are the following: (column temperature) 313.15, 333.15 K; (carrier gas) Ar; (trace gas) N2; (bridge current) 60 mA; (size of the column) Φ 4 mm × 1 mm × 17 mm; (size of the catalyst pellets) Φ1.6 mm (average); (number of pellets) 60. 3.3.3. Data Analysis and Results. The first absolute and the second central moments collected are indicated in Table 4. Hashimoto18 and Smith19 brought forward a linear method of data processing which requires blank, inert, and adsorption experiments at the same carrier gas velocity. It is difficult to execute this method, and the workload is heavy. Gao20 proposed a parameter estimation method of data processing. A lot of correlative parameters can be obtained through parameter estimation from a set of experimental data, and the results obtained are proved to be reliable. In addition, this approach only requires an adsorption experiment, so the experimental work is reduced remarkably. The parameter estimation method is applied in this paper and the objective function is shown in eq 15. Many parameters including Deff,A are calculated by the simplex algorithms. M
S)
∑ [(µ′ - µ′
1cal)
1
2
+ (µ2 - µ2cal)2]
(15)
i)1
Considering the effect of the pore radius distribution, the general diffusivity DE,A is computed through discretization of the eq 14 based on the pore radius distribution of A301 catalyst shown in Table 5. So, the tortuosity factor of irregularly shaped ammonia synthesis catalyst A301 is calculated from the eq 13, and the results obtained give an indication in Table 6. The tortuosity factor is a parameter reflecting the pore structure of a catalyst and is temperature-independent. From Table 4. Experimental Data of the First Absolute and Second Central Moments 313.15 K
313.15 K
V0 (cm3 s-1)
µ1′
µ2
V0 (cm3 s-1)
µ1′
µ2
0.07374 0.09434 0.1179 0.1460 0.1760 0.2002 0.2454 0.2827 0.3331 0.4156 0.4793 0.5356 0.6176 0.6881 0.7658 0.8784 0.9591 1.0384 1.1251
78.999 63.989 53.398 44.890 38.784 34.216 30.137 26.419 24.185 19.910 17.714 17.122 15.735 14.828 13.571 12.669 12.132 11.417 10.737
476.327 334.223 255.028 204.344 173.782 141.367 112.689 86.352 63.364 50.173 45.347 34.716 29.390 15.757 8.850 5.785 3.915 1.395 1.037
0.0411 0.0488 0.0541 0.0877 0.1117 0.1385 0.1679 0.1995 0.2335 0.2716 0.3140 0.3420 0.4076 0.4448 0.4956 0.5579 0.6140 0.6680
127.518 98.915 64.840 63.770 52.952 44.130 37.652 33.348 29.498 25.911 23.553 21.125 19.165 18.053 16.752 15.434 14.852 13.821
1304.382 799.063 412.427 312.734 233.710 187.423 147.495 116.828 90.936 61.269 52.727 34.838 25.340 21.645 15.848 8.926 6.712 4.396
Table 5. Pore Radius Distribution of A301 Catalyst (θ ) 0.3693) pore radius (nm)
3-4
4-5
5-10
10-20
20-30
jr (nm)
volume (%)
0.53
3.44
65.81
28.73
1.53
12.85
Table 6. Results Obtained from Experimental Data carrier gas Ar Ar
trace gas N2 N2
T/K
Deff,A (cm2 s-1)
DE,A (cm2 s-1)
δ
313.15 333.15
3.01 × 10-3 3.22 × 10-3
2.71 ×10-3 2.83 ×10-3
3.32 3.24
the result above, it is shown that the numerical values of tortuosity factor at 313.15 and 333.15 K are very similar. So, it is concluded that the SPSR method combined with parameter estimation is practicable to determine the tortuosity factor of a catalyst. 4. Model Solution Many of numerical methods such as the numerical integration shooting method, weighted allowance method, finite element method, finite difference method, singular perturbation method, and orthogonal collocation method are often used to solve this kind of ordinary differential equation set with boundary-value problems. According to orthogonal collocation method applied in this paper, a suitable test function is selected to fit the solutions of the differential equation. The method can overcome stiff problems of equation. The calculation speed is fast and results obtained are reliable. First, a dimensionless transform of the model equations is carried out. Let z ) (x - xd)/(1 - xd)(xd, 1) f (0, 1). z ) 0 when x ) xd and z ) 1 when x ) 1. d2yi dyi 1 2 ) + 2 2 (1 x )[(1 x )u + x ] (1 - xd) dz d d d dz ZRgT Rp2 (r ) F (i ) H2, N2, NH3) P Deff,i i w p
( )
(16)
The boundary conditions are (yH2)z)1 ) (yH2)s,
(yN2)z)1 ) (yN2)s, and (yNH3)z)1 ) (yNH3)s when z ) 1
(dyH2 /dx)z)0 ) 0,
(dyN2 /dx)z)0 ) 0, and (dyNH3 /dx)z)0 ) 0 when z ) 0
Assuming that the catalyst is a sphere, concentrations of all species in the catalyst are symmetrical with respect to the center of the sphere. Considering the boundary conditions, an even function z is chosen as test function to solve the model through orthogonal collocation method. N+1
∑
yH2 )
N+1
diz2i-2 ;
i)1
yN2 )
∑
N+1
eiz2i-2 ;
i)1
yNH3 )
∑fz
2i-2
i
i)1
(17) At collocation point zj, we have N+1
yH2,j )
∑dz
N+1 2i-2
i j
i)1
;
yN2,j )
∑ez
N+1 2i-2
i j
i)1
;
yNH3,j )
∑fz
2i-2
i j
i)1
(18) where j ) 1, ..., N, N + 1. Also, d, e, and f are undetermined coefficients. Table 7. Comparison of the Calculated Values with the Experimental Values of the Internal Effectiveness Factor no.
T/K
1 2 3 4 5 6 7 8 9
678.27 677.09 604.15 637.84 615.13 613.53 652.09 672.36 712.18
P/MPa V0/L h-1 yNH3/% 9.10 9.10 9.00 9.05 9.05 9.05 9.05 7.65 7.75
9.83 19.64 10.60 20.84 9.84 20.91 20.02 17.68 9.99
11.96 8.66 9.85 6.61 8.54 5.50 7.50 8.03 9.81
xd
ζNH3,cal
ζNH3,exp
err/%
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.6235 0.5510 0.7162 0.6072 0.7573 0.6322 0.5969 0.5750 0.4652
0.5997 0.5244 0.6866 0.5623 0.7330 0.6082 0.5585 0.5440 0.4695
-3.97 -5.07 -4.31 -7.99 -3.32 -3.95 -6.88 -5.70 0.92
Ind. Eng. Chem. Res., Vol. 48, No. 19, 2009
So the first-order and second-order derivatives of the collocation functions at collocation point zj are
N+1
∑ k)1
dyH2,j
N+1
∑ (2i - 2)d z
)
dz
2i-3
i j
;
i)1
dyN2,j dz
∑ (2i - 2)e z
2i-3
i j
;
i)1
N+1
∑ (2i - 2)f z
)
dz
dz2
}
1 2 A · yi,k ) Bjk + (1 - xd)[(1 - xd)z + xd] jk (1 - xd)2 ZRgT Rp2 (r ) F (i ) H2, N2, NH3) (28) P Deff,i i w,j p
( )
N+1
)
dyNH3,j
d2yH2,j
{
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2i-3
i j
(19)
i)1
The above nonlinear equation sets can be solved by the Broyden quasi-Newton method. The internal effectiveness factor of the catalyst is the ratio of actual overall rate of reaction to rate of reaction that would result if entire interior surface were exposed to the external pellet surface conditions.7,8
N+1
)
∑ (2i - 2)(2i - 3)d z
2i-4
i j
i)1 2
d yN2,j dz2
;
ζNH3 ) -
4 πR 3F (r ) 3 p p NH3 w,s
N+1
)
∑ (2i - 2)(2i - 3)e z
2i-4
i j
;
i)1
d2yNH3,j dz2
)
∑ (2i - 2)(2i - 3)f z
2i-4
i j
(20)
N+1
-3
i)1
jyH2 ) Q · dj ;
jyN2 ) Q · je ;
jyNH3 ) Q · jf
(21)
jyH′ 2 ) M · dj ;
jyN′ 2 ) M · je ;
jyNH ′ 3 ) M · jf
(22)
jyH′′2 ) N · dj ;
jyN′′2 ) N · je ;
jyNH ′′ 3 ) N · jf
(23)
Inversely transforming eq 21, we have dj ) Q-1 · jyH2 ;ej ) Q-1 · jyN2 ; jf ) Q-1 · jyNH3
(24)
Substituting eq 24 into eqs 22 and 23, we have jyH′ 2 ) M · dj ) M · Q-1 · jyH2 ) A · jyH2 ; A · jyN2 ;
(29)
Discretizing eq 29, we have
N+1
Deff,NH3
And then substituting eqs 25 and 26 into model eq 16, we have 2 A · jyi ) (1 - xd)[(1 - xd)z + xd] ZRgT Rp2 (r ) F (i ) H2, N2, NH3) P Deff,i i w p
( )
N+1,jyNH3,j
(30) Fp(rNH3)w,s
The model equations are solved based on experimental conditions of global kinetics, and then, the effective factor of the catalyst is obtained. 5. Verification of the Model The internal effectiveness factor ζNH3,cal of the catalyst under different experimental global kinetics conditions is computed based on solving model equations and compared with ζNH3,exp. The results are shown in Table 7. ζNH3,exp is directly calculated according to the eq 31.
jyN′ 2 ) jyNH ′ 3 ) A · jyNH3 (25)
jyH′ 2 ) N · dj ) N · Q-1 · jyH2 ) B · jyH2 ; jyN′′2 ) B · jyN2 ; jyNH ′′ 3 ) B · jyNH3 (26)
∑A
j)1 Rp2
ζNH3 )
Equations 18, -20 are expressed by matrices, so we have
B · jyi +
4πRp2Deff,NH3(dyNH3 /dR)s
ζNH3,exp )
(rNH3)global (rNH3)intrinsic
(31)
(rNH3)global in a gradientless reactor is also readily calculated from the NTyNH3 - N0yNH3
(rNH3)global )
wcat
(32)
Equation 33 is used to compute the relative error between ζNH3,exp and (rNH3)global. err )
(27)
ξexp - ζcal ξexp
(33)
The concentrations and concentration gradients of all reaction species are obtained by solving model equations at
Discretizing eq 27, we have
Table 8. Parameters in the Catalyst at Experimental Point 1 (T ) 678.27 K, P ) 9.1 MPa) x 0.0000 0.2154 0.4206 0.6063 0.7635 0.8851 0.9652 1.0000
yH2 0.6039 0.6053 0.6092 0.6157 0.6244 0.6342 0.6426 0.6465
yN2 0.1890 0.1897 0.1918 0.1952 0.1998 0.2050 0.2094 0.2115
yNH3 0.1839 0.1820 0.1762 0.1664 0.1533 0.1387 0.1261 0.1196
yCH4 0.0232 0.0230 0.0229 0.0228 0.0227 0.0226 0.0225 0.0223
rNH3 (mol/g s-1) -6
1.3629 × 10 1.4615 × 10-6 1.7700 × 10-6 2.3182 × 10-6 3.1297 × 10-6 4.1628 × 10-6 5.2027 × 10-6 5.7936 × 10-6
∆yH2/∆x
∆yN2/∆x
∆yNH3/∆x
0.0000 0.0065 0.0190 0.0350 0.0553 0.0806 0.1049 0.1121
0.0000 0.0032 0.0102 0.0183 0.0293 0.0428 0.0549 0.0603
0.0000 -0.0088 -0.0283 -0.0528 -0.0833 -0.1201 -0.1573 -0.1868
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Ind. Eng. Chem. Res., Vol. 48, No. 19, 2009
Notation
Figure 2. Schematic diagram of the pressure drop experiment: (1) air compressor; (2) air filter; (3) dryer; (4) buffer; (5) mass flow meter; (6) fixed-bed; (7) micropressure difference meter; (8) pressure measuring tap.
Figure 3. Schematic diagram of testing of tortuosity factor: (1) cylinder of carrier gas; (2) cylinder of trace gas; (3) six-way valve; (4) 1102 gas chromatograph; (5) soap bubble flow meter; (6) system of autoanalysis; (7) monitor.
the arbitrary experimental point of the global kinetics and are shown in Table 8. From the results obtained, it is seen that the calculated values from the model are in agreement with the experimental values on the whole. As the irregularly shaped A301 ammonia synthesis catalysts are thin in shape, the actual diffusion channels are shorter than those of the sphericized particles, so most of the experimental values are smaller than the model values. Even then it can also be concluded that the model developed on the multicomponent reaction-diffusion in an irregularly shaped catalyst is reliable and can be used to describe the concentration distributions in the catalyst. 6. Conclusions The method for developing a model of coupled reactiondiffusion processes in an irregularly shaped catalyst was set up. The one-dimensional reaction-diffusion of the irregularly shaped ammonia synthesis catalyst was developed. The intrinsic kinetics equation of the irregularly shaped ammonia synthesis catalyst under high pressure and temperature was derived from the experimental data. The method of measuring a shape factor of an irregularly shaped was developed, and the diameter ds of a sphere with same specific outer surface area of irregularly shaped ammonia synthesis catalyst was obtained. The single pellet string reactor method combined with parameter estimation was applied to get the tortuosity factor of the irregularly shaped ammonia synthesis catalyst. The orthogonal collocation method was used to solve the onedimensional reaction-diffusion model, and the internal effectiveness factor was calculated. The reliability of the one-dimensional reaction-diffusion model of the irregularly shaped ammonia synthesis catalyst was verified under high temperature and pressure. The integrative methodology of catalyst engineering presented in this paper can be extended to direct the design of optimal catalyst structure and reactor.
dp ) diameter of a sphere with the same volume of actual nonspherical particle [cm] ds ) diameter of a sphere with the same specific outer surface area [cm] dt ) diameter of fixed-bed [cm] DE ) general diffusivity [cm2 s-1] Deff ) effective diffusivity [cm2 s-1] Dij ) diffusivity of species i in species j [cm2 s-1] Dk ) Knusen diffusivity [cm2 s-1] EA ) axial diffusivity [cm2 s-1] Ea ) activation energy of reaction [J mol-1] f(r) ) distribution function of pore radius G ) mass flow rate [g cm-2 s-1] H ) height of fixed-bed [cm] Ji ) diffusion flux of species i [mol cm-2 s-1] Jj ) diffusion flux of species j [mol cm-2 s-1] k0 ) pre-exponential factor kT ) reaction rate constant Kf ) reaction equilibrium constant expressed by fugacity KA ) adsorption equilibrium constant of tracer gas [cm3 g-1] Kada ) adsorption rate constant [s-1] Km ) mass-transfer coefficient of tracer gas [cm s-1] L ) length of the chromatography column [cm] M ) number of experiment Mi ) molecular weight [g] N ) number of collocation N0 ) molar flow rate at reactor inlet [mol s-1] NT ) mole flow rate at reactor inlet [mol s-1] pi ) partial pressure of species i [Pa] P ) pressure [Pa] ∆P ) pressure drop [Pa] jr ) average pore diameter [nm] ri ) reaction rate per unit mass of catalyst [mol g-1 h-1] R ) radial distance of catalyst particle [cm] Re ) Renolds number Rg ) ideal gas constant [J mol-1 K-1] Rp ) radius of catalyst [cm] S ) objective function Ss ) outer surface area [cm2] Sp) outer surface area SP of a nonspherical particle [cm2] t0,A ) time of injecting tracer gas [s] T ) temperature [K] u0 ) flow rate of gas [m s-1] V0 ) flow rate of carrier gas [cm3(STP) s-1] Vp ) volume of a nonspherical particle [cm3] VSP2 ) space velocity based on flow rate at reactor outlet [h-1] (∑V)I ) molecule diffusion volume y ) mole fraction Z ) compressibility factor FP ) density of catalyst particle [g cm-3] Ff ) density of gas [g cm-3] φi ) fugacity coefficient θ ) porosity of catalyst pellet ζi ) internal effectiveness factor δ ) tortuosity factor ε ) voidage of bed µ ) dynamic viscosity [Pa s-1] µ1′ ) first absolute moments ′ ) first absolute moments originated from the dead volume µ1d µ2 ) second central moment µ2d ) second central moment originated from dead volume ζ1 ) overall resistance ζa ) adsorption resistance
Ind. Eng. Chem. Res., Vol. 48, No. 19, 2009 ζe ) external diffusion resistance ζi ) internal diffusion resistance Subscripts cal ) calculated value e ) equilibrium state exp ) experimental value global ) global kinetics intrinsic ) intrinsic kinetics s ) outer surface w ) weight
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ReceiVed for reView January 28, 2009 ReVised manuscript receiVed July 29, 2009 Accepted August 5, 2009 IE9001266
