Intraparticle Mass and Heat Transfer Modeling of Methanol to Olefins

Article pubs.acs.org/IECR

Intraparticle Mass and Heat Transfer Modeling of Methanol to Olefins Process on SAPO-34: A Single Particle Model Xiao-Min Chen,† Jie Xiao,‡ Ya-Ping Zhu,† and Zheng-Hong Luo*,†,§ †

Department of Chemical and Biochemical Engineering, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, China ‡ College of Chemistry, Chemical Engineering and Materials Science, Soochow University, Suzhou 215123, China § Department of Chemical Engineering, College of Chemistry and Chemical Engineering, Shanghai Jiao Tong University, Shanghai 200240, P. R. China S Supporting Information *

ABSTRACT: Light olefins such as ethylene and propylene are high value-added gases usually derived from petroleum refining. The methanol to olefins (MTO) process offers an attractive alternative route to produce olefins, as methanol can be prepared from a relatively cheap and widely available source of energy, i.e., coal or natural gas. In the present work, a comprehensive single particle model was developed to characterize detailed chemical and physical phenomena occurring within a SAPO-34 catalyst particle during the MTO process. This model allows prediction of the species mass fraction, temperature, reaction rate, and pressure distributions within the particle as a function of particle diameter, operating temperature, mean intraparticle pore diameter, and thermal conductivity. A thorough parametric study based on the validated model demonstrated that intraparticle phenomena have a great impact on the MTO process and thus cannot be neglected in MTO process modeling, especially for the cases where fast or strong exothermal reactions occurred in large catalytic particles.

1. INTRODUCTION Light olefins such as ethylene and propylene are key petrochemicals in the petrochemical industry.1 The conversion process of methanol to olefins (MTO) provides a promising alternative route for olefin production when cheap natural gas and coal are available.1−3 For production of light olefins, the MTO process does not rely on the methanol to gasoline process.4 Instead, it is based on the coal to olefin (CTO) process, whose industrialization was first accomplished in China in 2010.5,6 The MTO process that is in the core of the CTO process is also one of the promising methods to convert coal to olefins.6−9 Generally, the MTO reaction is a highly exothermic, catalytic reaction.10−15 The catalyst usually falls into the following categories: large-pore, medium-pore, and small-pore microporous/zeolite materials.1,8 In order to obtain C2−C4 olefins as the major products of methanol conversion, small-pore microporous/zeolite catalysts are generally applied.8 It has been shown that chabazite, erionite, zeolite-T, ZSM-34, SAPO34, etc., are active catalysts for the MTO process.16 Among them, SAPO-34 is widely used in industrial practices as well as experimental studies, and has been demonstrated to have a high selectivity toward light olefins at almost 100% methanol conversion.16−20 Up to now, most reports on the MTO process have focused on fixed-bed reactors.21−24 For the highly exothermic MTO reactions based on SAPO-34 in a fixed-bed reactor, they assumed that the intraparticle transfer limitation could be safely neglected due to the small catalyst particles (≤100 μm in diameter).21−24 Theoretically, highly exothermic and catalytic reactions can be implemented in various types of reactors,25 such as the autoclave, the fixed-bed reactor, or the fluidized-bed reactor © 2013 American Chemical Society

(FBR). Among them, the FBR has demonstrated excellent heat and mass transfer capabilities with a relatively simple structure and has been successfully applied for the MTO process in chemical industry.15,22 Since small catalyst particles (≤100 μm in size) tend to be blown out from FBRs26−29 and can lead to significant abrasion, catalyst particles with sizes larger than 100 μm are also used in FBRs.29 In this case, the transfer resistance does exist within the catalyst particles due to the molecule− molecule and molecule−wall collisions, as illustrated in Figure 1. However, it remains unclear whether the intraparticle diffusion can be safely neglected for a MTO process using large catalyst particles in a FBR. Therefore, there is an urgent

Figure 1. The single particle model schematic. Received: Revised: Accepted: Published: 3693

October 8, 2012 February 11, 2013 February 18, 2013 February 18, 2013 dx.doi.org/10.1021/ie302736b | Ind. Eng. Chem. Res. 2013, 52, 3693−3707

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far for certain catalytic systems cannot be adopted directly for the study of MTO reaction systems. In this work, a comprehensive single particle model is developed to describe the intraparticle transport phenomena in SAPO-34 during the MTO process. The model integrates the mass, energy, and momentum balances as well as the gas-state equations, lumped-species kinetics, and multicomponent diffusion equations. After validation, the model has been successfully used to predict most key process data including the species mass fraction, temperature, reaction rate, and pressure distributions within the single particle as a function of the particle diameter, rate constant, intraparticle diffusivity, and thermal conductivity coefficient.

need to develop a comprehensive single particle model that can predict the intraparticle transfer effects.29,30 Such a model is also expected to help the better design of FBRs and the optimization of catalyst particles and reaction conditions.30 In order to develop a comprehensive single particle model, the mass, energy, and momentum balances, together with lumped-species kinetics and multicomponent diffusion equations have to be solved simultaneously with the gas-state equation for the intraparticle gas flow.30 There are many open reports on the MTO process/reaction modeling.8,10−15,17,18,30−42 However, most of them focused on the MTO process catalyst10−15,17,18,31−35 and kinetics.34−48 For instance, Schoenfelder et al.22 suggested a circulating fluidized bed (CFB) reactor model for the MTO process based on the material and energy balance equations. A lumped kinetic model was incorporated. Their reactor model was used to predict the methanol conversion and product distribution.22 Soundararajan et al.8 also simulated the MTO process in a CFB reactor numerically. The simulation incorporated a SAPO-34 kinetic model and a core-annulus type hydrodynamic model.8 Recently, Alwahabi et al.15 adopted a SAPO-34 kinetic model12−14 and predicted the yield and selectivity of products in both the fixed reactor and the FBR. However, none of the above-mentioned works has addressed the effects of intraparticle mass and heat transfer. The authors have not identified any published works on the modeling of the MTO process with intraparticle transfer phenomena taken into account. On the other hand, a number of single particle models have been reported for the gas−solid catalytic reactions.29,49−61 Because the intraparticle transport is the molecular mass diffusion or molecular species transport involving the random and individual movements in nature, as described in Figure 1, these models are generally based on molecular diffusion models.54 There are four typical molecular diffusion models, namely, the Fick or Wilke model,55 the Maxwell−Stefan model,54 the dusty gas model,56,57 and the Wilke−Bosanquet model.54 Among them, the Fick or Wilke model and the rigorous Maxwell−Stefan model assume bulk diffusion, whereas the rigorous dusty gas model is used to describe the combined bulk and Knudsen diffusion fluxes. Simplified from the dusty gas model, the Wilke−Bosanquet model is proposed. In addition, a single particle model for describing the mass and heat transport in a porous pellet with chemical reactions can be formed on the basis of these molecular diffusion models with some simplifications or assumptions. For instance, Graaf et al.59 used the dusty gas model to investigate the relative importance of Knudsen and bulk diffusion for the methanol synthesis without considering the pressure and temperature gradients in the spherical catalyst particles. Salmi et al.61 assumed an isotropic pore structure of catalyst particles and applied the Maxwell−Stefan and Wilke models for the catalyst pellet without taking into account the effect of Knudsen diffusion and intraparticle pressure gradient. Recently, Solsvik et al.54 developed a comprehensive single particle model to characterize molecular diffusions in two cases: the steam methane reforming (SMR) and the methanol synthesis. Mass diffusion fluxes were described according to the rigorous Maxell−Stefan and dusty gas models, and the relatively simpler Wilke and Wilke−Bosanquet models. It was found that Knudsen diffusion hardly influences the results of the highly intraparticle diffusion limited SMR processes. On the basis of the above discussions, it can be concluded that the single particle models developed so

2. SINGLE PARTICLE MODELING AND SOLUTION DERIVATION In this section, a comprehensive single particle model specifically tailored for the MTO process is developed, which is followed by an effective solution derivation method. 2.1. Modeling of Intraparticle Mass and Heat Transfer. The model integrates a set of equations of the mass, energy, and momentum conservations as well as the gas-state, lumpedspecies kinetics, and multicomponent diffusion. Three assumptions have been adopted: (i) the catalyst particle is a spherical pellet, (ii) particle deformation can be neglected, and (iii) all intraparticle parameters vary with the radial position only. The following equations can be derived: Total mass balance in a particle: ε

vs ∂ ∂ρ + r2 (r 2ρ) = 0 ∂t r ∂r

(1)

where

ρ=

PM RT

(2)

Momentum balance in a particle:

vrs = −

B ∂P μ ∂r

(3)

s

where

B=

εd 0 2 32τ

(4)

Component material balance in a particle: ε

vs ∂ 1 ∂ ∂ (ρYi ) + r2 (r 2ρYi ) = − 2 (r 2ji , r ) + Si̅ ∂t r ∂r r ∂r

(5)

where ∂Yi ∂r

(6)

Si̅ = (1 − ε)ρcat R i

(7)

ji , r = −ρi Di ,eff

Heat balance in a particle: n

((1 − ε)Cpcat ρcat + ερ ∑ YCp i i) i=1 n s = −ρ ∑ YCpv i i r i=1

3694

∂T ∂t

∂T 1 ∂ − 2 (r 2Q r ) + S ̂ ∂r r ∂r

(8)

dx.doi.org/10.1021/ie302736b | Ind. Eng. Chem. Res. 2013, 52, 3693−3707

Industrial & Engineering Chemistry Research Q r = −λ Ŝ =

∂T ∂r

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Rei =

(9)

∑ R i·Hi

(10)

(26)

t * = t /t0 ,

(11)

Di , k

μi

A general model can be derived by transforming the above model into a dimensionless format using the following dimensionless parameters. The resulting model equations with the dimensionless format are listed in Appendix A.

where Di,eff is the diffusion coefficient and can be calculated via the Wilke−Bosanquet model.54 Namely, ε 1 Di ,eff = · 1 1 τ + Dim

d puρi

P* = P /P0 ,

T * = T /T0 ,

where

Dim =

r * = r /r0 ,

Y i* = Yi /Y i*0

M * = M /M 0 , (27)

1 − Yi M ∑nj = 1 j≠i

Yj MjDij

(12)

1.43 × 10−7 × (T )1.75

Dij =

2 PMij1/2[(∑v )1/3 + (∑v )1/3 i j ]

⎛ 1 1 ⎞⎟ Mij = 2⎜⎜ + Mj ⎟⎠ ⎝ Mi

(13)

−1

Di , k = 97

do 2

(14)

T Mi

Figure 2. The kinetic scheme. (15)

2.2. Reaction Kinetics. The MTO reactions that occurred within the catalyst particles are characterized by the kinetic model suggested by Gayubo et al.11 The reaction scheme can be found in Figure 2. The reaction rate of the ith step, ri, which is formulated as a function of the mass fractions of the reactants in that step, is given as follows:11

Boundary conditions: As described above, the single particle is spherical and symmetrical. Therefore, the following boundary conditions for the material transfer and gas flow at the outer surface of the particle can be applied:58−61 At r = 0:

∂Yi =0 ∂r

(16)

∂T =0 ∂r

(17)

For steps 1−5: r(1 − 5) = K(1 − 5)YCH3OH

For step 6: r6 = K 6YC3H6

At r = (dp/2): ki , g(Yis − Yi f ) = −Yi sDim

λ

∂T ∂r

s

∂Yi ∂r

− vrsYi s s

− ρCps T svr = hi , g (T f − T s)

r7 = K 7YC4H8

(18)

hi , g =

DimShi dp

For step 8: r8 = K8YC5H10

(19)

⎛ 1 1 ⎞ − K i = Ai exp Ei⎜ ⎟ 698 ⎠ ⎝ T0T *

(20)

(31)

ρi Dim

R CH3OH = −r1·2·MCH3OH /MC2H4 − r2·3·MCH3OH /MC3H6

(22)

− r3·4·MCH3OH /MC4H8 − r4·5·MCH3OH /MC5H10

ki , gNui dp

− r5·MCH3OH /MCH4

(23) (24)

Cpi ·μi λi

(32)

Therefore, the consumption or production rate of the j component, Ri, can be calculated as

(21)

μi

Nui = 2 + 0.6Pr1/3Rei0.5

Pri =

(30)

where

Shi = 2 + 0.6Sci1/3Rei 0.5

Sci =

(29)

For step 7:

where ki , g =

(28)

(25) 3695

(33)

R CH4 = r5

(34)

R C2H4 = r1 + r6 + r7 + r8

(35)

R C3H6 = r2 − r6 + r8

(36)



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R C4H8 = r3 − r7

(37)

R C5H10 = r4 − 2·r8

(38)

Therefore, the simplified single particle model must be validated first. Li et al.67 have applied an empirical formula for calculating the isotherm effectiveness factor, which is shown in eqs 40 and 41. On the basis of Gayubo et al.’s data11 and eqs 40 and 41, an estimated value of the isotherm effectiveness factor can then be obtained.

R H2O = −R CH3OH − R CH4 − R C2H4 − R C3H6 − R C4H8 − R C5H10

(39)

η=

The kinetic parameters are listed in Table S1 in the Supporting Information. 2.3. Solution Derivation. In this work, the orthogonal collocation method is used to solve the above governing equations. First, the radial derivatives of the dimensionless governing equations are discretized by the orthogonal collocation method. As a result, the above equations can be converted into a set of ordinary differential equations. Next, these ordinary differential equations are solved using the ODE23S function in the Matlab soft. All simulations are executed in a workstation with four 2.83 GHz Pentium CPUs and 4GB memory. Unless otherwise stated, all model parameters used in this work are listed in Tables S2−S5 in the Supporting Information.62−67 In order to make sure that the simulation results are independent of the number of collocation points, simulations with 4, 5, 6, 7, 8, and 9 points are performed in advance. It is found that six collocation points offer sufficiently fine results. Thus, all simulations in this work are based on six collocation points.

1⎛ 1 1 ⎞ − ⎜ ⎟ ϕ ⎝ tanh(3ϕ) 3ϕ ⎠

(40)

r0 3

(41)

where ϕ=

Kp De

Figure 3 compares the simulation data and the estimated data obtained from the above empirical equations. The total average relative deviation for all five particle diameters is 5.39%. The maximal and minimum relative deviations are 12.91 and 0.06%, respectively, which correspond to the maximal and minimum particle diameters, respectively (i.e., dp = 1 and 0.1 mm). With the decrease of the particle diameter, the relative deviation decreases. Since the empirical formula for calculating the effectiveness factor is based on experimental data, which incorporate the measurements of intrinsic kinetics data, effective diffusivity coefficient, and thermal conductivity coefficient, etc.,11,67 and these experimental measurements themselves have some errors, the simulated data shown in Figure 3 are in good agreement with the estimated data for small particles. On the other hand, Figure 3 also shows that temperature, pressure, and diffusivity within small particles are nearly constant, which fits well with the assumption of the above empirical formula. Larger relative deviation can be identified for large particles, which can be attributed to the larger gradients of temperature, pressure, and diffusivity within these particles. Our validated model can be used to predict intraparticle parameter distributions that are not easily obtained from experiments. 3.2. Intraparticle Transfer Phenomena. In this section, we focus on a comprehensive description of the flow field within a porous spherical pellet that includes intraparticle temperature, pressure, and species mass fraction distributions along time and spatial position. Figures 4 and 5 describe the intraparticle parameter distributions at different time instants and positions, respectively. With such a small particle in a reactor compared to the size of a reactor, the time from initial state to stability of the intraparticle parameter distributions is very short due to the high reaction rate during the intraparticle initial period. Accordingly, most previous efforts neglected this period. Indeed, it is shown in our work that the intraparticle parameters can quickly reach stability within ∼0.1 s, as illustrated in Figure 4. In the next analysis, only the distributions at steady state are analyzed. Figure 5 shows the intraparticle parameter distributions along the radial direction at steady state. It is shown that the mass fractions of CH3OH and C5H10 increase along the radial direction and the changes of other species are in the opposite direction. Since CH3OH is used as the reactant, its mass fraction decreases from the outer surface to the center inside the catalyst particle, which leads to the decrease of its gradient. Except for C5H10, the changes of all product concentrations are in the opposite direction, which can be owed to the MTO reaction scheme (see Figure 2). On the other hand, the

3. RESULTS AND DISCUSSION In this section, the model is first validated by comparing model predictions with some data calculated from the classical model.

Figure 3. The comparison between simulation and estimated data for the mean effectiveness factor of steps 1−5 (conditions: d0 = 5.35 × 10−9 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

Then, the intraparticle transfer phenomena are investigated thoroughly. Finally, the single particle model is applied to investigate the effects of some key factors on the intraparticle transfer limitation. Herein, we use the effectiveness factor to measure the diffusion limitation within the particle. The effectiveness factor is the ratio between the reaction rates with or without considering intraparticle transfer limitation. A deviation further from 1 of the effectiveness factor indicates a stronger influence by intraparticle transfer limitation. 3.1. Model Validation. Although the single particle model is based on the fundamental principles of flow and heat transfer, some assumptions are introduced to simplify the model. 3696



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Figure 4. The evolution of the intraparticle parameters at r = 0.125 mm with time (a, temperature; b, species mass fraction; c, pressure; d, reaction rate) (simulation conditions: dp = 5 × 10−4 m, d0 = 5.35 × 10−9 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

Figure 5. The evolution of the intraparticle parameters at steady state along the radial direction (a, temperature; b, species mass fraction; c, pressure; d, reaction rate) (simulation conditions: dp = 5 × 10−4 m, d0 = 5.35 × 10−9 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

formation and the cracking of C5H10 coexist in the MTO process (see Figure 2). Furthermore, the cracking reaction (see

step 8 in Figure 2) is mainly within the particle and the rate of the cracking reaction (i.e., 2r8) is close to that of the formation 3697



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Figure 6. The effects of the particle diameter on the intraparticle species mass fractions (a, CH3OH; b, CH4; c, C2H4; d, C3H6; e, C4H8; f, RH) (simulation conditions: d0 = 5.35 × 10−9 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

temperature, and for the pressure, this value is 1.2%. The very small intraparticle temperature/pressure gradients in this case are due to the small particle size (0.5 mm) and/or the MTO process that is not a strong exothermic process. Accordingly, Figure 5 proves that the intraparticle diffusion mainly affects the mass fraction of each species and the heat transfer limitation is weak in specific operation conditions. However, for some cases with a large particle size or a strong exothermic reaction in an industrial reactor, the intraparticle heat transfer limitation may not be negligible, which will be investigated in the next section. 3.3. Model Application. The model has been used to predict most key process data within the single particle of SAPO-34 during the MTO process as a function of the particle diameter, rate constant, intraparticle diffusivity, and thermal conductivity coefficient. The simulated results are shown in Figures 6−21. 3.3.1. The Effect of Particle Diameter. Figures 6−8 compare the intraparticle parameter distributions along the radial direction at steady state for cases with different particle

reaction of C5H10 (r4) due to the higher concentration of CH3OH near the surface as described above. In this case, a low concentration change for C5H10 can be observed near the surface. However, with the quick decrease of the CH3OH concentration from the surface to the center, the cracking reaction rate is obviously larger than that of the formation of C5H10, which leads to the decrease of the concentration of C5H10. Therefore, the similar change trend of CH3OH is observed for C5H10, as described in Figure 5. However, the maximum concentration difference for C5H10 inside the particle is very small. In summary, the species mass fraction distributions along the radial direction (see Figure 5) are due to the quick reaction rate that is higher than the diffusion rate, which implies that the MTO process inside the particle is diffusion-controlled. The same trend happens for the temperature and pressure parameters. It is also observed that the intraparticle temperature gradient is very small. The maximum temperature difference inside the particle is less than 0.03% of its outer surface 3698



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means that the intraparticle mass transfer resistance can be neglected. Furthermore, Figure 6 shows that these fractions in the particle outer surface are nearly the same at different particle sizes. It means the external particle mass transfer resistance is nearly constant at different particle sizes. Accordingly, the intraparticle MTO reactions are mainly controlled by the intraparticle diffusion as described above. As indicated by eq 5, the species mass fractions are mainly determined by the mass diffusion flux (ji,r) and the mass source term (S̅i). Furthermore, ji,r and Si̅ have positive and negative effects on the CH3OH fraction, respectively. Therefore, when applying these catalyst particles with large diameter, an obvious intraparticle mass transfer resistance exists, which leads to a low mass fraction in the center of these particles and a significant intraparticle fraction gradient. As to the external mass transfer resistance, it can be related to eqs 18 and 20. The increase of the particle size can lead to the increase of Re, which gives a higher kg; however, eq 20 shows that kg decreases with the increase of the particle size. Therefore, kg keeps nearly constant at different particle sizes. In other words, the particle size does not change the external mass transfer resistance, although it can lead to the change of the intraparticle mass transfer resistance. Furthermore, as described in Figure 6, for the intraparticle transfer resistance, the particle size has different influences on different species. For instance, the species mass fraction ratio (i.e., Yi/Yi,o) decreases for CH3OH and C5H10, while it increases for the other species with the increase of the particle size. The degree of influence of the particle size is the most obvious for CH3OH and the smallest for C5H10, which indicates that the MTO selectivity may be changed by the catalyst particle size. These results prove that the MTO process inside the particle is diffusion-controlled, and with the increase of the particle size, the diffusion limitation increases, as shown in Figure 5. In addition, it implies that the influence of the particle size is also related to other factors such as the effective diffusivity coefficient of the species within the particle. Figure 7 shows that the intraparticle temperature gradient increases with the increase of particle size, which can be explained by eq 8. The intraparticle temperature is mainly determined by the heat diffusion flux (Qr) and the heat source term (Ŝ). For instance, lower Qr and higher Ŝ both have positive effects on the intraparticle temperature. A larger particle size makes the heat diffusion resistance increase, which leads to a larger intraparticle temperature gradient and a higher temperature in the center of these particles. In addition, Figure 7 shows that the temperatures of the outer surfaces of these particles increase with the increase of the particle size, which means that the external heat transfer resistance increases with the increase of particle size. The nearly constant kg mentioned above and the increase of particle size can lead to a lower hg (see eq 23). In turn, it leads to the increase of the outer surface temperature of these particles according to eq 19. Figure 8 gives the intraparticle pressure distribution profiles along the radial direction at different particle sizes. It can be observed that the pressure distribution profiles are similar to those profiles in Figures 6 and 7. In practice, at steady state condition, calculating the partial derivative of ideal gas law (eq 2) leads to eq 42.

Figure 7. The effects of the particle diameter on the intraparticle temperature (simulation conditions: d0 = 5.35 × 10−9 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

Figure 8. The effects of the particle diameter on the intraparticle pressure (simulation conditions: d0 = 5.35 × 10−9 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

Figure 9. The effects of the particle diameter on the effectiveness factors for different reaction steps (simulation conditions: d0 = 5.35 × 10−9 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

diameters. Figure 9 illustrates the effect of the particle diameter on the values of the effectiveness factor for reaction steps 1, 6, 7, and 8 shown in Figure 2. From Figure 6, the mass fraction gradients of all species increase with the increase of particle size. The fraction gradients are very small and can be neglected at dp < 0.1 mm, which

∂P 2 P ∂T P ∂M = P+ − ∂r r T ∂r M ∂r

(42)

Since ∂P/∂r is a linear function of both ∂T/∂r and ∂M/∂r, the changes of these quantities should have a similar trend. 3699



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Figure 10. The effects of the operated temperature on the intraparticle species mass fractions (a, CH3OH; b, CH4; c, C2H4; d, C3H6; e, C4H8; f, RH) (simulation conditions: dp = 5 × 10−4 m, d0 = 5.35 × 10−9 m, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

Figure 11. The effects of the operating temperature on the intraparticle temperature (simulation conditions: dp = 5 × 10−4 m, d0 = 5.35 × 10−9 m, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

Figure 12. The effects of the operating temperature on the intraparticle pressure (simulation conditions: dp = 5 × 10−4 m, d0 = 5.35 × 10−9 m, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427). 3700



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Figure 13. The effects of the operated temperature on the effectiveness factors for different reaction steps (simulation conditions: dp = 5 × 10−4 m, d0 = 5.35 × 10−9 m, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

Figure 15. The effects of the mean pore diameter on the intraparticle temperature (simulation conditions: dp = 5 × 10−4 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

Figure 14. The effects of the mean pore diameter on the intraparticle species mass fractions (a, CH3OH; b, CH4; c, C2H4; d, C3H6; e, C4H8; f, RH) (simulation conditions: dp = 5 × 10−4 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427). 3701



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obvious diffusion effect, which is identical to the above results and analyses. 3.3.2. The Effect of Operating Temperature. Figures 10−12 illustrate the intraparticle parameter distributions along the radial direction at steady state under intrinsic rate constants. Figure 13 shows its effect on the effectiveness factor. A higher operating temperature leads to a more significant mass fraction gradient for all species (see Figure 10), which indicates that the intraparticle diffusion resistance increases with the increase of the rate constant. It is due to the increase of the average Si̅ with the increase of the operating temperature according to eqs 7, 19, 28−38, which leads to a lower methanol mass fraction at the particle center and more obvious fraction gradients. Furthermore, from Figure 10, the operating temperature has different influences on different species due to various kinetic parameters for different species. The degree of influence is the most obvious for CH3OH, the second for C2H4, and the least for C5H10. Meanwhile, Figure 10 also shows that all species fraction ratios approach 1 with the decrease of the temperature, which implies the shift from a diffusion-controlled process to a kinetics-controlled process within the particle. According to Figure 11, one knows that the intraparticle temperature gradient and the temperature at the particle center exhibit an upward tendency with the increase of the operating temperature due to the increase of the average Ŝ according to eqs 10 and 30. As shown in Figure 12, the intraparticle pressure gradient increases with the increase of the rate constant. Figure 13 shows that the effectiveness factor of step 1 decreases with the increase of the rate constant and those of reaction steps 6, 7, and 8 increase with the increase of the operating temperature. In summary, the diffusion effect becomes obvious with the increase of the operating temperature. The same analysis approach applied in section 3.3.1 can be used to explain all of these results. 3.3.3. The Effect of Intraparticle Average Pore Diameter. Since the mean intraparticle pore diameter can influence the intraparticle transfer resistance through affecting the diffusion coefficient (see eqs 11 and 15), its effect is also simulated and these simulation results are shown in Figures 14−17. Figure 14 illustrates that the species mass fraction gradients decrease and the CH3OH mass fraction in the center of these particles increases with the increase of average pore diameter, which means that the intraparticle diffusion effect decreases with the increase of average pore diameter. It is due to the increase of the average ji,r with the increase of the diffusion coefficient (eq 6), which is caused by the increase of average pore diameter. In addition, from Figure 15, one knows that the intraparticle temperature gradient increases with the increase of average pore diameter. Similarly, it is due to the increase of the average heat source term according to eqs 10, 19, 28, and 29. Figure 16 shows that the intraparticle pressure gradient decreases with the increase of average pore diameter. The diffusion-controlling feature of the MTO process can be further confirmed by Figure 17, which shows that the effectiveness factors of steps 1 and 8 decrease with the increase of average pore diameter but the changes of effectiveness factors for reaction steps 6 and 7 are in the opposite direction. 3.3.4. The Effect of Thermal Conductivity. Figure 18 shows that the intraparticle temperature gradient decreases with the increase of the catalyst thermal conductivity due to the increase of the heat conduction term caused by a higher thermal conductivity. Figure 18 also shows the particle outer surface temperature decrease with the increase of the thermal

Figure 16. The effects of the mean pore diameter on the intraparticle pressure (simulation conditions: dp = 5 × 10−4 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

Figure 17. The effects of the mean pore diameter on the effectiveness factors for different reaction steps (simulation conditions: dp = 5 × 10−4 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

Figure 18. The effects of the thermal conductivity coefficient on the intraparticle temperature (simulation conditions: dp = 5 × 10−4 m, d0 = 5.35 × 10−9 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

The intraparticle resistance is determined by the gradient of mass fraction, because of the very small gradients of the intraparticle temperature. The effectiveness factors for reaction steps 1−5 are almost identical, and we only show the effectiveness factors of reaction steps 1, 6, 7, and 8 in Figure 9. Figure 9 also implies that a larger particle can lead to a smaller effective factor, namely, a larger particle causes a more 3702



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Figure 19. The effects of the thermal conductivity coefficient on the intraparticle species mass fractions (a, CH3OH; b, CH4; c, C2H4; d, C3H6; e, C4H8; f, RH) (simulation conditions: dp = 5 × 10−4 m, d0 = 5.35 × 10−9 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

conductivity. Figure 19 shows that the species mass fraction gradients keep nearly constant under different thermal conductivities. Figure 20 shows that the intraparticle pressure gradient decreases with the increase of the thermal conductivity. Furthermore, Figure 21 illustrates that the effectiveness factor decreases with the increase of the thermal conductivity until the thermal conductivity increases to a certain value. The reason is that a higher temperature and nearly constant intraparticle species fraction with the decrease of the thermal conductivity during a certain range can lead to a higher mean reaction rate inside the particle. Therefore, the effectiveness factor increases with the decrease of the thermal conductivity. When the thermal conductivity increases to a certain value, the intraparticle temperature is controlled by Ŝ.

equations as well as the multicomponent diffusion equation, a comprehensive single particle model was developed to describe the intraparticle transfer phenomena of SAPO-34 during MTO reactions. The single particle model, after validation, was then used to investigate the influences of some key factors on the intraparticle transfer limitations. The following conclusions can be drawn on the basis of the simulation results: (1) The unsteady state period within particles was very short, and it could be ignored for an industrial process. Furthermore, the intraparticle diffusion mainly affected the intraparticle mass fraction of species for the MTO reactions. (2) The intraparticle diffusion was not conducive to the cracking reaction of CH3OH (i.e., steps 1−5) and could benefit the cracking reaction of C3H6 and C4H8 (i.e., steps 6 and 7). For the cracking of C5H10, the influence could change under different conditions. (3) The intraparticle mass and heat transfer influences increased with the increase of particle diameter or operating temperature and decreased with the increase of the average

4. CONCLUSIONS In this work, based on the mass, energy, and momentum balances, together with gas-state, lumped-species, and kinetic 3703



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APPENDIX A. MODEL EQUATIONS WITH DIMENSIONLESS FORMAT

Total mass balance in a particle: s

ε

v ∂ρ* ∂ *2 (r ρ*) = 0 + r2 t0∂t * r0r * ∂r *

(A1)

where ρ* = ρ /ρ0 =

vrs = −

Figure 20. The effects of the thermal conductivity coefficient on the intraparticle pressure (simulation conditions: dp = 5 × 10−4 m, d0 = 5.35 × 10−9 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

PM P0M 0 P*M * / = RT RT0 T*

BP0 ∂P* μr0 ∂r *

(A2)

(A3)

s

εd 0 2 32τ

B=

(A4)

Component material balance in a particle: s

ε

v ∂ *2 ∂ (ρ*Y i*) + r 2 (r ρ*Y i*) * t0∂t * r0r ∂r * =−

Si̅ ∂ *2 1 (r ji , r ) + 2 * ∂ * r Y r0Yi ,0ρ0 r i ,0ρ0

ji , r = −

Yi ,0

ρi Di ,eff

r0

∂Y i* ∂r *

(A5)

(A6)

Heat balance in a particle: n

Figure 21. The effects of the thermal conductivity coefficient on the effectiveness factors for different reaction steps (simulation conditions: dp = 5 × 10−4 m, d0 = 5.35 × 10−9 m, T0 = 723.15 K, P0 = 101.325 kPa, YCH3OH,0 = 0.1877, YCH4,0 = 0.0081, YC2H4,0 = 0.1541, YC3H6,0 = 0.1211, YC4H8,0 = 0.0243, YRH,0 = 0.0620, YH2O,0 = 0.4427).

((1 − ε)Cpcat ρcat + ερ0 ρ*(∑ YCp i i )) i=1 n s = −ρ0 ρ* ∑ (YCp i i )vr i=1

+

pore diameter. However, for the thermal conductivity coefficient, the transfer influence for reaction steps 1−5 and 8 increased with the increase of thermal conductivity coefficient, and the transfer influence for reaction Steps 6 and 7 decreased with the increase of thermal conductivity coefficient. The simulation results also confirmed that the intraparticle transfer influence could not be ignored for fast or strong exothermic reactions with large catalytic particles. Therefore, it is necessary to consider the effect of intraparticle diffusion limitation when using FBR. In future work, we will extend the single particle model by coupling it with an FBR-scale CFD model. Some key bulk flow parameters in the reactor can be first predicted using the FBR model. Next, these parameters are handled as the boundary conditions to solve the single particle model for calculating the reaction rate. Finally, the reaction rate will be transferred back to the FBR model for the next iteration. With the above approach, we are able to characterize the behavior of an FBR more accurately.

∂T * t0∂t *

∂ *2 ∂T * 1 − (r Q r ) 2 * ∂ r0∂r * r* T0r0r

Ŝ T0

(A7)

Equations 12, 13, and 15, which are related to Di,eff, can be converted into the following equations: Dim =

1 − Yi0Y i* M 0M * ∑nj = 1 j≠i

Dij =

Yj0Y *j MjDij

(A8)

1.43 × 10−7 × (T0T *)1.75 2 P0P*Mij1/2[(∑v )1/3 + (∑v )1/3 i j ]

Di , k = 97

do 2

T0T * Mi

(A9)

(A10)

Boundary conditions At r = 0:

3704

∂Y i* =0 ∂r *

(A11)

∂T * =0 ∂r *

(A12) dx.doi.org/10.1021/ie302736b | Ind. Eng. Chem. Res. 2013, 52, 3693−3707


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At r = (dp/2): ki , g(Yis * − 1) = −Yi s *Dim

λ

■

∂T * r0∂r *

Yi0∂Y i* r0∂r *

− vrsYi s * s

− ρ0 ρ*Cps T s *vr = hi , g (1 − T s *) s

(A13)

(A14)

ASSOCIATED CONTENT

* Supporting Information S

The tables listed in the text are given as Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +86-21-54745602. Fax: +86-21-54745602. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the National Natural Science Foundation of China (No. 21076171, 21276213), the National Ministry of Science and Technology of China (No. 2012CB21500402), and the State-Key Laboratory of Chemical Engineering of Tsinghua University (No. SKL-ChE-10A03) for supporting this work.

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NOMENCLATURE Ai = kinetic parameter of the ith step, kg·h−1·kg(catalyst)−1 B = permeability factor Cpcat = mass heat capacity of catalyst, kJ·kg−1·K−1 Cpi = mass heat capacity of i component, kJ·kg−1·K−1 d0 = catalyst average pore diameter, m dp = catalyst particle diameter, m Di,eff = effective diffusion coefficient of the ith component, m2·s−1 Dim = Fick diffusion coefficient of the ith component, m2·s−1 Dik = Knudsen diffusion coefficient of the ith component, m2·s−1 Ei = kinetic parameter of the ith step, K f = fanning coefficient hi,g = heat transfer coefficient, J·m−2·s−1·K−1 Hi = enthalpy of the ith component, kJ·kg−1 ji,r = mass diffusion flux, kg·m−2·s−1 ki,g = mass transfer coefficient, m/s Ki = intrinsic rate constant of the ith step based on species mass fraction, kg·h−1·kg(catalyst)−1 Kp = the sum of the intrinsic rate constants for steps 1−5 based on species concentration, s−1 M = mixture fluid molar mass, kg·kmol−1 Mi = molar mass of the ith component, kg·kmol−1 M0 = mixture fluid molar mass at bulk, kg·kmol−1 M* = dimensionless mixture fluid molar mass P = pressure, kPa P0 = pressure at bulk phase, kPa P* = dimensionless pressure Pr = Prandtl number Qr = heat flux, J·m−2·s−1 r* = dimensionless catalyst particle radius

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r0 = catalyst particle radius, m ri = reaction rate of the ith step, kg·h−1·kg(catalyst)−1 Rei = Reynolds number of the ith component R = ideal gas constant, kJ·kmol−1·K−1 Rj = consumption or production rate of the ith component, kg·h−1·kg(catalyst)−1 Shi = Sherwood number of the ith component Sc = Schmidt number Si̅ = mass source of the ith component, kg·m−3·s−1 Ŝ = heat source, J·m3·s−1 T = temperature, K T* = dimensionless temperature T0 = temperature at bulk phase, K Ts = temperature at particle outer surface, K Ts* = dimensionless temperature at particle outer surface u = apparent gas velocity, m·s−1 vsr = gas velocity at particle outer surface, m/s Yi = mass fraction of the ith component Yi* = dimensionless mass fraction of the ith component Ysi = mass fraction of the ith component at the particle outer surface Ys* i = dimensionless mass fraction of the ith component Yi,0 = mass fraction of the ith component at bulk ε = catalyst porosity ρ = mixture fluid density, kg/m3 ρ0 = mixture fluid density at bulk, kg/m3 ρ* = dimensionless mixture fluid density ρcat = real catalyst density, kg/m3 μ = mixture fluid viscosity, Pa·s ρi = the ith component density, kg·m−3 (∑v)i = diffusion volume of the ith component, cm3·mol−1 λ = thermal conductivity coefficient, J·m−1·s−1·K−1 η = effectiveness factor of the ith step ϕ = Schiller modulus τ = curvature factor Nui = Nusselt number of the ith component

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Intraparticle Mass and Heat Transfer Modeling of Methanol to Olefins

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