Mathematical modeling of a slurry reactor for DME direct synthesis from syngas

Journal of Natural Gas Chemistry 21(2012)148–157

Mathematical modeling of a slurry reactor for DME direct synthesis from syngas Sadegh Papari,

Mohammad Kazemeini∗ ,

Moslem Fattahi

Department of Chemical and Petroleum Engineering, Sharif University of Technology, Azadi Avenue, P. O. Box 11365-9465, Tehran, Iran [ Manuscript received July 25, 2011; revised August 23, 2011 ]

Abstract In this paper, an axial dispersion mathematical model is developed to simulate a three-phase slurry bubble column reactor for direct synthesis of dimethyl ether (DME) from syngas. This large-scale reactor is modeled using mass and energy balances, catalyst sedimentation and single-bubble as well as two-bubbles class flow hydrodynamics. A comparison between the two hydrodynamic models through pilot plant experimental data from the literature shows that heterogeneous two-bubbles flow model is in better agreement with the experimental data than homogeneous single-bubble gas flow model. Also, by investigating the heterogeneous gas flow and axial dispersion model for small bubbles as well as the large bubbles and slurry (i.e. including paraffins and the catalyst) phase, the temperature profile along the reactor is obtained. A comparison between isothermal and non-isothermal reactors reveals no obvious performance difference between them. The optimum values of reactor diameter and height were obtained at 7 m and 50 m, respectively. The effects of operating variables on the axial catalyst distribution, DME productivity and CO conversion are also investigated in this research. Key words modeling; large-scale slurry bubble column; optimization; dimethyl ether synthesis; single-bubble class; two-bubbles class; isotherm and non-isotherm

Water gas-shift

1. Introduction As the simplest ether, DME may be easily liquefied and becomes a good substitute for fossil fuels which are harmful to environment, because DME doesn’t cause pollution such as solid particulates or toxic gases. DME is utilized as LPG substitute, transportation fuel, propellant, chemical feedstock and fuel cells feed [1]. There are usually two common techniques used to synthesize DME. In the first method, methanol is converted into DME which is called indirect DME synthesis or double-stage process. In the other method, it is synthesized from direct syngas conversion through a bifunctional catalyst which is called single-stage method. The main advantage of the single over double stage process is the lower cost of it. The reactions of a single-stage process for producing DME might be divided into the following steps: Methanol synthesis CO + 2H2 ←→ CH3 OH

ΔH = −90.85 kJ/mol

(1)

Methanol dehydration 2CH3 OH ←→ CH3 OCH3 + H2 O ΔH = −23.4 kJ/mol (2) ∗

CO + H2 O ←→ CO2 + H2

ΔH = −41.1 kJ/mol

(3)

Carbon dioxide hydrogenation CO2 + 3H2 ←→ CH3 OH + H2O

ΔH = −50.1 kJ/mol (4) Syngas to DME conversion is better conducted in a slurry bubble column reactor due to several reasons, including i) simple reactor construction, ii) uniform reactor temperature during reaction for a highly exothermic reaction, iii) convenient adding and removing of the catalyst convenient and iv) easily controlled reactor temperature to avoid sintering of the catalyst. In recent decades, numerous studies concerning flow regime and computational fluid dynamics (CFD) investigations [2−6], gas hold-up and bubble characteristics [7−10], volumetric mass transfer coefficient [11,12] and heat transfer coefficient measurements [13,14] have been performed in bubble column reactors. Up to now, no commercial scale syngas to dimethyl ether plant has been developed (i.e. constructed and undergone production) [15], therefore, it might be very important to utilize

Corresponding author. Tel: +98-21-66165425; Fax: +98-21-66022853; E-mail: [email protected]

Copyright©2012, Dalian Institute of Chemical Physics, Chinese Academy of Sciences. All rights reserved. doi:10.1016/S1003-9953(11)60347-2

Journal of Natural Gas Chemistry Vol. 21 No. 2 2012

the bubble column slurry reactors as a design tool to achieve such a goal. Recently, a few researchers modeled commercial slurry bubble column for direct synthesis of DME from syngas. Liu et al. [15] simulated a 100000 t/y DME reactor using single-bubble class hydrodynamic under isothermal conditions. They investigated the effects of temperature and pressure on CO conversion and DME productivity and found that with the increase of operating parameters of both temperature and pressure, the bubble column performance is improved. Chen et al. [16] modelled a large-scale DME reactor using single-bubble class hydrodynamic in which the slurry phase was considered to be perfectly mixed while the reactor assumed isothermal. They studied the effects of temperature, pressure and reactor diameter on DME selectivity and CO conversion and found that DME selectivity varied with operating parameters and reactor dimensions rather slowly. Zhang et al. [17] simulated and optimized the DME direct synthesis by considering plug flow for gas phase and perfectly mixed model for slurry phase, single-bubble class hydrodynamic and an isothermal column. They found that CO conversion was an ascending functions of temperature in the range of 500−550 K, but started to decline after 550 K. In addition, volumetric mass flow rate was usually considered constant along the column in the literature. In the present paper, a mathematical model was proposed using mass conservation, heat balance, catalyst particle sedimentation, global kinetic model, two-bubble class hydrodynamic in addition to the variable volumetric flow rate along the bubble column. Then, the simulation results were compared with those of a single-bubble class model and pilot plant experimental data available in the open literature. The effects of reactor dimension, superficial gas velocity, catalyst concentration, temperature and pressure were investigated upon the axial dispersion of the catalyst, CO conversion and DME productivity. Furthermore, the temperature profile along the column was determined while the isothermal and non-isothermal

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reactors were compared. Ultimately, the optimum dimensions of the reactor were calculated. 2. Hydrodynamics In the bubble column reactors, the choice of flow regime is very important due to the fact that for thorough modeling of a reactor, hydrodynamic parameters including gas holdup, volumetric mass transfer coefficient, superficial gas and slurry phase velocities, axial dispersion of liquid, solid-gas heat transfer coefficient are all required. The flow regime in slurry bubble column reactors depends upon the superficial gas velocity. In other words, when superficial gas velocities is lower than 0.05 m/s, the flow regime is called homogeneous regime through which there are single or small bubbles with diameter of 1−7 mm in the column. By increasing this velocity, the small bubbles might be unified and form larger ones called heterogeneous (i.e. churn-turbulent) regime. In this regime, two kinds of bubbles exists, including small and large ones with diameter of 20−70 mm [18−20]. The variation of gas velocity from bottom to top of the reactor is attributed to gas consumption which can be estimated by defining gas volume contraction factor and overall syngas conversion calculated through the following equation: UG = UG0 (1 + αc XCO+H2 )

(5)

where, αc is the gas volume contraction factor. Although αc should be calculated from experimental data, it may also be obtained from reference values published by Levenspiel and co-workers [21]. For the present study, an estimated constant value for αc = −0.525 was assumed according to the aforementioned reference which is a common value. A schematic representation of a slurry bubble column reactor which is divided into three separate phases (i.e. the large, small and slurry) is shown in Figure 1.

Figure 1. Hydrodynamics schematic of the slurry bubble column reactor

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3. Mathematical modelling Large bubbles are presented by the following equation:   ∂Cj,LB ∂ ∂(ULB Cj,LB ) εLB ELB − − ∂z ∂z ∂z

(6)

∂Cj,LB kl aLB (Cj − Cj,L ) = εLB ∂t

(7)

∂Cj,SB kl aSB (Cj∗ − Cj,L ) = εSB ∂t and the slurry phase is determined through:   ∂ ∂Cj,SL ∂(USL Cj,SL ) (1 − εG )ESL − + ∂z ∂z ∂z ∗

kl aLB (Cj − Cj,SL ) + kl aSB (Cj − Cj,SL )+ Nr

(1 − εG) ∑ Mcat. νj,i ri = (1 − εG ) i=1

ρSL USL T − (1 − εG)ρSL ESL

(8)

in addition, the particle mass balance is given by:

(9)

∂ ∂CS {[(1 − εG)UP − USL ] CS } = (1 − εG) ∂z ∂t while the energy balance is provided through:   ∂T ∂(ρSL CP,SL USL T ) ∂ (1 − εG)ρSL CP,SL ESL − + ∂z ∂z ∂z Nr

UHeat (T − Tcool) + (1 − εG) ∑ ΔHDME,i rDME,i i=1

= (1 − εG)

∂T ∂t

(10) where, j = H2 , CO, CO2 , H2 O, CH3 OH, CH3 OCH3 . The boundary conditions at the inlet of the slurry bubble column reactor are Danckwertz’s types, i.e: ULB Cj,LB − εLBELB

∂Cj,LB = ULB Cj,0 ∂z

(11)

USB Cj,SB − εSB ESB

∂Cj,SB = USB Cj,0 ∂z

(12)

∂Cj,SL =0 USL Cj,SL − (1 − εG)ESL ∂z

∂T = (ρSL USL T )0 ∂z

∂Cj,LB =0 ∂z ∂Cj,SB =0 ∂z ∂Cj,SL =0 ∂z ∂CS =0 ∂z ∂T =0 ∂z and the initial conditions are taken to be:

∂Cj,SL ∂t

  ∂CS ∂ (1 − εG)ES − ∂z ∂z

(14)

(15)

The boundary conditions at the outlet of the slurry bubble column reactor include:

while the small bubbles are presented by:

∗

∂CS + [(1 − εG)UP − USL ]CS + ∂z USL Cave = 0

∗

  ∂ ∂Cj,SB ∂(USB Cj,SB ) εSB ESB − − ∂z ∂z ∂z

(1 − εG)ES

(13)

(16) (17) (18) (19) (20)

Cj,LB = Cj,0

(21)

Cj,SB = Cj,0

(22)

Cj,L = Cj∗

(23)

CS = Cave

(24)

T = Tcool

(25)

The empirical correlations utilized for the gas hold-up, volumetric mass transfer coefficient, superficial gas velocity of small bubbles, hindered sedimentation velocity of particles, dispersion coefficient of small and large bubbles, as well as for liquid for predictions of DME production and CO conversion in large-scale slurry bubble column reactor are summarized in Table 1. The inlet feed composition were set to yH2 = 0.65, yCO = 0.32 and yCO2 = 0.03. Temperature of the cooling pipes was taken to be 503 K. Other slurry bubble column reactor parameters are provided in Table 2. 4. Kinetics In the present study, the chemical kinetics of methanol synthesis, water gas shift and DME synthesis as independent reactions were all taken from the Guo et al. [32]. Commercial Cu-based methanol synthesis catalyst C301 and dehydration catalyst gamma-alumina were used [32]. In other words, reaction rate of methanol synthesis employed in the current model is given by: r1 =

1.954 P 0.9174 k1 PCO H2 0.1795 2.26 1.501 + K (1 + KCOPCO CO2 PCO2 )

(26)

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Table 1. Gas hold-up, volumetric mass transfer coefficient, superficial gas velocity, hindered sedimentation velocity, dispersion coefficient and overall heat transfer coefficient Description

Equations

0.48

ρG 0.7εs ref 1 − ref εSB = εSB ρref ε G df 0.5 0.8 ρG 1 (Ug − USB ) εLB = 0.3 0.18 DR (Ug − USB )0.22 ρref G

References

Total gas hold-up

εG = εLB + εSB (1 − εLB )

[22]

Volumetric mass transfer coefficient of large bubbles

3 ref ρref G = 1.3 kg/m and εdf = 0.27   Dj 0.5 (kl a)LB,j = 0.5εLB Dref   Dj 0.5 (kl a)SB,j = 1.0εSB Dref

[23]



Small bubble gas hold-up Large bubble gas hold-up

Volumetric mass transfer coefficient of small bubbles

USB

Superficial gas velocity of small bubbles

[22] [22]

[23]

Dref = 2 × 10−9 m2 /s

0.8εs ref ref 1 + ref , VSB = VSB = 0.095 m/s VSB

[23]

ULB = UG − USB

0.8   gc Dp2 (ρG − ρl ) Cs,ave 3.5 1− Up = 1.1U G 0.026 18μl ρp

Superficial gas velocity of large bubbles Hindered sedimentation velocity of particles (m/s) Dispersion coefficient of liquid (m2 /s) Dispersion coefficient of small bubbles (m2 /s) Dispersion coefficient of large bubbles (m2 /s)

ELB

E l = 0.35DR1.33 (gUG )0.33

[25]

ESB = E l   UG 3.56 1.33 = 5.64 × 10−3 DR εLB

[26] [27]

UHeat = 11710UG0.445 (μSL × 103 )−0.060 (P × 10−6 )0.176

Overall heat transfer coefficient

[24]

[28]

Table 2. Slurry phase properties Catalyst properties Density (kg/m3 )

Liquid phase properties [29] 171 × 0.1617−(

1987

Viscosity (Pa·s)

−

Heat capacity (J/(kg·K))

993

0.9402 k2 PM 1 .739 (1 + KM PM + Kw Pw2.243 )0.4415

(27)

Analysis of online gas chromatography showed that water content in exit stream was very low [33,34]. Thus, it was a foreseen conclusion that water gas shift reaction was at equilibrium. Hence, the H2 O partial pressure can be obtained from the following equation: Pw =

PCO2 PH2 Pco KP,WGS

)

2 7

103 ln(μL ) = −3.0912 + 1.7038 × T 2927

While the reaction rate of methanol dehydration is taken to be: r2 =

T 1− 916.18

(28)

  E0 where, ki = k0 exp −RT , k0 (Arrhenius constant) and E0 (activation energy) are all listed in Table 3.

Slurry phase properties [30,31] ρSL = εs ρs + (1 − εs )ρL μSL = μL (1 + 4.5εs ) Cp,SL = ws Cp,S + (1 − ws )Cp,L

Table 3. Arrhenius constant and activation energy of the global kinetic model Parameters k1 k2 KCO KCO2 Kw KM

k0 (mmol/(gcat ·h)) 2.034×108 7680 66.63 139507 1.225×10−2 2.3362×10−3

E0 (J/mol) 14061 23533 −1588.2 17551 −45601 −39572

5. Gas solubility The solubility of syngas, CO2 , methanol, water and dimethyl ether in the liquid phase are obtained using Henry’s law [35,36]:

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b (29) H = a × exp RT Values for constants a and b for each component are listed in Table 4. H j is used for prediction of P∗j value as follow: Pj∗ = Hj × Cj∗

with ARD of 6.8% and 4.5%, respectively. Thus, the twobubble class model was utilized in this paper to investigate the effect of operating parameters.

(30)

Table 4. Parameters for Henry’s constant Components Hydrogen Carbon monoxide Carbon dioxide Methanol Water Dimethyl ether

a (bar·L/mol) 78.192 109.87 429.52 8190.6 7936.7 2765.4

b (J/mol) 2854 939.7 −8969.7 −30410 −37421 −17377.1

6. Results and discussion In general, the semi-batch and continuous modes are two types of valid operation for slurry reactors. In this study, the semi-batch mode was simulated. All partial non-linear equations were solved through the partial substitution finite difference method using MATLAB software 2010a. The followed flow chart in this research for the purpose at hand is shown in Figure 2. The value of “w” was adjusted to achieve stable convergence which in this program was chosen to be 0.5. Figure 3 illustrated the parity plot of CO conversion and DME STY (i.e. DME production rate per catalyst weight) for comparison between the two models of hydrodynamics (i.e. two-bubble class and single-bubble class) with experimental data [37]. The catalyst with the size of 10-microns order was utilized, which is a mixed catalyst containing CuO, ZnO and Al2 O3 [37]. It may be seen from the figure that a good agreement between the two-bubble class models and experimental data is observed and the average relative deviations (ARD) of this model were lower than the other model. The twobubble class model predicted CO conversion and DME STY

Figure 2. The flow chart of the simulation in this research

Figure 3. Validation of single and two bubbles axial dispersion models with pilot plant experimental data: T = 260 ◦ C, P = 5.0 MPa, εs = 0.3, L = 22 m, DR = 2.3 m

Journal of Natural Gas Chemistry Vol. 21 No. 2 2012

Figure 4 depicted the effect of superficial gas velocity on the axial catalyst concentration distribution. It is clear from this figure that the enhancement of the superficial gas velocity led to lowering of the slope of the catalyst concentration versus the reactor height, which can be attributed to the increase of the slurry recirculation. Consequently, by rising up the recirculation slurry phase, the behavior considered tended to-

153

wards a perfectly mixed model. In addition, Figure 4 demonstrated the effect of the aspect (i.e. height to diameter) ratio on the axial catalyst concentration distribution. The results revealed that the increase of the reactor diameter or decrease of the reactor height resulted in more slurry recirculation and less sedimentation of catalyst. Hence, the slurry phase in the bubble column might be considered as a perfectly mixed reactor.

Figure 4. Axial catalyst concentration profile vs. dimensionless height for different superficial gas velocity and height over diameter ratio

On the other hand, the dense phase temperature of the bubble column reactor was controlled by the tube heat exchangers. The temperature profiles along the reactor at different operating temperatures are illustrated in Figure 5. As indicated, the axial temperature profiles from bottom to top of the slurry reactor changed very little related to that of the paraffin liquid having a high heat capacity as well as desirable heat carrying characteristic. Therefore, direct DME synthesis by syngas and carbon dioxide in the slurry reactor might be considered as an isothermal reactor. Also, it may be concluded from this figure that the hot region for this system is situated at height of about 7.5−10 m above bottom of slurry reactor due to the catalyst grain sedimentation and high syngas partial pressure which led to the increase of methanol synthesis and methanol dehydrogenation rate thus causing

Figure 5. Axial slurry temperature profiles: ture = 503 K

cooling water tempera-

more heat generation. Furthermore, it demonstrated that the variation of temperature along the reactor at higher operating temperatures was smaller than at lower operating temperature. It might be a foregone conclusion that at fixed cooling water and high slurry temperatures the driving force for heat removal is higher than the state at which the reactor operated at lower temperatures. Therefore, the heat lost was higher and the slurry temperature became uniform. DME mole flow production and CO consumption in isothermal and nonisothermal slurry reactors are illustrated and compared with experimental results available in the open literature in Figure 6. It can be concluded that there is no obvious difference between the isothermal and non-isothermal reactors. The axial distribution of DME production per catalyst weight is shown in Figure 7. Initially, the DME production rate along the column height increased to achieve a maximum

Figure 6. A cooperation between non-isotherm and isotherm slurry reactors

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Figure 7. DME production rate per catalyst weight along the bubble column

indicated in this figure, CO conversion and DME productivity increased initially and achieved maxima at 270 ◦ C, then started to decline. This trend was confirmed by other data available in the open literature [17]. The ascending function of temperature for CO conversion and DME productivity in the range of 240−270 ◦ C might be interpreted as that the rising temperature improved the syngas solubility in paraffin liquid, hence its volumetric mass transfer coefficient enhanced [36] which accelerated methanol synthesis and its dehydration rates. Descending function of temperature after 270 ◦ C was rationalized due to the occurrence of methanol synthesis and dehydration at high exothermic reactions. Thus, 270 ◦ C was determined as the optimum temperature for single-stage DME synthesis. In addition, Figure 9 represented the effect of pressure on CO conversion and DME productivity. The effect of column height on pressure was neglected in the dispersion model because the operating pressure was considerably higher than the slurry height in the reactor. The increase of pressure from 4 to 6 MPa resulted in the increase of CO conversion from 62% to 86% and DME productivity from 3214 to 4257 t/d at constant temperature of 260 ◦ C. The water gas shift and methanol dehydration have the same number of total moles on both sides of the reaction. However, the methanol

The effects of height over diameter ratio (H/D) on CO conversion and DME productivity operating at constant superficial gas velocity U G = 0.2 m/s, catalyst concentration CS = 30 wt.% and pressure of 5 MPa are illustrated in Figure 8 at constant reactor volume. As shown in this figure, CO conversion and DME productivity increased with enhanced H/D ratio. When this ratio increased to 7, enhanced residence time caused a rise in CO conversion and DME productivity as well. However, any further increase in H/D ratio raised the catalyst grain sedimentation and did not have any significant effect on CO conversion or DME production. Thus, the H/D ratio equal to 7 was chosen as an optimum value for the DME production reactor. This reactor height and diameter were thus determined to be 50 m and 7 m, respectively. CO conversion and DME productivity versus average temperature at different pressures are shown in Figure 9. As

Figure 8. CO conversion and DME productivity vs. height over diameter ratio: T = 260 ◦ C, P = 5.0 MPa, εs = 0.3, U G = 0.2 m/s

value, then started to decline. Higher syngas partial pressure at the inlet of the slurry reactor resulted in higher methanol synthesis rate which led to an increase of methanol dehydration rate. Also, as mentioned above, the slurry temperature in this section of the reactor hit a maximum value which can be attributed to the high methanol synthesis and methanol dehydrogenation rate.

Figure 9. CO conversion and DME productivity vs. average slurry temperature for different pressures: DR = 7 m, L = 50 m, εs = 0.3, U G = 0.2 m/s

Journal of Natural Gas Chemistry Vol. 21 No. 2 2012

synthesis is a stoichiometrical contracting reaction, hence the increased pressure accelerated the rate of this reaction. Thus, it might be considered that the methanol synthesis is the limiting step of the overall reaction [33]. Moreover, the rising pressure led to a decrease in the bubble size as well as an increase in the mass transfer area which in turn caused an increase in the volumetric mass transfer coefficient. Hence, mass transfer between the two phases of slurry and syngas increased at higher pressures [36]. It should be noted, however, that operating at higher pressures results in high costs. Therefore, 50 bar might be an accepted value. CO conversion and DME productivity versus superficial gas velocity at different catalyst concentrations (mass fraction of catalyst in the gas free slurry) are illustrated in Figure 10. As clearly seen from this figure, the increased superficial gas velocity decreased CO conversion due to a lowered contact time (i.e. mean residence time) between the slurry and syngas phases, however, DME production demonstrated a different behavior. At the superficial velocity lower

155

than 0.2 m/s, DME productivity was an ascending function of velocity for all catalyst concentrations. Nonetheless, after 0.2 m/s, DME productivity at lower catalyst concentrations (lower than 40 wt.%) declined, and at higher catalyst concentrations (i.e. higher than 40 wt.%) it enhanced. Therefore, considering the decrease in CO conversion and DME productivity, the optimum value of superficial gas velocity was chosen to be 0.2 m/s. Furthermore, investigation of catalyst concentration on CO conversion and DME productivity in Figure 10 indicated the positive effect of catalyst concentration on the aforementioned conversion and production. However, according to the empirical correlation of volumetric mass transfer coefficient utilized, further increase of the catalyst concentration led to a decrease in the mass transfer coefficient in the slurry phase. Besides, rising in the heat generation of reactions made the reaction temperature control difficult under such conditions. In other words, the enhancement of the slurry concentration had its limiting value as well.

Figure 10. CO conversion and DME productivity vs. superficial gas velocity for different catalyst concentration: DR = 7 m, L = 50 m, T = 260 ◦ C, P = 5.0 MPa

7. Conclusions In this paper, a large-scale slurry bubble column reactor for direct synthesis of dimethyl ether from syngas and carbon dioxide is mathematically modelled. The results of this simulation indicate that for 3800 t/d at temperature of 260 ◦ C, pressure of 5 MPa and catalyst concentration of 30 wt.%, the optimum reactor diameter and height requirements are 7 m and 50 m; respectively. Also, the investigation of axial temperature profiles and a comparison between the isothermal and non-isothermal reactor behaviors reveal that the slurry bubble column reactor might be considered isothermal without incorporating many errors into the simulation. In addition, the results of axial catalyst dispersion indicate that with increasing superficial gas velocity and decreasing reactor aspect ratio, the slurry phase might be considered as a perfectly mixed reactor. CO conversion and DME productivity are demonstrated to rise with increasing temperature up to 270 ◦ C. After 270 ◦ C, due to highly exothermic nature of reactions involved, these values

are lowered. On the other hand, the rising pressure and catalyst concentration enhance the reactor performance. Nonetheless, they have their limits as well before hitting their respective inflection points. Ultimately, the optimum value for the superficial gas velocity is chosen to be 0.2 m/s. It is reminded that the developed model in this research is also applicable for design of other large-scale slurry bubble column reactors. Nomenclatures a Henry constant (bar·L/mol) b Henry Constant (J/mol) Cj,LB Molar concentration of j component in large bubble phase (mol/m3 ) Cj,SB Molar concentration of j component in small bubble phase (mol/m3 ) Cj,L Molar concentration of j component in liquid phase (mol/m3 ) CS Catalyst concentration or catalyst density (kg/m3 )

156 Cj∗ Cp,S Cp,L Cp,SL DR Dj ELB ESB ESL g (kl a)LB (kl a)SB k1 k2 KCO2 Kw KP,WGS L Mcat. PCO PH2 PCO2 PM Pw P r1 r2 R T ULB USB USL USL UG UG0 UP UHeat WS XCO+H2 w

Sadegh Papari et al./ Journal of Natural Gas Chemistry Vol. 21 No. 2 2012

Equilibrium molar concentration in liquid (mol/m3) Catalyst particle heat capacity (J/kg·K) Liquid paraffin heat capacity (J/kg·K) Slurry phase heat capacity (J/kg·K) Reactor diameter (m) Diffusion coefficient (m2 /s) Large bubble dispersion coefficient (m2 /s) Small bubble dispersion coefficient (m2 /s) Slurry phase dispersion coefficient (m2 /s) Acceleration gravity constant (m2 /s) Volumetric mass transfer coefficient for large bubble (1/s) Volumetric mass transfer coefficient for small bubble (1/s) Rate constant of methanol synthesis Rate constant of methanol dehydration Adsorption constant of CO2 on the methanol synthesis catalyst Adsorption constant of water on the methanol dehydration catalyst Equilibrium constant of water-gas shift reaction Reactor height (m) Mass of catalyst (kg) Partial pressure of CO (MPa) Partial pressure of H2 (MPa) Partial pressure of CO2 (MPa) Partial pressure of methanol (MPa) Partial pressure of water (MPa) Operating pressure (MPa) Methanol productivity (mmol/(gcat·h)) DME productivity (mmol/(gcat·h)) Gas constant (J/(mol·K)) Reaction temperature (K) Superficial velocity of large bubbles (m/s) Superficial velocity of small bubbles (m/s) Superficial velocity of slurry phase (m/s) Inlet superficial velocity of slurry phase (m/s) Superficial gas velocity (m/s) Inlet superficial gas velocity (m/s) Hindered sedimentation velocity (m/s) Overall heat transfer coefficient (W/(m2 ·K)) Weight percent of catalyst in paraffin Overall syngas conversion convergence criteria

Greek symbols αc Expansion factor εSB Small bubbles gas hold-up

εG εLB εs ρG ρSL vj,i μSL

Total gas hold-up Large bubbles gas hold-up Solid concentration Gas phase density (kg/m3 ) Slurry phase density (kg/m3 ) Reaction coefficient Slurry viscosity (Pa·s)

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