Author
 M. Bayat ^{}
Department of Chemical Engineering, Faculty of Engineering, University of Bojnord, Bojnord, Iran
Abstract
In the first part of two section paper, a mathematical model of the fluidized bed reactor in the presence of insitu water adsorbent for methanol synthesis is assessed. The bubbling twophase regime is applied to model the fluidization concept. The binary adsorbent and catalyst particles system can be separated from each other based on their density difference. The heavy catalyst particles tend to sink and the light adsorbent particles tend to rise. The inducement for in situ water vapor removal by using adsorbent particles (Zeolite 4A) is to displace the watergas shift equilibrium to improve methanol productivity. This is accomplished through the methanol synthesis process.
Simulation result indicate that selective adsorption of water in the fluidized bed configuration leads to 46.36% and 41.88% enhancement in methanol production and 26.143% and 25.63 % in selectivity, compared with the zero solid mass ratio condition and conventional reactor, respectively. This model is applied in a multiobjective optimization and decision making method to be presented in the upcoming study, Part II.
Keywords
Main Subjects
1. Introduction
Energy is a crucial constituent in human survival. Currently, a great volume of the energy is yield from nonrenewable fossil fuel. The synthesis gas can commercially be converted into methanol through CuO/ZnO/Al_{2}O_{3} catalyst. In general, a small improvement in main chemicals properties like in methanol would lead to significant energy conservation, environmental protection and thereupon profit increment (Schack et al., 1989). In order to improve the efficiency of industrial methanol synthesis reactor, several configurations are proposed (Rahimpour et al., 2010; Rahimpour and Elekaei, 2009; Rahimpour and Lotfinejad, 2008; Vakili et al., 2011).
In methanol production from synthesis gas, mainly three overall reactions are involved:
(1) 

(2) 

(3) 
In this article, the reaction expression rates is from Graaf et al. (1990).
In the methanol reactor, only certain perpass conversion of reactant is yield because of thermodynamic limitations. Hence, the reactant recycles loop and product separator are introduced to obtain a reasonable degree of reactant conversion. These methods are mostly expensive and cumbersome (Bayat et al., 2014b).
In methanol synthesis process, the sorptionenhanced process run through the fluidized bed reactor is found to be an appropriate method to overcome the thermodynamic limitation. Equilibrium can be shifted towards the formation of more products by selective adsorption of byproducts.
Johnsen et al. (2006) revealed that the contact of gassolid in a bubbling fluidized bed lab scale reactor is enough to reach near equilibrium conditions. Almost 50 years ago, the suggestion of contacting gas and flowing particles inside a fluidized bed configuration was presented (Everett and Retallick, 1963).
Bayat et al. (Bayat et al., 2014a; Bayat et al., 2014b, c; Bayat et al., 2014d; Bayat et al., 2013, 2014e; Dehghani et al., 2014) have run studies on methanol synthesis and FischerTropsch synthesis in the presence of water adsorbent particle. They have assessed the steady state packed bed reactor (Bayat et al., 2014a), membrane reactor (Bayat et al., 2014b), coupling reactor (Bayat et al., 2014e), unsteady state packed bed reactor (Dehghani et al., 2014) and dual bed reactor (Bayat et al., 2014c) for methanol synthesis process. Furthermore, Fischer–Tropsch synthesis process of gas flowing solid fixed bed configuration with in situ water adsorbent (Bayat et al., 2014d) and thermally coupled reactor (Bayat et al., 2013) are of concern. There exist a big volume of studies on sorption enhanced process (Bayat et al., 2014a; Bayat et al., 2014b, c; Bayat et al., 2014d; Bayat et al., 2013, 2014e; Dehghani et al., 2014) which prove the importance of how the utilization of zeolite 4A adsorbent is consumed in reversible reactions with water vapor formation, while, their results prove that the conversion of reactant is enhanced by removal of water from the reactor.
A mathematical model of fluidized bed reactor in the presence of regenerative water vapor adsorbent for methanol synthesis is presented in this article Part I. In Part II, uses this model is adopted to determine the optimal operating conditions for simultaneous maximization of methanol production and selectivity in SEFMR.
2. Process Description
2.1. Conventional Methanol Reactor (CMR)
Methanol synthesis from syngas is run in a vertical nonisothermal heat exchanger reactor, where the tubes are packed by commercial CuO/ZnO/Al_{2}O_{3} catalyst and surrounded by boiling water (Bayat et al., 2012).
2.2. Fluidized Bed Methanol Reactor (FMR)
The process in FMR configuration is similar to that of CMR with the exception of some changes: 1) in the inner tube side, fluidization is applied instead of the fixed catalyst bed through applying small catalyst size and 2) the feed synthesis gas enters the bottom of the reaction side to fluidize the catalyst bed. The parameters: tube diameter, length of reactor, feed operating condition are the same as that of the conventional reactor.
2.3. Sorption EnhancedFluidizedbed Methanol Synthesis Reactor (SEFMR)
This configuration is almost similar to that of the FMR. The difference between them is attributed to consumption of zeolite 4A as water adsorbent. The chemical reactions over the fluidized catalyst begin and convert the feed synthesis gas into methanol along the sorptionenhanced reaction. In the next step, the adsorbent solid particles regenerate and fresh solids that enter the inlet of SEFMR at every regeneration cycle. In this apparatus, the equilibrium is shifted toward the formation of more products by implementing the additional selective water adsorbent. In order to have a realistic assessment regarding industrial fixedbed reactors, the same operating conditions of an actual industrial reactor are set. The inlet conditions of feed are extracted from study run by (Bayat et al., 2012). A conceptual schematic of sorption enhanced fluidized bed reactor is shown in Fig. 1.
Figure. 1. A schematic Diagram of SEFMR.
3. Mathematical Modeling
The following assumptions are considered in developing a mathematical model of SEFMR:
(a) The equality between temperature of emulsion and bubble phase is assessed.
(b) Emulsion phase consist of adsorbent solids particle
(c) The constant average value of velocity for rising of bubbles is considered
(d) The PengRobinson EOS is assumed for PVT calculation
(e) The average size of spherical bubbles is assessed
(f) The temperature difference between catalyst phase and other phases is neglected
An element of length dz Fig. 2, is considered here. On the basis of the aforesaid assumptions, the equations of emulsion and bubble phase mass conservation are expressed as follows:
Figure. 2. Elemental Volume of SEFMR.
3.1. Emulsion Phase:


(4) 

where, K_{bei} is the mass transfer coefficient between bubble phase and emulsion phase, y_{i}^{e}and y_{i}^{b} are the emulsion phase and bubble phase mole fraction, respectively. Parameter is assumed to be zero for the methanol, carbon dioxide, carbon monoxide, hydrogen, nitrogen and methane components and one for water component.
3.2. Bubble Phase:

(5) 
where γ is the volume fraction of catalyst bed occupied by solid particles in bubble phase. The F_{i}^{b}and F_{i}^{e} are expressed as follows:
, 
(6) 
The energy balance equation in emulsion and bubble phase is expressed as follow:

(7) 
3.3. Adsorbent Solid Phase
The mass balance of the adsorbent solid particle is calculated as follow:
(8) 
where, is the water concentration in the solid particle adsorbent, and is the real velocity of zolite4A solids. The water adsorption/desorption isotherm on zeolite 4A particles is developed as Unilan equation presented by Zhu et al. (2005):
(9) 
The heterogeneity of the system is assessed based on parameter s. The parameter of water saturation capacity in solid particles is considered base on q_{m}. parameter (adsorption affinity) and is calculated through van’t Hoff equation as follow(Zhu et al., 2005):
(10) 
where, is the adsorption affinity at the reference temperature and is the average energy of adsorption. The parameter has the following functional form of temperature dependence (Zhu et al., 2005).
(11) 
The main parameters of this model are tabulated in Table 1 (Bayat et al., 2014a; Bayat et al., 2014b, c). For the solid phase adsorbent, one term for the heat transfer between the gas phase and solid adsorbent, and the other term state the released heat of adsorption ( ) in energy balance equation:

(12) 
3.4. Boundary Conditions
, , at z=0
Table 1. Main Parameter of Mathematical Modeling
Parameters 
Value 
Catalyst diameter (m), 
7.51×10^{4} 
Number of tubes () 
2962 
Length of reactor (m), 
7.022 
Diameter of tube (m), D 
38 ×10^{3} 
Catalyst density (kg.m^{3}), 
2721 
Adsorbent particle density (kg.m^{3}), 
1245 
Flowing solids diameter (m), 
3 ×10^{4} 
Specific heat of adsorption (kJ.mol^{1}), 
64 
Mass flux of water adsorption (kg.m^{2}.s^{1}), 
0.1 
Mass of catalyst in FBR (kg), (M_{cat}) 
2689.1 
Mass of catalyst + Mass of adsorbent in SEFR (kg) 
2689.1 
Thermal conductivity of solid wall (Wm^{1} K^{1}) 
48 
Saturation capacity (kmol/kg), q_{m} 
15.81 × 10^{3} 
Adsorption affinity at reference temperature,K_{0} 
4.722 kPa^{1} 
Average energy of adsorption, 
56.23 kJ/mol 
Reference temperature, T_{0 }(K) 
340.6 K 
s_{0} 
4.67 
Here, the other useful correlations for heat transfer and physical properties are considered in solving the set of these developed models. The reported empirical correlations of the hydrodynamic parameters for SEFMR in Table 2 are extracted from the study run by (Davidson and Harrison, 1963; Deshmukh et al., 2005; Goossens et al., 1971; Holman, 1989; Kunii and Levenspiel, 1991; Mori and Wen, 1975; Perry et al., 1977; Tatlıer and ErdemŞenatalar, 2000). The mathematical modeling is explained in Appendix A.
Table 2. Properties of Physical Parameter, the Phenomenological Relationship for the Hydrodynamic Parameters, Heat and Mass Transfer Correlation
Parameter 
Equation 
Heat capacity of each component 

Heat capacity of mixture 

Heat capacity of adsorbent solids 

Viscosity 

Archimedes number 

Superficial velocity at minimum fluidization 

minimum fluidization bed voidage 

Mixture density and diameter 

Bubble diameter 

Mass transfer coefficient (bubbleemulsion phase) 

Bubble rising velocity 

Volume fraction of bubble phase to overall bed 

Specific surface area for bubble 

Density for emulsion phase 

Solid velocity 

Fluidized bed heat transfer coefficient 

Heat transfer coefficient of boiling water in the shell side 

Overall heat transfer coefficient 
3.5. PengRobinson Equation of State
The wellknown equation in thermodynamics is Peng and Robinson (1976), that is, the equation of state (EOS), formulated as:
(13) 
The compressibility factor for both phases (liquid and vapor phases) is developed as follows:
(14) 
where,
(15) 

(16) 
For multicomponent when the gas phase contains methane, carbon dioxide and nitrogen, the phase equilibrium is calculated as:
(17) 
4. Numerical Solution
The mathematical model consists of the mass and energy balance equation in bubble, emulsion and adsorbent phase, rate expression of the kinetic model, transport property and other supplementary correlation. This set of DifferentialAlgebraic Equations (DAE’s) is solved through the Gear's method (ode15s) in MATLAB programming environment.
5. Model Validation
Wagialla and Elnashaie (1991) simulated the methanol synthesis process in a fluidizedbed reactor at steady state based on bubbling regime. This validated data is applied in comparing with the simulation results of SEFMR. As it observed in Table 3, the results of simulation data are in perfect agreement with the Wagialla’s data.
Table 3. Comparison between Simulation and Wagialla’s Model
Component 
Wagialla’s model 
Fluidized bed reactor model 
Deviation (%) 
Carbon monoxide Hydrogen Methanol Carbon dioxide Water Nitrogen Methane 
0.01881 0.73512 0.04744 0.02838 0.01809 0.02356 0.1286 
0.0179 0.7538 0.0492 0.0312 0.0168 0.0231 0.1121 
4.84 2.54 3.71 9.93 7.131 1.95 12.8 
6. Results and Analysis
The following definition is introduced to assess the mass ratio:
(18) 
In this part, steadystate simulation of SEFMR is assessed for the base case of mass fraction ratio (M_{ratio}), which is 0.2.
6.3. Simulation Results
The temperature profiles of reacting gas in the axial direction of CMR, FMR and SEFMR are shown in Fig. 3. The thermal equilibrium can be worse at high temperature in exothermic methanol synthesis process. The risk of catalyst deactivation through coking and sintering is diminished at minor temperatures, hence, the control of temperature in the fluidized bed reactors is easier than that of fixed beds. In excellent heat transfer, fluidized beds dominate the limitations of packedbed methanol reactors; the temperature profile does not rise suddenly at first 2 m of the reactor. In the methanol synthesis process, the upper limit temperature for the CuO/ZnO/Al_{2}O_{3} catalyst must be kept at about 543 K to avoid deactivation. Gas phase temperature profile of SEFMR is higher than FMR, and this is due to H_{2}O adsorption as zeolite 4A adsorbent that boosts the reaction rate, thus discharging more heat in the interior of SEFMR.
Figure. 3. Gas Phase Temperature in Axial Direction of CMR, FMR and SEFMR
The carbon dioxide, carbon monoxide, hydrogen and water molar flow rate of SEFMR, FMR and CR are shown in Figs. 4 (a)(d). As observed in Fig. 4(a), the molar flow rate of CO_{2} in this new configuration is lower than the other ones. The WGS reaction at reverse direction becomes faster with the removal of insitu H_{2}O. Consequently, CO_{2} and H_{2} consumption rate are improved, Figs. 4 (a) and (d)) and CO production rate is increased along the SEFMR rather than the CMR and even FMR (Fig. 4 (b)). As illustrated in Fig. 4 (c), the water production as well as catalyst recrystallization of this new configuration decreases duo to adsorption of water by adsorbent solid particles.
(a) 
(b) 
(c) 
(d) 
Figure. 4. Molar Flow Rate of (a) Carbon Dioxide, (b) Carbon Monoxide, (c) Water and (d) Hydrogen in Axial Direction of Three Different Configurations 
The methanol production rate and selectivity of CR, FMR and SEFMR are shown in Fig. 5 (a) and (b). The predicted SEFMR reveal that the selectivity and methanol production rate is noticeably higher than the without adsorption. The idealistic effect of consuming zeolite 4A adsorbents in the SEFMR enhances the reaction of CO_{2} hydrogenation toward methanol production.
(a) 
(b) 
Figure. 5. Profiles of (a) Methanol Production Rate and (b) Selectivity in CMR, FMR and SEFMR.
6.4. Influence of the Adsorbent/ Catalyst Mass Ratio
The methanol production rate and selectivity profile at different bed compositions is illustrated in Fig. 6 (a) and (b), where, although adding adsorbent to the reactor influence the weigh factor in enhancing the methanol production rate and selectivity, but it can have an adverse effect on the reactor length required for the complete methanol synthesis. It is observed that there exists a bed composition beyond which the methanol synthesis cannot be accomplished. Here it is evident that the composition of the bed should be well controlled, therefore, in part II, the mass fraction of adsorbent is selected as the decision variable.
Figure. 6. Influence of the Adsorbent/ Catalyst Mass Ratio on (a) Methanol Production Rate and (b) Selectivity of SEFMR
6.5. Influence of the Adsorbent Solid Diameter
The variation of adsorbent solid diameter over SEFMR performance is studied in Fig. (7). In Figs. 7 (a) and (b) exhibit the effect of adsorbent solid diameter on reaction rate at the reactor exit. By increasing the adsorbent solid diameter, the amount of reaction rate increases per unit catalyst volume, and then decrease Fig 7 (a) and (b). Hydrogenation of carbon monoxide (r_{1}) and dioxide (r_{3}) are two factors influencing the methanol concentration. According to Eqs. (1) and (3) here the overall methanol reaction rate is equal to methanol formation through the CO hydrogenation (r_{1}) plus methanol formation through CO_{2} hydrogenation (r_{3}). The adsorbent solid diameter versus overall reaction rate of methanol is shown in Fig (7) (c). An increase in the adsorbent solid diameter from 0.1 to 1.1 mm increases the net of methanol formation reaction and then decreases. Hence, the overall reaction rate of methanol is maximized when the adsorbent solid diameter is 0.22 mm. The influence of adsorbent solid diameter on the methanol production rate is shown in Fig. 7 (c). The trend of the overall reaction rate and production rate of methanol profiles are comparable. The overall maximum reaction rate and the methanol production rate are located at the same adsorbent solid diameter, 0.22 mm.
The effect of adsorbent solid diameter on selectivity at the outlet section of reactor is shown in Fig. 7 (d). In Fig. 7 (c) and (d) follow the same trend. An increase in the adsorbent solid diameter from 0.1 to 1.1, cause a sharp increase in selectivity up to 0.8762 and maximize the selectivity of methanol and then decreases smoothly to 0.8729, Fig. (7) (d).
(a) 
(b) 
(c) 
(d) 
Figure. 7. Influence of the Adsorbent Solid Diameter on (a) Reaction Rate of CO and CO_{2} Hydrogenation, (b) Summation Reaction Rate of them, (c) Methanol Production Rate and (d) Selectivity of SEFMR 
7. Conclusion
The methanol synthesis process is exothermic and restricted due to the thermal equilibrium. The fixed bed methanol reactor has disadvantages like low effectiveness factors coefficient and low heat transfer. Hence, one of the best manners to remove this disadvantage is applying a sorption enhanced fluidizedbed reactor, with an inherently low pressure drop through the catalytic bed. A steady state two phase theory of bubbling regime is developed for the simulation of (SEFMR). The simulation results indicate that there is favorable profile of temperature in reaction side with 46.36% and 41.88% enhancement in the methanol productivity in comparison with FMR and CMR, respectively. Enhancements of 26.143% and 25.63% in the selectivity relative to the other configuration are observed as well. Finally, the effects of adsorbent on catalyst mass ratio and adsorbent solid diameter on the methanol production rate are obtained.
In the companion contribution (Part II), the mathematical model is integrated into the optimization procedure using NSGAII algorithm and decision making method, in order to optimize key operating condition of SEFMR.
Appendix A. Developing the Governing Equations
A differential element as shown in Fig. 2 is considered along the axial direction where the corresponding equations of mass and energy balance are developed.
A.1. Mass Balance:
A.1.1. Emulsion phase:
(A1) 

(A2) 

(A3) 

(A4) 

(A5) 

(A6) 
here, k'_{g} is the gas–sorbent mass transfer coefficient (m/s), ρ_{ads} is the adsorbent density (kg of ads/m^{3 }ads), f_{ads} is the volume fraction of adsorbent in emulsion phase (m^{3 }ads /m^{3 }emulsion), d is the volume fraction of bubble phase rather than total bed volume (m^{3 }bubble /m^{3}), q is the concentration of water in the adsorbent solids (mol/kg of adsorbent), q_{e} is the equilibrium concentration of water on the adsorbent (mol/kg of adsorbent), ΔS_{b }and ΔS_{s} are the external bubble and adsorbent solid surface area, respectively, contained in the volume element. The steadystate mole balance then converts into:
(A7) 
Dividing through by ΔV_{t} = A_{c}.Δz yields:
(A8) 
The effective area for mass transfer, and , is the bubble and adsorbent particle surface area per unit bed volume, respectively:
(A9) 

(A10) 
where, is the emulsion phase volume as a fraction of total bed volume (m^{3 }emulsion /m^{3}).
As the volume element becomes small, it approaches the differential volume element and the final mole balance equation becomes:
(A11) 
Eq. (A11) can be rewritten as:
(A12) 
Where
A.1.2. Bubble Phase:
(A13) 

(A14) 

(A15) 

(A16) 

(A17) 
The steadystate mole balance then becomes
(A18) 
Dividing through by ΔV_{t} = A_{c}.Δz yields:
(A19) 
where, r_{p} is the catalyst particle density (kg cat/m^{3} cat), g is the volume fraction of catalyst bed occupied by solid particles in bubble phase (m^{3 }cat /m^{3 }bubble), r_{ij,b } is the reaction rate in bubble phase (mol/kg cat.s) and is the bubble phase volume as a fraction of total bed volume (m^{3 }bubble /m^{3}).
(A20) 
A.1.3. Adsorbent Solid Phase:
(A21) 

(A22) 

(A23) 
Inserting Eqs. (A22) and (A23) into Eq. (A21) yields:
(A24) 
Dividing by ΔV and considering the limit as ΔV → 0 yields:
(A25) 
By applying Eq. (A10), (A25) can be simplified into:
(A26) 

(A27) 

(A28) 

(A29) 
where, is the real flowing solids velocity (m . s^{1})
A.2. Energy Balance:
A.2.1. Emulsion and Bubble Phase:
(A30) 

(A31) 

(A32) 

(A33) 

(A34) 

(A35) 
By inserting Eqs. (A31)  (A35) into Eq. (A30) and dividing by ΔV, the following is yield:
(A36) 
where, c_{p} is the specific heat capacity of the gaseous state at constant pressure (J/(kg·K)), T is the bulk gasphase temperature (K), T^{shell} is the saturated water temperature (K), U is the overall heat transfer coefficient between the two sides (W/(m^{2}·K)), h'_{f} is the gas–sorbent heat transfer coefficient(W/(m^{2}·K)), and T'_{s} is the temperature of the flowing solid (K).
The area for heat transfer in the elemental volume is the circumference of the channel multiplied by length:
(A37) 
where, D_{i} is the tube internal diameter (m).
By considering the limit as ΔV → dV, we obtain the ordinary differential equation is obtained. Then Eqs. (A10), (A37), and in Eq. (A36) are substituted with each other:
(A38) 

(A39) 
Thus:
(A40) 
Equation (A40) is expressed as:
(A41) 
A.2.2. Adsorbent Solid Phase:
(A42) 

(A43) 

(A44) 

(A45) 

(A46) 

(A47) 

(A48) 
where, _{ }is the specific heat of the flowing solid at constant pressure ( J kg^{1} K^{1}) , DH_{ads} is the specific heat of adsorption ( J mol^{1}), is the mass flow rate of adsorbent solid particle (kg s^{1}) and is the mass flux of water adsorption ( kg m^{2} s^{1}).
At this stage, Eqs. (A43), (A47) and (A48) into Eq. (A42) are substituted with each other to yield an expression for the energy balance in adsorbent solid particle:
(A49) 
Dividing by ΔV =A_{c}. Δz and taking the limit as Δz→ 0:
(A50) 

(A51) 
Substituting with in
Eq. (A51) the following is yield:
(A52) 

(A53) 
Appendix B: Nomenclature
Bubble surface area, m^{2} m^{3} 

_{ } 
Tube cross section area, m^{2} 

Adsorbent solid surface area, m^{2} m^{3} 

Specific heat of the gas at constant pressure, J mol^{1 }K^{1} 

Specific heat of the gas at constant pressure, J kg^{1 }K^{1} 
Specific heat of the adsorbent solid at constant pressure, J kg^{1} K^{1} 

_{ } 
Total concentration, mol m^{3} 
Reactor diameter, m 

Bubble diameter, m 

_{ } 
Diameter of inside tube, m 
Diameter of outside tube, m 

Diameter of catalyst, m 

Diameter of adsorbent solid, m 

_{ } 
Molar flow of species i, mole s^{1} 
Molar flow in emulsion side, mole.s^{1} 

Molar flow in bubble side, mole.s^{1} 

DH_{f,i} 
Enthalpy of formation of componenti, J mol^{1} 
Gassolid heat transfer coefficient, W m^{2} K^{1} 

h_{i} 
Heat transfer coefficient between fluid phase and reactor wall, W m^{2} K^{1} 
h_{o } 
Heat transfer coefficient between coolant stream and reactor wall, W m^{2} K^{1} 
Mass transfer coefficient for component i in fluidizedbed, m.s^{1} 

Gassolid mass transfer coefficient, m s^{1} 

Length of reactor, m 

Total pressure, bar 

_{ } 
Concentration of water adsorbed in flowing solids, mol kg^{1} 
_{ } 
Equilibrium concentration of adsorbed water, mol kg^{1} 
Universal gas constant, J mol^{1} K^{1} 


Reaction rate of component i, mol kg^{1} s^{1} 
Reaction rate of component i in bubble phase, mol.kg^{1}.s^{1} 

Mass flux of water adsorption, kg m^{2} s^{1} 

Bulk gas phase temperature, K 

Temperature of catalyst phase, K 

Temperature of flowing solids, K 

T_{shell} 
Temperature of coolant stream, K 
W 
water content 
_{ } 
Overall heat transfer coefficient between coolant and process streams, W m^{2} K^{1} 
u_{b} 
Velocity of rise of bubbles, m.s^{1} 
u_{g } 
Superficial gas velocity, m s^{1} 
Velocity at minimum fluidization, m s^{1} 

Real flowing solids velocity, m s^{1} 

Mole fraction of component i in the fluid phase, mol mol^{1} 

Mole fraction of component i in the bubble phase 

Mole fraction of component i in the emulsion phase 

Axial reactor coordinate, m 

Z 
Compressibility factor 
Greek letter 

e_{mf} 
Void fraction of catalytic bed at minimum fluidization 
β 
Solid holdup 
r 
Density of fluid phase, kg m^{3} 
r_{B} 
Density of catalytic bed, kg m^{3} 
r_{e} 
Emulsion phase density, kg m^{3} 
Particle density, kg m^{3} 

r_{ads} 
Adsorbent density, kg m^{3} 
Bubble phase volume as a fraction of total bed volume 

Effectiveness factor 

flowing solid holdup 

Dynamic viscosity, Pa s 

Flowing solid density, kg m^{3} 

Volume fraction of catalyst occupied by solid particle in bubble 
Abbreviations
SEFMR 
Sorption enhanced fluidizedbed methanol reactor 
FMR 
Fluidizedbed methanol reactor 
CMR 
Conventional methanol reactor 