Modeling of Catalytic Fixed-Bed Reactors for Fuels Production by

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Modeling of catalytic fixed bed reactors for fuels production by Fischer-Tropsch synthesis Cesar I Mendez, Jorge Ancheyta, and Fernando Trejo Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b01431 • Publication Date (Web): 11 Oct 2017 Downloaded from http://pubs.acs.org on October 12, 2017

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Modeling of catalytic fixed-bed reactors for fuels production by FischerTropsch synthesis César I. Méndez1, Jorge Ancheyta2, Fernando Trejo1 1. Instituto Politécnico Nacional, CICATA-Legaria. Legaria 694, Col. Irrigación, Mexico City 11500, México 2. Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas Norte 152, Col. San Juan Bartolo Atepehuacan, Mexico City 07730, México Address correspondence to Jorge Ancheyta, Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas Norte 152, Col. San Bartolo Atepehuacan, Mexico City 07730, México. Email: [email protected] Abstract A comprehensive review of the various studies reported in the literature up to date on the mathematical modeling of fixed bed reactors for the production of fuels by the Fischer-Tropsch synthesis (FTS) was carried out. It is quite clear that most of the proposed models are based on a set of assumptions that allow their wide simplification by reducing the models into forms of low complexity, due to the fact that in most cases the effects of phase equilibrium are neglected, and relatively simple Fischer-Tropsch kinetics of the power-law type are used. In addition, most of the proposed modeling schemes neglect the effects of resistances to gas-liquid and liquid-solid mass transfer. On the other hand, few reports consider the energy effects under the consideration of a non-isothermal operation assuming a plug-flow behavior and a gas-liquid system. A generalized model of a fixed-bed FTS reactor is proposed which takes into account all the mass and heat transfer phenomena, as well as hydrodynamics and vapor-liquid equilibrium (VLE), based on the information given in the literature. It is evident that for fixed-bed reactors for fuel production using Fischer-Tropsch technology, there is little experimental information for validation and a need to explore different types of reactor models, such as reactor models under a trickle-flow regime considering the effects of phase distribution and dispersion under transient state conditions.

1 Introduction Today much of the world faces the need to meet the high energy demands, for which various fields of study have been opened to explore useful alternatives that are economical and friendly to the environment. Fischer-Tropsch technology has become a viable option for the production of commercial fuels (mainly gasoline, diesel and airplane fuels), which has attracted the attention of the scientific and industrial community [1]. The Fischer-Tropsch 1 Environment ACS Paragon Plus

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process is quite complex due to the difficulty of predicting the product distribution generated by the synthesis. The use of different catalysts has led to the proposal of various reaction mechanisms that attempt to accurately describe the steps involved in this catalytic reaction [2]. On the other hand, there is a great diversity of kinetic models that have been developed for FTS, which arise due to the different reaction mechanisms that have been proposed, the operating conditions, and catalyst type. In addition, different reactor technologies exist to carry out the FTS process that have been used to design, modeling and optimization studies, and are applied according to the Fischer-Tropsch technology considered depending on the catalyst type and operating temperature used such as: 1) high temperature Fischer-Tropsch (HTFT) at temperatures of 290-360°C with iron catalyst (in a fluidized bed reactor, or circulating fluidized bed reactor); and 2) low temperature (LTFT) at temperatures of 180260°C with cobalt catalyst (fixed-bed reactor, slurry bubble column, microchannel reactor) [3]. Despite the advantages of fluidized bed reactor, slurry bubble column reactor, and the novel microchannel reactor compared with those exhibited by the fixed-bed reactor (FBR), the latter has been preferably employed in commercial applications [4,5], because it shows superiority over others in both operational and economic aspects, since it can be scaled-up from a single tube to industrial plants [6] as can be seen in large commercial scale developed by Sasol [2] and Shell [7]. Due to the high exothermicity of the FTS reaction, and that there is experimental evidence that a complex mixture of hydrocarbons ranging from methane to wax is normally produced in a fixed bed reactor [8, 9], it is important to model phase equilibrium as well as to take into account the following aspects for reactor modeling purposes: (i) reduce pressure drop, (ii) consider the effects of mass and heat transfer limitations, (iii) intraparticle diffusion limitations and, (iv) have a good control of the temperature to achieve the elimination of the heat generated by the process thus avoiding the formation of hot spots, runaway, and catalyst wear due to overheating. In this work, a comprehensive review was made on the main issues that need to be considered for the modeling of FTS reactor, which are hydrodynamics, thermodynamics, and mass and heat transfer phenomena in order to establish the appropriate conditions for a generalized FTS reactor model.

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2 Modeling of Fixed-Bed FTS Reactors Although different types of reactors have been reported to carry out the FT synthesis for fuels production (diesel and gasoline), the FBR has increased the interest of the scientific and industrial community for the following reasons [10, 11]: 1. Strong potential over slurry bubble column because it operates with a concentration gradient of plug flow type. 2. High catalyst holdup compared with other reactor configurations. 3. It does not require a separation step and recovery of the catalyst, which is a major advantage from the point of view of time and reduction of operating costs. 4. Easy to scaling-up. In addition, it is possible to produce different types of hydrocarbons due to the following reasons: 1. The wide range of catalysts that can operate at low and high temperature. 2. Different operating conditions. 3. Type of material and geometry employed in the reactor design. Mathematical modeling is a tool with high potential to accurate description of the different types of reactors that are used by FTS technology [12]. The importance to develop models is to predict the effects of operating conditions on the thermal behavior of the reactor, especially in industrial application reactors or so-called large-scale reactors [13]. In the case of fixed-bed reactors there are some factors that complicate their modeling [9, 14, 15]: 1. Multidimensional distributed configuration. 2. The nature of the two phase (gas and liquid) involved in the reactor bed. 3. Nonlinear dependence of the reaction rate on temperature. 4. Uncertainties of both the packaging and flow through reactor. 5. The uncertainties of the parameters for heat and mass transfer. 6. Parameters variation depending on the place within the catalyst bed.


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7. The complex nature of the reaction kinetics involving many chemical species. 8. The time-varying reaction rate due to catalyst deactivation. For modeling of FTS catalytic fixed bed reactor the transport parameters of heat and mass, hydrodynamics, effects of pressure drop, holdups of both phases (gas and liquid), phase equilibrium, chemical kinetics, selectivity and product distribution must be taken into account to obtain a generalized mathematical model to predict the dynamic behavior under a wide range of operating conditions. In addition, the model must be fitted to data obtained in experimental runs for validation purposes and scaling-up. To give robustness to the model it is required further verification, which is that the model must accurately describe other experimental data. 2.1 Classification of fixed-bed FTS reactor models It is well-known that a multitubular fixed bed FTS reactor is similar to conventional shell and heat exchanger tube reactor with the FTS reaction that occurs in the tube side. Water is generally used as coolant, which circulates through the shell side to keep the process under isothermal regime. The proposed mathematical model for such a system must take into account the interactions that arise between the shell and catalyst tube where highly exothermic reaction occurs. Therefore, the phenomena occurring in the catalytic fixed-bed FTS reactor can be characterized according to [16]: •

The effects of heat and intraparticle mass diffusion.

•

The heat and mass convective transport in the fluid phase.

•

Heat and mass effects occurring between the fluid and solid phases.

•

The effect of heat exchange between the side of the exothermic FTS reaction and tube wall.

•

The process of thermal conduction and mass diffusion in the solid phase.

The typical modeling approaches widely used for accurate description of fixed-bed reactors are the continuum models, which are classified according to Figure 1 [17, 18]:


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Continuous Models Heterogeneous

Pseudohomogeneous

1-D plug-flow 1-D with axial dispersion

2-D plug-flow with radial dispersion and 2-D axial and radial dispersion

1-D axial dispersion

1-D plug-flow 2-D axial and radial dispersion Integration effectiveness factor

Interparticle effects

Figure 1. Schematic representation of models commonly used for the description of catalytic fixed-bed reactors [17, 18]. 2.2 One- and two-dimensional pseudohomogeneous model In the pseudohomogeneous model the presence of solids is not considered, i.e., the resistances to heat and mass transfer occurring between the solid phase and fluid phases are negligible, assuming that the catalyst surface is exposed to bulk fluid conditions and intraparticle diffusion effects are ignored. Such a model is classified in one-dimensional plugflow, one-dimensional plug-flow with axial dispersion, two-dimensional plug-flow with radial dispersion and two-dimensional with axial and radial dispersion models [18]. Tables 1 and 2 show a summary of advantages and disadvantages of one and twodimensional pseudohomogeneous models for fixed-bed FTS reactors reported in literature, respectively. In Figure 1 and 2, the results of some fixed-bed FTS reactor one and two dimensional-models are reported. It is important to mention that Figures 1 and 2 represent in most of cases the simulation runs under a set of operating conditions, and report only the reactor outlet values. In addition, in some cases calculated and experimental values are compared.


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Figure 2 represents the results of some research works in the literature based on onedimensional pseudohomogeneous models focused on CO conversion, where only the final values achieved in each of the simulation cases are reported. Models detailed in [19], [20], [21], [23] and [27] show higher CO conversions compared with others in the same figure due to different operating conditions and catalyst type.


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Table 1. Advantages and disadvantages of one-dimensional pseudohomogeneous models for fixed-bed FTS reactors reported in the literature. Ref. [19] -

-

Advantages Ease of use of the model due to: (1) steady-state operation, and (2) saturation state was assumed between liquid phase and synthesis gas at the reactor inlet. The efficiency comparison between the reactors used is representative even though it is performed under different characteristic diffusion length (ratio of catalyst volume and its external surface area), catalyst load per reactor volume, and catalyst and reactor geometry, which lead to the modeling approach to be different. Trickle-flow regime, which accurately reflects the actual behavior of a fixed-bed reactor operating at sufficiently low gas and liquid rates.

-

-

[20] -

-

[21]

-

[22] [23]

Use of large values of catalyst diameter ratio to lab-scale reactor tube diameter leads to almost negligible mass transfer limitations. 1D pseudohomogeneous scheme. Internal pore diffusion limitation evaluated with Weisz-Prater criterion, whose value was lower than 0.01 for all experimental conditions, which means that there are no diffusion limitations in the pore. Simulation results with the estimated kinetic parameters satisfactorily predicted the effects of operating conditions, such as temperature, pressure, space velocity and H /CO ratio, on the CO conversion and the entire hydrocarbon distribution. Although the 1D pseudohomogeneous model proposed does not contemplate the terms of external mass transfer, the effects of external mass diffusion were evaluated by the use of dimensionless parameter of Mears whose value was below 0.15, indicating that such effects are negligible. Negligible internal pore diffusion limitation because of the Weisz-Prater criterion is below 0.01 under all experimental conditions. Model applied to FBR to analyze the productivity of C + and compare with that achieved by using a slurry bubble column reactor. 20% improvement in productivity per reactor volume in FBR compared with slurry reactor. -

Disadvantages Behavior of gas phase described with the ideal gas law, and Henry's law. Reaction properties considered constant and independent of temperature. Constant flow velocity of liquid, no temperature and concentration gradients along the radial direction. FBR showed efficiency losses due to deviation of isothermal conditions even at relatively low catalyst activities, and to the reactor operation under trickle-flow regime in which the resistance to external mass transfer is neglected, which is attributed to absence of formation of a film of liquid on surface of catalyst particle. Scarce numerical simulations in liquid phase because they only take into account the formation of a single compound. Kinetic equations of FTS based on first-order approach with respect to H2. Novel kinetic model, however it is important to remark the effect of the radial dispersion of concentration in the reactor for scaling commercial purposes.

Study oriented to validate the proposed kinetic model, but not the reactor model.

Research oriented to study a three-step sintering mechanism for Co-based catalysts under FTS reaction conditions. Temperature control is assured by a very simple criterion and not by considering a temperature gradient in the axial and radial directions in the model used. Work oriented to modeling of FTS reaction kinetics and not to the reactor.

[24]

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-

-

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Focused on evaluating reaction conditions on FTS Co-catalyst deactivation FTS.

[25]

Table 1. Advantages and disadvantages of one-dimensional pseudohomogeneous models for fixed bed FTS reactors reported in the literature (Cont’d) Ref. [26] -

[27] -

[28] [29] [30] [31] -

[32]

-

Advantages Model allowed to evaluate effects of pellet size and geometry on performance of a wall-cooled multitubular FBR in terms of catalyst effectiveness, bed void fraction, and overall heat transfer coefficient. Model allows to evaluate effect of size and shape of catalyst pellet on reactor pressure drop, showing that ∆P per unit length decreases with shape in the order: spheres>cylinders>trilobes>hollow cylinders (aspect ratio=3.5). An increase in pellet diameter causes increase in heat transfer coefficient, which allows to improve the mixing gas within the bed and especially at reactor wall. Model validated under 16 experimental conditions with respect to CO conversion and product selectivity. Negligible error between predictions and experiments. Model simulations and experiments indicate that total flow velocity is a key factor to achieve high conversion rates. Decrease in temperature and/or increase in total pressure improves the performance of FTS process in terms of formation of C . Model largely predicts syngas conversion and products selectivity that are experimentally obtained at five different operating conditions. Mean absolute residuals between model and experimental data was 13.29%. Model uses detailed kinetics from literature to explain limitations of heat and mass transfer on catalyst performance by use of particle diffusion model. Results showed that membrane reactor provides better performance than conventional FBR due to efficient removal of water formed in reaction mixture, which prevents the progress of gas-water shift reaction. The reactor model employs a comprehensive FTS kinetics expressed in terms of the fugacity, which makes the predictions fit the experimental data. Peng-Robinson equation used to predict fugacity coefficients of gas-phase species. Used for FBR near PFR, and applied to others such as slurry and micro-channel. Model used to size an industrial reactor from data obtained from a laboratoryscale.

Disadvantages The model does not predict the reactor temperature profile in the axial direction, nor does it show how the catalyst size and shape of affects the performance of the FTS reactor.

It is shown a case of experimentation in which the temperature is varied from 485-540 K, an increase in the CO conversion was observed which makes evident the use of a non-isothermal model that allows predict the CO conversion and the products selectivity (light gases and liquid hydrocarbons).

Results show that increase in temperature by 57°C has negative effect on produced synthetic bioliquid selectivity by which reduction of 20.8% was observed. Model does not predict thermal behavior of reactor under conditions used. Simulations show that reactor temperature increases as reaction occurs along the catalytic bed. Comparison with experimental data for validation is not shown.

-

The effect of the pressure drop is not considered for reactor modeling effects. Products distribution and evaluation of reactor model not studied.

-

Results particularly oriented to validate the proposed FTS kinetic model and not the reactor model.

-

Great need to have a temperature control of reactor to maintain both reaction rates and wax production. Proposed model be applied under isothermal regime. Uncertainty about extrapolation of kinetics to other operating conditions.

-


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[33] -

180,000 simulations conducted to investigate the performance of a randomly packed bed reactor and a reactor with packed closed cross flow structure (CCFS), to determine optimum operating and design conditions for each configuration. CCFS can benefit from higher catalyst activities due to improved heat transport properties of the CCFS internal.

The catalyst particle size must be minimized to maximize reactor productivity; however, the pressure drops and limitations on the heat transfer must be limited. Both reactors tend to improve the space time yield of C by using optimal inlet syngas ratio H /CO of approximately 1. Maximum CO conversion limited by 50%.


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100

80

60

% mol

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40

20

0 Reference P0 (MPa) H2 /CO uG s (m/s) T 0 (°C)

[19] 2 0.01 247.6

[20] 1.0-2.5 1.0-2.5 210-240

[21] 210-240

[23] 3.0 1.0 225

[27] 1.0 230

[29] 2.0 240

[32] 2.5 230

[33] 3.0 1.0 0.55 207

Figure 2. Results of some one-dimensional pseudohomogeneous models for fixed-bed FTS reactors reported in the literature. Simulated values: full symbols, experimental values: void symbols: (■, □) CO conversion (mol%); (▲, ∆) selectivity (mol%); (●, ○) olefins (mol%).


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Table 2. Advantages and disadvantages of two-dimensional pseudohomogeneous models for fixed-bed FTS reactors reported in the literature. Ref. [34] [35] -

-

[36]

-

[37] -

-

Advantages Simulation performed for industrial reactor using laboratory data to investigate the influence of certain process variables on optimum reactor performance. Under the operating conditions used for simulation the formation of a relatively small amount of oxygenates is shown. Model simulations indicate that nitrogen plays an important role in removal of heat generated by the FTS reaction in multitubular FBR. A stable and safe operation of the reactor is guaranteed in terms of temperature control when a diameter of tubes of 70 mm is used for nitrogen-rich syngas, while for the case of nitrogen-free syngas the diameter of tubes is 45 mm. Production rate of diesel and wax per tube is about three times higher in the case of using a nitrogen-rich syngas, which could reduce the number of tubes and investment costs during industrial operation of a reactor multitubular. Results predicted by 2 D pseudohomogeneous model for production of gasoline with high octane number and low sulfur content were compared with those predicted by 1 D pseudohomogeneous model considering variation of operating parameters such as H /CO molar ratio in syngas and reactor tube diameter. 2 D model is more accurate and reliable in the temperature runaway prediction within the reactor compared with 1 D model. Model able to verify that an increase in tube diameter has relatively low effect on C hydrocarbons productivity. Model allowed to observe that an increase in cooling medium temperature, large quantity of reaction heat is released, which leads to temperature runaway behavior. It is not significant since it adversely affects C and CO production rates. An increase in H /CO molar ratio does not favor an increase in overall yield of C products, but a value of the H /CO ratio of 0.8 improves reactor performance. Model showed that thermal properties of catalyst support exhibit strong influence on reactor performance for low fluid velocities and/or large tube diameters. For high surface gas velocities (> 0.15 m/s) there is no significance influence of catalyst support on CO conversion and product selectivity profiles. For low velocities (A 12 G%>A GIJA %>A G %>A A A Gas phase +FA LOMP Q SR TU F = − +F L A A MN 13 R GR GR GH GK GK 14 Liquid phase: 1 G G%>W 15 G%>W GIJW %>W G %>W W W volatile species +F L Q SR TU 1 F − , = − +F L W OMP Z. W W MN 16 R GR GR GH GK GK 17 18 1 G G%>W Liquid phase: G%>W GIJW %>W G %>W W W +F L Q SR TU 1 F − , = − +F L W OMP Z. W W MN 19 nonvolatile species R GR GR GH GK GK 20 21 Stagnant liquid G%>W 22 ,Z. FW = GH 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 27 44 45 46 ACS Paragon Plus Environment 47 48

G-L transfer −V>AW XYAW %>A −V>AW XYAW %>A

− −

G-S transfer %>W %>W

−V>AW XYAW [%>A

−1 − ,D V>AZ XYAZ Z [%>A − %> \]^ _

-

−

Z. %W,> _

-

LS-transfer +,D V>WZ XYWZ [%>W − %>Z\]^ _

+,D V>WZ XYWZ [%>W − %>Z\]^ _ +,D V>WZ XYWZ Z [%>W − %> \]^ _

+,D V> à XYWZ Z Z. [%W,> − %> \]^ _ W Z

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The dynamic holdup of the liquid phase must be considered as that of the gas phase, because as the FTS reaction occurs along the catalytic bed the gas and liquid fraction inside the reactor changes, which is mainly due to the consumption of the syngas (convection) and production of liquid that are a function of time and of the axial and radial positions. In addition, if the liquid produced has a partial volatilization effect this will reflect a variation of the dynamic holdup along the catalytic bed [61]. It is well-known that inside fixed-bed reactors involving a liquid phase there may be some areas of liquid stagnation, the so-called dead zones. Regions of stagnant liquid regularly occurs mainly in several regions of the catalytic bed affecting the holdup dynamics both the the liquid and gas holdup. For this reason, in this paper an equation of mass balance for stagnant liquid is proposed taking into account the gas-liquid mass transfer effects in the liquid side and mass transfer from stagnant liquid to solid catalyst. To develop the mass and energy balances in solid catalysts the following assumptions were proposed based on the schematic diagram depicted in Figure 7 [44,65,66]:

•

Fischer-Tropsch reaction is carried out inside pores as commonly a heterogeneous catalytic reaction occurs.

•

The catalyst pores are considered to be full-filled with liquid.

•

Since inner catalyst pores are filled with waxy liquid, the liquid film thickness is commonly neglected.

•

Due to the formation of liquid within the catalyst, the concentration profile agrees with the two-film theory.

•

The wax formed inside pores is liquid by which solid waxes in pores are discarded.


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Reactants diffusing into the pores

Profile of concentrations during the Fischer-Tropsch reaction

132

142

Gas i Liquid filled catalyst pores

8 12 9:;

Resistance to internal mass transfer

Pore

12 2< 8

Liquid i

Solid Resistance to gas-liquid external mass transfer (usually negligible)

Figure 7. Schematic diagram of the Fischer-Tropsch reaction in the fixed bed system on the catalyst particle. For performing the mass balances in the solid catalyst particle, two cases were taken into account: (i) a dry surface of the catalyst and, (ii) a partially wetting. In both cases the internal mass gradients in the solid phase must be evaluated using the effectiveness factor [61], which is defined as the ratio of the overall reaction rate to the reaction rate that arises when the whole inner surface is exposed to the external conditions [60]. The mass gradient occurred within the catalytic particle result from the effective diffusion, which depends mainly on two important factors: (i) the porosity of the catalyst and, (ii) the size of the molecule diffusing through the catalytic pores. In order to achieve maximum FT catalyst effectiveness, it is recommended to operate under conditions where the mass transfer limitations at the liquid-solid interface are neglected [61]. There is evidence that reducing the

size of the catalyst particle allows increasing the effectiveness factor (b> ) by reduction of the

path lengths inside the particle [14]. However, the catalyst pellet size plays an important role in reactivity and selectivity during FTS process because they are size-dependent [44, 65]. According to the reactivity and selectivity analysis provided by Wang et al. [14], a large size of catalyst has a negative effect on the reactivity and selectivity of desired products because strong diffusion effects occur and small size particles are preferred instead. In industrial scale applications, catalyst pellets with 2-4 mm in diameter are commonly used to keep the pressure drop as low as possible and for removing the generated heat efficiently [67]. However, according to Jess and Kern [44] and [66] a particle diameter >3 mm is suitable for obtaining a high production rate of higher hydrocarbons. Moreover, under effective internal mass-transport limitations, which lead to an increased H /CO ratio inside the

conditions (i.e., with particles of 1 mm as used in fixed-bed reactors), the FTS is affected by

particle, which has an impact on the local reaction rate and selectivity [65]. 29 Environment ACS Paragon Plus

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On the other hand, the effect of the particle size of the catalyst on the reaction rate can also be quantified with the effectiveness factor, as well as with the Thiele modulus [70]. A simple expression for the effectiveness factor and the Thiele modulus valid for spherical particles and first-order kinetics is [68, 69]:

b =

1 1 1 d − g ΦZ HX+ℎ3ΦZ 3ΦZ

V>h ΦZ = S T L0ii

*.

(2)

(3)

In industrial operation is impossible to reduce the catalyst pellets at small sizes without having higher pressure drops; however, high yields of products are required [7]. Furthermore, according to Jess and Kern [44] for typical FTS conditions, the particle size suitable to avoid an excessive pressure drop in the catalyst bed must be higher than 1 mm. It is therefore imperative that research on the development of novel FT catalysts strives to design materials that take into account the advantages and disadvantages that may arise in their direct application [70]. However, it is noteworthy that pore diffusion greatly affects the effective rate constant within a typical temperature range of 200-250 °C for particle diameters above 0.5 mm [44]. Post et al. [68] reported insignificant external diffusion limitations caused by the resistance of the film of gas or liquid around the catalyst particle that was confirmed by observing no changes in synthesis gas conversions when changing the bed length at the same space time values.


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Table 6. Generalized mass balance equations for gas-solid phase and liquid-solid phase. Accumulation Solid phase/wetsurface Solid phase/dry Surface

1 − kl FW/

G%W,>\]^ Z

GH

G%A,>\]^ /

1 − kl FA

Z

GH

Intraparticle diffusion

G-S transfer

L-S transfer

-

-

W \]^ _ +V>mWZ XYWZ [%Z.,> − %Z.,>

= =

Z

+1 − ,D V>AZ XYAZ [%>A − %A,>\]^ _ Z

-

Generation

-

uZ

+nl o p>q bq rst,q [%> \]^ , wZ\]^ _ qv

Z

Spherical catalyst

A G%A,>xy\ L0ii,> G + {z | z Gz Gz Z

Cylindrical catalyst

Solid phase/inner

G%A,>xy\ /

FA

Z

GH

uZ

A G%A,>xy\ L0ii,> G + {z | z Gz Gz Z

=

-

Hollow cylinder catalyst

+

G%A,>xy\ G [z}~ + R>hh _ Gz [z}~ + R>hh _ Gz A L0ii,>

Z


-

+n Z o p>q bq rst,q [%> xy\ , wZ\]^ _ qv

Z

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To develop the mass balance within the catalyst particle, pores have uniform properties and are completely filled with liquid is to be assumed. A close-up of the catalyst particle is shown in Figure 7 in which the concentrations profile occurring within the catalyst particle with mass transfer resistance effects can be seen. The equations that model the behaviour of the FTS process inside the solid catalyst particle arise from the mass balance on an infinitesimal volume element of catalyst particle considering different geometric forms (Table 6).

3.1.2 Energy balance It is well-known that the temperature control of a fixed bed reactor deserves special attention for modeling and simulation. There are several studies fixed bed reactors modeling in which an isothermal regime is assumed. However, this assumption could not be reliable since the model could generate erroneous predictions of the actual thermal behaviour in a reactor and st = −165/ for paraffins formation and st = −204/ for olefins

especially at industrial scale. It may be noted that the FTS reaction is highly exothermic, with

formation [71]. Then, due to the high exothermicity of the FTS reaction it is necessary to develop an effective temperature control of the fixed bed reactor. According to Jager and Espinoza [72], the following circumstances may occur:

•

Generation of hot spots (temperature peaks) along the catalyst bed is common.

•

Axial and radial temperature profiles arise inside the tubes.

•

In order to maximize the synthesis gas conversion, the maximum average temperature must not exceed the maximum permissible temperature peak (hot spot) to prevent carbon deposition on the catalyst surface.

•

Carbon deposition deactivates the catalyst, which results in activity loss and catalysts replacement.

It is also important to evaluate the temperature profiles as well as the heat transfer rates to ensure control and performance of the reactor [73]. As aforementioned, the FTS reaction is highly exothermic and good control of the heat released during the process is required. In addition, the formation of undesirable methane as well as the deactivation of the catalyst usually occurs [74]. It is also possible that the handling of high temperatures leads to a considerable loss of selectivity and thermal runaways. Therefore, an efficient alternative is the primary mechanism for removing the generated heat by radial heat transfer, for which it is necessary to match the radial heat transfer rate to the reaction rate of the catalyst to ensure that the catalyst bed temperature can be controlled as reported elsewhere [75]. 32 Environment ACS Paragon Plus

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Energy & Fuels

It has been determined by Pölhmann and Jess [76] that in exothermic reactions such as that of the FTS process, the generation of temperature gradients can occur within the particle, which may lead to an over-heating of the particle, which would not be beneficial to reactor performance.


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Table 7: Generalized heat balance equations for fluid phases (gas and liquid). Fluid-interface Accumulation

Convective

Axial dispersion

Conductive

Radial dispersion transfer

Gas phase

Liquid phase

A A G[nA %,> w _ FA = GH

FW

W G[nW %,> wW _ = GH

A A G[IJA nA %,> w _ − GK

−

W G[IJW nW %,> wW _ GK

Gw A +FA AMN GK

+FW WMN

Gw W GK

G w A 1 Gw A +FA AOMP S + T GR R GR

+FW WOMP S

G w W 1 Gw W + T GR R GR

−ℎ~AW XYAW w A − w Y

−ℎ~AW XYAW w Y − w W

Fluid-solid transfer

Gas phase

Liquid phase

u

@>A A w Y − w A − Δ> − %>W T %,> C>

u

@>A W w Y − w W + Δ> − %>W T %,> C>

AW AW + o W,> XY S >v

AW AW + o W,> XY S >v

Fluid wall transfer −1 − ,D ℎ~A

−1 − ,D ℎ~AZ XYAZ w A − wZ\]^

−,D ℎ~W

−,D ℎ~WZ XYWZ w W − wZ\]^


^

^

w A − w

w W − w

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Energy & Fuels

Table 8: Generalized heat balance equations for the solid phase (catalyst).

Accumulation

Solid phaseisothermal

F n

Z Z

%Z

GwZ\]^ = GH

Accumulation

Solid phase-Nonisothermal

GwZxy\ n Z %Z = GH

Axial dispersion +F Z ZMN

G wZ\]^ GK

Radial dispersion +F Z ZOMP

G-S transfer

G wZ\]^ 1 GwZ\]^ S + T GR R GR

LS-transfer −,D ℎ~WZ XYWZ w A − wZ\]^

−1 − ,D ℎ~AZ XYAZ w A − wZ\]^

Axial dispersion +Z0ii

Generation nl Qo

uZ

[−Δst ,q _p>q bq rst,q [%> \]^ , wZ\]^ _U

qv

Generation

G wZxy\ 2 GwZxy\ S + T Gz z Gz

u

n ¡o[−Δst ,q _rst , ¢[%> xy\ , wZxy\ _£ Z


qv

Z

Z

Energy & Fuels

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The heat balance equations for gas and liquid phases considering isothermal and nonisothermal cases are reported in Table 7. The heat generated by axial and radial dispersion, the fluxes of fluid-interface convection heat are observed where the driving force is produced by temperature gradient among temperature in the bulk fluid phase (gas and liquid) and temperature at the interface. Transport of energy by conduction from gas film side at the gasliquid interface occurred through transfer of interfacial mass. The conductive heat flux term for both phases takes into account the heat generated by evaporation/condensation that occurs between the gas and liquid phases. Convective heat transfers of both the gas and liquid phases on the external catalyst surface are also involved. is considered. Here, Δ> represents the latent heat corresponding to the heat which is In the case of heat balance in the liquid phase, the heat evolved by vaporization/condensation

consumed by vaporization effects only for the products of the FTS reaction.

Wang et al. [14] investigated the transfer and reaction phenomena in a catalyst pellet for FTS, and the interaction between diffusion and reaction in the pellet and its effects on product external surface and the center of the pellet were less than 0.02 due to the excellent thermal selectivity. The results of simulations showed that the temperature differences between the

conductivity of Fe-Cu-K catalyst. Accordingly, it is possible to assume in certain cases of fixed bed FTS reactor modelling an isothermal behaviour of the catalyst particle.

In Table 8 the heat balance equation for the catalyst is established considering the isothermal case in which concentrations and temperatures on the solid surface must be used. It should be

noted that the sign of the heat of reaction (Δst ,q ) is negative because the FTS reactions are

highly exothermic.

Furthermore, the resistances that exhibit the films (gas and liquid) greatly influence on the heat transfer, whereas heat transport within catalyst particle is normally fast. Thus, heat transfer phenomena occurring within the particle pores can be described with the Fourier´s law in terms of partial differential equation (PDE) with respect to temperature for the nonisothermal case as illustrated in Table 8.

3.2 Boundary conditions of the proposed generalized model The generalized model for the fixed bed FTS reactor consists of mass balances of each and boundary conditions to be K = 0 and K = ¤l at the reactor inlet and outlet, respectively.

species and of the heat balance, which imply a system of PDE's that require a set of initial


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Energy & Fuels

As indicated by Wärna and Salmi [77], in a fixed bed reactor modeled by an axial dispersion model, appropriate boundary conditions to describe steady state fluxes are needed [14, 78]. It is important to indicate that boundary conditions at the reactor outlet for the case of a dispersion model is needed because backmixing takes place. The set of initial and boundary conditions that define the proposed generalized model for the FTS reactor is given in Tables 9 and 10.

At K = 0, the concentration of gas phase is in physical equilibrium with the inlet temperature and pressure and often the gas concentration at the reactor inlet is expressed as a function of %>A = @>A /r¥ w

conversion considering the following ideal gas ratio ([43, 27, 46, 79]): @>A = ¦t §>A

(4) (5)

Effects of axial and radial dispersion of mass and heat in the dispersion model give second order differential equations with two boundary conditions as follows [78]:

•

The Danclwerts' boundary condition at the reactor inlet at K = 0 is: −ki LMN i

G%> i i i ¨ = IJ «[%> _* − [%> _©v*ª ¬ GK ©v*ª

i Gw −ki MN

i

i

GK

which is simplified to

•

¨

©v*ª

= IJ n i % w i * − wi ©v*ª ® i

i

%> = wi [%> _* ; wi = wi * i

i

(6)

(7)

(8)

Such boundary conditions are acceptable with reliability since the axial dispersion of mass and heat are relatively small and concentration and temperature gradients at the The boundary condition at the exit of the reactor at K = ¤l is: reactor inlet are usually fairly flat [80].

•

G%> Gwi = =0 GK GK i

(9)

In addition, Pearson [81] showed a way of manipulating the boundary conditions of Danckwerts to impose the continuity of the reactant concentration in continuous flow reactors 37 Environment ACS Paragon Plus

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that suggests different boundary conditions can be applied at the inlet and outlet in packed bed systems, as well as in more complex situations involving problems of diffusion of multicomponent systems with variation in temperature.


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Energy & Fuels

Table 9. Initial conditions (H = 0) of generalized mass and heat balance equations of the model. Condition

K=0 0≤R≤r

0 ≤ K ≤ ¤l 0≤R≤r

0≤z≤r 0 ≤ K ≤ ¤l 0≤R≤r

Gas phase: ) = , %&, %& , % and light hydrocarbons: paraffins and olefins (% − % and % ) A %>A = %>,* w A = w*A = w*

%>A = 0 A w = w*A = w*

Liquid phase: ) = light and heavy hydrocarbons: paraffins and olefins (% − % and % ) W %>W = %>,* W w = w* = w* = 0

Solid phase: Surface (all components)

Solid phase: inside (all components)

%i,>\]^ = 0

%i,>xy\ = 0

Z

W

%>W = 0 w W = w*W = w* = 0

Z

wZ\]^ = w* \]^ = w*

wZxy\ = w* xy\ = w*

%i,>\]^ = 0

%i,>xy\ = 0

Z

Z

wZ\]^ = w* \]^ Z

Z

Z

wZxy\ = w* xy\ = w* Z

%i,>xy\ = 0 Z

wZxy\ = w* xy\


Z

Stagnant Zone

%i,>\]^ = 0 Z

Z %Z.,> =0

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Page 40 of 86

Table 10: Boundary conditions of generalized mass and heat balance equations of model.

Condition K=0 0 ≤ R ≤ r K = ¤l 0 ≤ R ≤ r R=0 0 ≤ K ≤ ¤l R=r 0 ≤ K ≤ ¤l z=0 0 ≤ K ≤ ¤l

-/0 2 0 ≤ K ≤ ¤l z=

Gas phase

Liquid phase

A %>A = %>,* w A = w*A = w* G%>A =0 GK Gw A =0 GK G%>A =0 GR Gw A =0 GR A G%> =0 GR A Gw −AOMP = ℎ~A w A − w GR

%>W = 0 wW = 0 G%>W =0 GK Gw W =0 GK G%>W =0 GR Gw W =0 GR A G%> =0 GR

-

Solid phase (surface)

−ZOMP

Gw W +WOMP = ℎ~W w Z − w W GR

Solid phase (interior)

GwZ\]^ = ℎ~AZ wZ\]^ − w GR

G%i,>xy\ Z

-

W −L0ii

-

A −L0ii

-

G%A,>xy\ Z

Gz

−Z0ii


Gz

Z G%W,>xy\

Gz

=

GwZxy\ =0 Gz

= ,D V>WZ XYWZ [%> \]^ − %>W _ Z

Z. +V> à [%> \]^ − %W,> _ W ±Z

Z

= 1 − ,D VYAZ XYAZ [%> \]^ − %>A _ Z

u²`

= nl kl o p>q bq rst [%> \]^ , wZ\]^ _ qv

Z

Gw AZ Z\]^ = 1 − ,D ℎAZ − wA ~ XY w Gz Z

+ ,D ℎ~WZ XYWZ wZ\]^ − w W

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Energy & Fuels

3.3 Pressure drop In designing and modeling of a fixed bed FTS reactor it is mandatory to predict the pressure drop through the catalyst bed. In the case of the LTFT process for the production of clean fuels, the main products are heavy oil and waxy material whose molecular weights are relatively high and in consequence the effect of flow velocities occurring through packed bed could cause severe attrition of catalyst. Among various mathematical models presented in previous sections, there are not many of them that consider the effect of pressure drop through the bed and many of them make use of the classical Ergun correlation for its prediction. However, the classical model of Ergun to estimate the pressure drop in packed bed reactors is inefficient when various geometries of catalyst are used. In different fields of engineering, it has been employed semiempirical Ergun equation for predicting pressure drop across packed beds with particles made of various materials [82]. In the case of FTS, there are several models for fixed bed FTS reactors that use such a correlation to describe the effect of pressure drop through the packed bed. However, various research papers have been conducted in which a comparison between modified correlations of Ergun and contemporaneous correlation has been made and obtaining as results a deficiency in predicting pressure drop by the classical model of Ergun compared to the modified correlations [82-84]. It has been established in the literature that deviations from the predicted pressure drop across packed beds that exhibit the classical correlation of Ergun may be due to the high sensitivity of the equation to variations in the bed porosity by which it is necessary to estimate accurately the packed bed porosity especially when dealing with non-spherical geometry [85]. It has been found that several factors influence on the bed porosity as the particle diameter, particle size distribution, surface roughness, Reynolds number and others [84, 86]. In summary, the literature reports give both theoretical and experimental evidence that pressure drop depends on flow velocity and physical properties (viscosity and density), the average bed porosity, shape and surface of the particles packaging, bed height and contribution of the ratio of particle to the container diameters [87-89]. Flow regimes in catalytic fixed bed reactors are quite important as they are directly related to pressure drop. Four flow regimes in fixed bed reactors are reported in literature [90]:

•

The Darcy or creeping flow regime at rC> < 1: At this flow rate the pressure drop varies linearly with flow rate and interstitial flow is controlled by viscous forces. 41 Environment ACS Paragon Plus

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Page 42 of 86

•

In a laminar flow regime in stationary Reynolds interstitial number range of 10
< 150 pressure drop shows a nonlinear relationship with interstitial flow rate.

In a laminar flow regime in non-stationary Reynolds interstitial number range of

•

form and in the number of rC> = 250.

150 < rC> < 300 oscillations of laminar wake are present in the pores and vortices In a flow regime considered highly nonstationary at Reynolds interstitial number above 300 eddies arise like turbulent flow in pipes.

The effects of pressure drop involving both the gas and liquid phases must strictly be considered separately to get better accuracy and approach to the real pressure drop across the catalytic bed. Bai et al. [86] proposed an empirical correlation to predict the pressure drop of a two-phase flow through pebble beds in a gas-water system by considering the use of the relative permeability in gas phase while the bed is subdivided into two regions: (i) near walls and (ii) another one in the central zone of the bed to consider the effects of wall and improvements to low ratios of tube diameter to the particle. The correlation used to calculate the pressure drop in two-phase flow was established in dimensionless terms to cover the whole range of Reynolds and Galileo numbers assuming that the relative permeability is a dependent on the void fraction bed and the subdivision of the bed in two regions to consider the effects of wall. Bai et al. [86] based on previous studies reported by Eisfeld and rC i -/0 rCi = µ ¶ + µ ¶ D D i ·X i ·X i n i [I _ ¤ Δ¦i

Schnitzlein [90] and Reichelt [91], proposed the following equations: Ψi =

where

J

IJ n i -/0 -/0 kl ni rCi = i ; ·X i = S i T ¹~ d g ¸ 1 − kl ¸ 1 − kl

i

-/0 =

6º/ »/

(10)

(11)

(12)

µ and µ are Ergun constants type, and ¶D and ¶D are the coefficients that consider the

When small packed beds are used (L¼P /-/0 < 10 ), the central region of the bed

effect of wall under the following restrictions:

(i)

∗ -/0 ¿ 1 − kD

is negligible and the following wall effect coefficients must be considered: ¶D = ½X + ¾


(13)

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Energy & Fuels ∗ ¶D = [À-/0 _ ∗ -/0 =

(ii)

-/0 -¼

±

(14) (15)

When large packed beds are used (L¼P /-/0 ± = ,ÞÛ = o »* ÞÛ Yv>

= 0.45445 + 0.62675 × 10± ÞÛ − 0.93697 × 10± ÞÛ

(37)

! + 0.37387 × 10±( ÞÛ

The equilibrium between the gases (reactants and products) and the Fischer-Tropsch liquid wax is established as a function of the fugacities of each component in its respective gas or ,> = ,>W

liquid phase as follows:

,> = ¦§> í> i

i

i

(38) (39)

where í> is the fugacity coefficient for component ) in , phase which can be estimated with i

¾> Ñ − 1 − Ö+Ñ − ç ¾Û

the SRK equation of state as follows: Ö+[í> _ = i

» ¾> 2 ç + − o §> äZ å X>q Ö+ d1 + g ç ¾Û äZ å,> X> Ñ

(40)

q

where XÛ and ¾Û are the parameters for mixtures which are estimated by the following

mixing rule:

XÛ = o o §> §q èX> Xq [1 − V>q _ >

q

¾Û = o §> ¾> >


(41)

(42)

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Energy & Fuels

In addition, the solubility of the gases in the Fischer-Tropsch waxy product can be determined by means of:

Ò> =

î> DMN 1 − ∑> î> ÞÛ

(43)

DMN where ÞÛ is the molecular weight of the wax, and xi is the molar fraction of each

component in wax.

It should be noted that the critical properties (temperature and critical pressure) with respect to the components of the FTS must be determined for VLE calculations; however, the information for heavy hydrocarbons is scarce since there are only data reported in the literature for hydrocarbons up to C20. Thus, the following Gasem and Robinson [152]

correlation including data of normal boiling points and number of carbon atoms (á ) is used:

±Ê % % ± ð = ½ − d − %! Ê g Cî@ñ−% á − 11 − % ò¿ % %

(44)

where ð represents either ¦~ or wì /w~ , and the constants C1 to C4 are indicated Table 16: Table 16. Gasem and Robinson [152] correlation constants. óô (bar)

õö õô

%

8.0936

0.036175

0.99288

-0.47775

%

54.555

0.58524

8.0034

0.043358

Constants %

%!

Mikhailova [146] modified SRK equation and in this case the molecular weight distribution of the FTS products was estimated by the ideal Anderson-Schulz -Flory polymerization model:

ÞÛ = 1 − äh±

(45)

However, it should be noted that the molecular weight distribution can be estimated by nonidentical Anderson-Schulz-Flory models [153-155].


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Page 56 of 86

3.4.4 Catalyst particles parameters In the case of the catalyst particles it is necessary to determine the different properties involved as well. A review of the different correlations published in the literature was done for the parameters estimation of the catalyst particle used in the FTS process. Due to the formation of a film or thin layer of waxy liquid product on the catalytic particle surface, the external mass transport effects may be negligible, but the influence of pore diffusion is important [47]. Thus, the Thiele modulus is an important dimensionless parameter that allows determining the presence of diffusion limitations within the particle, which for the case of a pseudohomogeneous first reaction order is defined as follows [44, 156]: ΦZ = ¤/ ÷

V

L0ii,W

= ¤/ SV>h.

A %

*.

W T L0ii,W %

= ¤/ øV>h.

1

w L0ii,W r¥ C

ù

*.

(46)

where ¤/ is the characteristic length of the reaction system also known as form factor, V is

the kinetic rate constant for the homogeneous first-order reaction and L0ii,W is the effective

diffusivity for CO at the liquid phase.

One approach given in the literature to determine the mass transfer limitations of the catalyst, consists in solving the mass balance of a differential shell taken into the catalyst particle by considering a spherical geometry of the pellet or catalyst particle and in which a reaction of first order occurs is as follows: [157]: %>

i

Z %i,>\]^

-/0 = 2R

2R g -/0 sinh3ΦZ

sinh d3ΦZ

(47)

whose modified form for a reaction of order + was established as [158, 159]: ΦZ = ¤/ ý+ + 1

V>h. Ä%i,>\]^ Å Z

2L0ii i

h±

(48)

The shape factor ¤/ was used to modify the solution of different geometries, which in the

case of a cylinder pellet turns into -/ /4, and -/ /6 for a spherical pellet. V>h. is denoted by

the intrinsic reaction rate coefficient per volume unit for the +th order reaction considered, 56 Environment ACS Paragon Plus

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Energy & Fuels

and L0ii is the effective diffusivity given as follows: i

L0ii = L~×Ûì i

k/ =

k/ þ

(49)

º/×O0 nl 1 − kl

(50)

On the other hand, about the determination of the effective diffusivity, the following correlations are used in modeling a fixed bed configuration in which the FTS occurs [160, 161]:

L0ii,MN,> = Lt d1 + i

1 ¦C g 192

-. IJA r¥ ℎCRC ¦C = X+- L>q = A L>q 6Ø¸Û>N -/ L0ii = i

(51)

kZ 1 A þ 1 + 1 i i Lx Lå,>

(52)

whose restrictive factor A is given by Iliuta et al. [162] and RJ×¥.0 is the hydrodynamic

molecular radius of the solute, R/Û is the mean radius of the pore, and Ñ is a dimensionless constant used in the Bosanquet's formula:

A = 1 − A ; A =

RJ×¥.0 R/Û

(53)

The effectiveness factor of the catalyst can be estimated by the correlations reported in Table 23. Mederos et al. [61] reported other equations for determining the effectiveness factor and the Thiele modulus for different shapes of catalyst. Table 17. Effectiveness factors for conventional geometry of catalyst particles and their concentration profiles according to analytical solution. Geometry of the system Rectangular

Cylindrical

¤/

¤l -/ 2

b

tanhΦZ ΦZ 2Ö ΦZ ΦZ Ö* ΦZ


Observations

For pellets of cylindrical catalyst with a particle diameter of less than 200 µm (-/ < 200 µm). Where Ö* and Ö are the typical Bessel functions of first and second order respectively.

Energy & Fuels

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3 1 1 ½ − ¿ ΦZ HX+ℎΦZ ΦZ

-/ 6

Spherical

Page 58 of 86

Spherical particles with a diameter greater than 200 µm (-/ > 200 μm).

necessary adjustment for an +-order reaction (Equation 49). However, it is important to It is possible to obtain close approximations if Thiele modulus is used by making the

mention that there are no analytical methods to determine the effectiveness factor for cases involving non-linear reaction rates and arbitrary catalyst particle shapes. Thus, its approximation based on the consideration of non-linear reaction rates and catalysts of an arbitrary shape can be calculated by the equations shown in Table 17 using the generalized Thiele modulus [16]: ΦZ = ¤/

rst [%i,>\]^ , wZ\]^ _ Z

è2-/0

â

`

È,x\]^

*

Z Z rst [%i,>\]^ , wZ\]^ _-%i,>\]^

±/

(54)

In the case where b tends to 1, the shape adopted by the catalytic particle becomes an

important factor that influences the estimated value of the effectiveness factor. Wijngaarden et al. [163] proposed a novel approach to calculate the effectiveness factor in the case where chemical reactions whose rate is nonlinear and catalytic particles with an arbitrary geometry are contemplated that involve two dimensionless groups: Zeroth Aris number: »h* =

¤/

rst [%i,>\]^ , wZ\]^ _

First Aris number: »h

Z

è2-/

â

`

È,x\]^

*

rst [%i,>\]^ , wZ\]^ _- %i,>\]^

Z

\]^ Z Γ Grst [%i,> , w \]^ _ = ¤/ Z -/0 G% \]^

Z

i,>

Z

` È,x\]^

(55)

(56)

where Γ is the geometric factor which depends only on the particle shape. Γ is 2/3, 1 and 6/5 for a slab, an infinite cylinder and a sphere, respectively. The zeroth Aris number was introduced to calculate the effectiveness factor in a low region of η, whereas the first Aris number determines η when its value approaches 1. The following expression is a generalized 1

form of the effectiveness factor that implies the two numbers of Aris already defined: b=

è1 + 1 − b»h* + b»h

which can be solved by an iterative procedure.


(57)

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When V>h. is unknown, it is not possible to use the equation 49 for the Thiele modulus

already described, so if only information on the observed experimental reaction rate is available, the Wagner-Weisz-Wheeler modulus (ÞD ) must be used [164]: ÞD =

¤/ R n Z + + 1 = ΦZ b Z i L % \]^ 2 0ii

i,>

(58)

where n Z is the density of the pellet, R is the observed rate of reaction of the species ), which also translates as the %& depletion rate in the FTS reaction per unit mass of the catalyst.

In equation (59), the diffusivity L~×Ûì represents the combination between the bulk

diffusivity L>q and Knudsen diffusivity Lå that can be estimated from the correlations 1

reported by Pollard and Present [165]:

L~×Ûì

=

1 1 + L>q Lå

(59)

In addition, the diffusivity of each component involved in the gas mixture is given by the following expression [166]:

A L>,Û>N =

1 − §> § ∑>q > L>q

(60)

where L>q (where ) diffuses through the liquid wax ¢) was estimated using the following correlations which are valid for a wide range of temperatures and pressures [167-169]: L>q =

L>q =

0.945√w

*.! *.Ü( ÞÛ ÞÛ [> q _ x

10±Ü w .Ü ÷

.!

1 1 + ÞÛx ÞÛ

A nÛ>N Ä~> + q Å /!

/!

(61)

(62)

where ÞÛx and ÞÛ are the molecular weight of the solute (either CO or H ) and the

molecular weight of the solvent (waxy liquid product) respectively, boths in g mol-1, > and

q are the hard-sphere diameter for the solute (either CO or H ) and hard-sphere diameter for

the solvent (waxy liquid product) respectively, boths in Angstroms, ~> and ~q are the critical

volume of component i and j, respectively, and ºæ is the specific volume of different species

in m3 mol-1. Meanwhile the diffusivity of Knudsen Lå is calculated by the following 59 Environment ACS Paragon Plus

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correlation obtained from the kinetic theory of the gases [170]: Lå = 4850-/ ÷

w ÞÛx

(63)

Hallac et al. [170] indicate that when Lå exceeds L>q by more than two orders of magnitude

under used a set of operating conditions, the effects of diffusion resistance can be neglected.

It should be noted that the diffusivity of CO in the produced wax is usually lower than that of H2 and it is considered as the limiting diffusion rate. On the other hand, when synthesis gas feed stream contains low concentrations of hydrogen, it is convenient to consider that the diffusivity of hydrogen in the waxy product is the limiting diffusivity [171]. Table 18 summarizes several correlations used to determine some parameters such as species concentration, viscosity, syngas mixture molecular weight, Reynolds number and tortuosity, all of them being necessary for constructing the generalized model for FTS reactor. Table 18. Correlations for determining some parameters of the FTS reactor. Parameter

Reference

Parameter

Reference

Mixture molecular weight

[130]

Species viscosity

[134], [173], [174]

Molecular weight of

[130]

Reynolds number

[18]

[134]

Solids Reynolds

[101]

component i Fluid density

number Initial superficial velocity

[134]

Tortuosity

[175], [176], [177], [178]

Mixture viscosity

[130], [172]

Molecular volume (La

[148]

Bas method)

In case of using catalyst pellets with irregular geometry, the equivalent particle diameter (-/0 ) must be used to use the aforementioned correlations.

The equivalent particle diameter is referred to as the diameter of a sphere whose surface area (or volume) is equal to that of the catalyst particle, being a characteristic property of the particle that depends on both its size and shape. One way to determine the equivalent particle diameter for fixed bed systems is provided by Cooper:


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-/0

º/ 6d g Ò/ = íZ

(64)

There are some papers related to one dimensional heterogeneous models for fixed bed FTS reactor in which catalyst particle is modelled on the effectiveness factor (see subsection 2.4). The geometric forms of catalysts are ideally spherical but it would be possible to use any catalyst shape; however, correlations for effectiveness factor are not able in all cases mainly for irregular geometry. Therefore, there are other parameters involved in calculating the particle size also known in the literature as form factor ¤/ . It should be noted that for the

effectiveness factor that can be estimated for other types of catalyst particles such as the

construction of the model of an irregular geometry catalyst particle in the FTS process it would be more complicated.

Nowadays, the design of novel FTS catalysts is of utmost importance for purposes of improving the synthesis gas conversions, product selectivities and overall performance of the reactor [179-180]. In the research of catalyst design not only applied to the FTS process but also for hydrotreating processes, it is very important to improve the geometries of the catalyst

mainly to diminish the mass and heat transfer intraparticle limitations. The form factor (¤/ )

defined as the volume ratio to the catalyst particle surface (º/ /Ò/ ) is frequently used to

correlate certain characteristics such as the size and shape of catalyst with catalytic activities, as well as to perform studies on the characteristics of catalyst, packed bed, intraparticle diffusion limitations, catalytic pore size, pressure drop restrictions, etc. [181]. Ancheyta et al. [181] proposed some equations for determining the area and external volume for some porous calculations of º/ and Ò/ for different types of hydrotreating catalysts; however, its and nonporous particles of different shapes and sizes. This study is mainly oriented to application can be extended to catalysts used in FTS process. Calculation of º/ and Ò/ of º/ = +ØR~ ¤ − » ¤

lobe-shaped catalyst particles are as follows:

Ò/ = +2ØR ¤ + 2ØR~ ¤ ± 2» − +»

(65) (66)

where + is the number of lobes, » is the lateral area of the geometric shape between the

and a frame for tetra-lobular geometry, while » is the common area between each cylinder

lobes which represents a rhombus for a bi-lobular geometry, a triangle for a tri-lobular shape,

and between each side of the shape between the lobes. In equation 66 the term after 61 Environment ACS Paragon Plus

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parenthesis is negative for bi-lobular forms and positive for tri and tetra-lobular shape. Table 25 shows the parameters involved in equations (65) and (66) for cylinders, spheres, pellets, bi-lobular, tri-lobular and tetra-lobular shapes.

Table 19. Definition of parameters to determine the volume and surface of particles of different geometry. Shape

+

ë

Cylinder and pellet

1

-

Bi-lobular

2

45°

Tri-lobular

3

60°

4

45°

Tetralobular

R~

-/ /2

-/ 3.4142 -/ 4

-/ 4.8284

»

» 0

«

(

¬ = 8.88348 × 10±! -/

Ê PÎ ±Ê

(

Ê ±Ê PÎ

-/ Q

= 3.86751 × 10± -/

2cosë − 1 U = 2.94373 × 10± -/ 1 + 2 cosë

Ø R ¤ 2 ~ Ø R ¤ 3 ~ Ø R ¤ 2 ~

An important aspect in catalyst designing, simulation and modeling of a fixed bed reactor is the analysis of transport limitations at particle level. Mears [182] indicated that concentration and temperature gradients can exist in three domains: (i) intraparticle: refers to the gradients that occur within the individual catalytic particles; (ii) interface: here the gradients are generated between the external surface of the catalytic particles and the adjacent layer of the fluid; (iii) interparticle: the axial and radial gradients within the reactor are considered. It is of interest to have precise criteria that allow predicting the existence of intraparticle mass transports, since it is important in determining suitable pathways for model simplification of the catalyst particle, such as neglecting or not of the effects of internal diffusion in the pore. On the other hand, it is stated that it is only possible to obtain information related to the intrinsic kinetics in the absence of effects caused by resistances to mass and heat transport in the catalytic particle [183]. There is a criterion given in the literature to determine if the presence of intraparticle diffusion limitations exists, this is the so-called Weisz-Prater −rst,×ìJ n Z ¤/

criterion given by the dimensionless parameter: % =

L0ii %>A i


(67)

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where rst,×ìJ is the observed reaction rate (/¹~M. ∙ ), n Z is the density of the solid

catalyst particle, and %>A is the limiting reactant concentration in the syngas mixture. This criterion establishes that if the value of % becomes much larger than 1, the effects of

limiting the internal pore diffusion are significant, i.e., there is a strong resistance to intraparticle diffusion resulting in a concentration gradient from the catalyst surface to its pores [183]. Akpan et al. [184] showed experimentally that the diffusion resistance is null when the particle size (form factor) is less than 0.3 mm. The criterion of Mears [182] is often considered for the determination the onset of external mass transport limitation that is given by:

R×ìJ nl ¤/ + < 0.15 Vst

(68)

To determine that external mass transfer resistance has no significant effect on the reaction rate is by using the following criterion relating the observed reaction rate to the reaction rate observed rate −rst,×ìJ = ; ) = reactants: CO and H i rate if film resistance controls V~ %

if the resistance to the film controls [183]:

>

(69)

However, Levenspiel [164] points out that the film resistance to mass transfer should not influence the rate of reaction. Ibrahim an Idem [183] stablished two criteria: 1) the heat transfer resistance within the pore and, 2) the limitations of heat transfer across the gas film, to be: ∆wÛMN,/MO.>~0 =

L0ii [%> \]^ − %> _−∆ st

ΔwÛMN,i>Û =

i

Z

~/

0ii

(70)

¤[−rst,×ìJ _−Δ st ℎ

(71)

where ∆wÛMN,/MO.>~0 and ΔwÛMN,i>Û is the upper limit to temperature variation between

pellet center and its surface and the upper limit to temperature difference between the gas bulk and the pellet surface, respectively, ∆ st is the heat of reaction, and %> \]^ and %> Z

~/

are

the concentrations at the pellet surface and center, respectively, L0ii is the effective mass diffusivity, and ¤ is the characteristic length.

i

One way of assessing the thermal severity resulting from the FTS process in fixed bed systems is by using the Mears [185] rigorous criterion, which allows to compare the heat 63 Environment ACS Paragon Plus

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generated by the reaction with the capacity of the catalytic bed to transfer heat from the reaction zone towards the wall of tube. This criterion was used in pseudohomogeneous -/ wM -. 8OMP !Δ st,q ! ∙ ∙ d1 + g < 0.4 2 40ii,OMP wD ÕO -.

modeling of a fixed bed FTS reactor proposed by Philippe et al. [37]: 1 − kl

(72)

When this criterion is fulfilled it can be assumed that the radial temperature profile is almost flat, which means that the reaction heat is eliminated in the radial direction. However, this does not ensure that there is no generation of hot spots along the catalytic bed [37]. The following criteria can be applied in cases where the resistance to heat transfer at the wall is significant as well as its impact on catalyst particle:

!∆ st,q !rst,l -. "M 1 0.4r¥ wD ø ù< -/ 40ii,OMP wD r¥ wD "M 1 + 8 d g ç)D .

(73)

where rst,l is the reaccion rate per unit bed volume, and ç)D is the dimensionless number of

Biot defined as ℎD -. / 0ii,OMP . It should be noted that the term 0ii,OMP / 8ℎD -. represents

tubes to diameter of the particle are used (-. /-/ ≈ 10) [182, 185]. Otherwise, Mears [182]

the heat transfer resistance in the wall that cannot be ignored when low ratios of diameter of

pointed out that the magnitudes of heat transfer resistance in experimental reactors behaves as follows: interparticle> interphase> intraparticle.

3.4.5 Catalytic bed parameters An extremely important parameter in the modeling of fixed bed systems involves the determination of the bed void fraction (or bed porosity), as this indicates the space within the fixed bed that is filled with catalyst particles, i.e., it states the fraction of the catalyst bed volume that is available for the phases involved in a multiphase multicomponent system. Alternatively, having a reliable estimation of the bed void fraction is important for minimizing the pressure drop across the catalyst bed. Table 20 reports some correlations for calculating the void bed fraction applicable to many problems involving fixed bed configurations. Table 20. Correlations for the calculation of the void fraction bed. Correlation


Reference

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Energy & Fuels kl = 0.1504 +

0.2024 1.0814 + í/ d . + 0.1226g -/0

(1) kl R = Q1 + Ä

±É É

Å $1 − Cî@ d−2

for R>hh ≤ R ≤

(2) kl R = Q1 + Ä

±É

for

É

Å $1 − Cî@ d−2

O%]a ±Oxyy

O±Oxyy

gU

O%]a ±O

gU

PÎ

O%]a ±Oxyy

≤ R ≤ R×¥.

[186]

PÎ

±

[187]

±

( d . − 2g+ -/0 ' * kl = 0.38 + 0.073 + '1 + * ' d .g * -/0 & )

[18], [188]

Antwerpen et al. [189] reported a wide range of correlations to estimate the porosity of the bed of structured packages that can be used in fixed bed reactors modeling. Once bed void fraction is determined, the density of the solid catalyst particle can be calculated according to the next equation proposed by Tarhan [190]: nl nZ = 1 − kl

(74)

Based on the consideration of volume conservation, the bed void fraction is the sum of gas and liquid holdups [191]:

kl = FW + FA

(75)

FW + FA + FZ = 1

(76)

kZ = FW + FA

(77)

Relationships between holdups within the solid catalyst particle are given by: /

/

FW + FA + FZ = 1 /

/

/

(78)

The external surface area of the catalyst particles per unit volume of the fixed bed reactor can be calculated as:


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XZ =

Page 66 of 86

61 − kl -/0

(79)

It is also convenient to estimate the pore size to determine the pore filling of the catalyst 2> nZ r w @ @> − @JM. − > ¥ ln d > g ÞÛx @JM.,>

particle with water and liquid hydrocarbons using the Kelvin equation [59]: -/×O0 =

(80)

Because it is possible that the formation of waxy liquid is carried out inside the catalytic reactor, and that the velocities reached by the liquid flow are relatively low in the range of a trickle regime, it is necessary to determine the liquid holdup for generalized modeling of the fixed bed FTS reactor. This parameter is useful in designing of fixed bed reactor under trickle flow regime [192]. Liquid holdup has been shown to greatly affect the pressure drop across the bed, the amount of catalyst wetting, and the thickness of the liquid film surrounding the catalyst particle [193]. In addition, the holdup knowledge is essential in the case of high exothermicity of the reactions such as the FTS, since it allows avoiding the formation of hot spots in the catalytic bed and the prevention of reactor runaways. On the other hand, it is a parameter that affects the wetting efficiency of the catalyst and hence the selectivity, which depends on whether the reaction is carried out on the wet catalyst area or in the wet and dry catalyst regions alike [194]. Bazmi et al. [195] proposed a correlation for the prediction of liquid holdup where the effects of the catalyst shape (extruded trilobe particle) on the pressure drop are considered, as well as the behaviour of the liquid holdup to the different methods of loading the catalyst into trickle bed systems: FW,P where

,CW*. kl! = 0.07 S T -W 1 − kl

!.

*.Ü

rCW d g

rCA

,CW*. kl! Cî@ . S T -W 1 − kl

IJW nW *. -W = A A *. IJ n IJW -/0 nW ,CW = W

!.

d

rCW g / rCA

(81)

(82)

(83)

The equation is valid within the range of gas superficial velocities from 0.01 to 0.1 m s-1 and liquid superficial velocities from 0.001 to 0.01 m s-1 included in the range of trickle flow 66 Environment ACS Paragon Plus

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regime. Such a correlation was compared with the correlation proposed by Larachi et al. [192] as well as with experimental data and showed acceptable improvements in predicting liquid holdups. Al-Dahhan et al. [194] also reported a comprehensive review on various correlations given in the open literature for the prediction of the transition from trickle flow regimes to pulse, for liquid holdups, two-phase pressure drops, catalyst wetting efficiency (liquid-solid contacting efficiency), among others in high-pressure trickle bed reactors.

4 Concluding remarks The search for new and better alternatives to produce improved clean fuels that meet the environmental constraints more stringent is a worldwide aim. The scientific and industrial community is increasingly studying the implementation of the Fischer-Tropsch synthesis as it has been considered as one of the most viable options to replace conventional technologies for producing fuels of commercial value. In addition, the design of more efficient catalysts and novel reactor technologies (microchannel reactor and monolith reactor) have been a field of research currently active to greatly improve the overall FTS process. However, the fixed bed catalytic reactor has been preferably used because of their marked advantages over other reactor technologies that include the relatively easy scaling up from a single tube. On the other hand, the FTS process is considerably complex since the distribution of FTS products has not been accurately known because of products are quite broad and difficult to be accurately predicted. FTS behaviour is attributed to the effect of (i) the catalyst type and/or support-promoter, (ii) the type of reactor used, among others. Despite the large number of mathematical models published in literature up until now, it is necessary to propose a generalized modeling scheme of the fixed bed reactor for the FTS obtained from simplified versions reported in literature. However, solving the generalized model is a rather complex task, since it requires an in-depth study of hydrodynamics, the analysis of the phases distribution in the catalytic bed (vapor-liquid equilibrium), mass and heat transport phenomena in the bulk and particle phases, a robust model for pressure drop, and the search for precise correlations that estimate the several parameters of the model. Therefore, it becomes evident the need to study different modeling schemes that can be derived from this proposed generalized model and that must be validated through comparisons between simulated and experimental data.


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Acknowledgments César I. Méndez thanks to CONACYT for the scholarship granted and also thanks to Instituto Mexicano del Petróleo for supporting this research project.

Nomenclature Symbols X

XYAW XY

iZ

XÛ XZ »

»>hh

»×¥. »D ¾Û ç

Attractive term parameter, XH ∙ ' />

Gas-liquid interfacial area per unit reactor volume, W /O!

Gas (or liquid)-solid interfacial area per unit reactor volume, Z /O!

Parameter related to mixing rule

Specific surface area of the solid, Z /O!

Dimensional parameter of Soave-Redlich-Kwong EOS Internal reactor area, O

External reactor area, O

Parameter that takes into account the effect of the reactor wall, (dimensionless) Parameter to related to mixing rule

çD

Dimensional parameter of Soave-Redlich-Kwong EOS

%>,*

Initial concentration of component ) in the , phaser, [> /i! _

%>

i

i

%>∗

%> xy\ Z

%i,>\]^ Z

%

i

%x i

Parameter that takes into account the effect, (domenionless)

Molar concentration of ) component in the , phase, [> /i! _

Concentration of compound ) in the liquid in equilibrium with the gas phase, [> /i! _ Molar concentration of component ) inside the solid filled with , phase, [> /i! _

Molar concentration of component ) at surface of the solid covered by , phase, [> /i! _

Specific heat capacity of , phase, Ä/[¹i °_Å

Specific heat capacity of component ) in the , phase, Ä/[¹i °_Å 68 Environment ACS Paragon Plus

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%ÙxÚ,

Specific heat capacity of the gaseous mixture, Ä/[¹i °_Å

% -¼

The Weisz-Prater criterion, (dimesnsionless)

-×¥. -/

Outer diameter of the reactor, O

Catalyst particle diameter, Z

∗ -/0

The dimensionless equivalent particle diameter, Z

->hh -/0 -.

LMN i

L~×Ûì

L0ii,MN i

Hydraulic diameter, O

Inner diameter of the reactor, O Equivalent particle diameter, Z

Reactor diameter, O

Mass axial dispersion coefficient of , phase, O /

Diffusion combined coefficient (molecular and Knudsen diffusion), Äi! /Z ∙ Å

Axial effective fickian diffusivity of component ) inside a porous catalyst, Äi! /Z ∙ Å

L0ii,OMP Radial effective fickian diffusivity of component ), Äi! /Z ∙ Å i

L>q

Binary diffusion coefficient (molecular diffusion), [i /_

L>Û

Molecular diffusion coefficient of component ) in the , phase, Äi! /Z ∙ Å

"&Ò

Equation of states

Lå,> i

"M

Knudsen diffusion coefficient of component ) in the , phase, Äi! /Z ∙ Å

Activation energy according to the Arrhenius equation, / t

,>W

Liquid phase fugacity of component ), (dimensionless)

,O

Friction factor, (dimensionless)

,> ,/

,Z. ,D ,©

Vapor phase fugacity of component ), (dimensionless) Radial volume force, [á/i! _

Fraction of liquid which is stagnant, (dimensionless)

Catalyst wetting efficiency, [Z,D0. /Z _

Axial volume force, [á/i! _


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ℎ~AW

Heat transfer coefficient at the gas side of the gas-liquid interface, [/Y ∙ ∙ °_

ℎ~WZ

Liquid-solid heat transfer coefficient, Ä/[i ∙ ∙ °_Å

ℎ~AZ

Gas-solid heat transfer coefficient, Ä/[i ∙ ∙ °_Å

ℎ~W

Liquid-wall heat transfer coefficient, Ä/[i ∙ ∙ °_Å

Entalphy at the temperature w and pressure ¦, /¹>

ℎD

Wall heat transfer coefficient, Ä/[i ∙ ∙ °_Å

C>

Henry’s law coefficient of component ), Þ¦X ∙ W! />

×¥.

Out reaction enthalpy, [/¹i _

>hh ß

à

V>h

Inlet reaction enthalpy, [/¹i _

Chilton-Colburn ¢ −factor for mass transfer, (dimensionless) Chilton-Colburn ¢ −factor for heat transfer, (dimensionless)

Intrinsic reaction rate constant, (for first order reaction, ± )

V>AW

Mass transfer coefficient from to gas-liquid, [A! /Y ∙ _

V>WZ

Liquid-solid mass transfer coefficient, [W! /Z ∙ _

V>AZ

Gas-solid mass transfer coefficient, [A! /Z ∙ _

W ±Z

Stagnant liquid-solid mass transfer coefficient, [W! /O ∙ _

VD

Thermal conductivity od the reactor wall, [/O ∙ ∙ °_

V> à

VÛ

VXAW >

VXAZ > VXWZ >

Average of the heat transfer coefficients of the fluid and solid, [/Z ∙ °_

Volumetric coefficient of gas-liquid mass transfer, ± Volumetric coefficient of gas-solid mass transfer, ± Volumetric coefficient of gas-solid mass transfer, ±

>

Adsorption equilibrium constant of component ) on catalyst active sites, W! />

¤l

Length of catalyst bed, O

W,> ¤/

Overall G-L mass transfer coefficient of component ) in the liquid phase, [W! /Y ∙ _ Characteristic length or shape factor of the catalyst particle, Z 70 Environment ACS Paragon Plus

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¤O

Reactor length, O

Soave-Redlich-Kwong EOS parameter, (dimensionless)

ÞÛx

Molecular weight of component ), ¹> />

@MN

Probability of axial displacement, (dimensionless)

ÞÛÙxÚ, Molecular weight of gas mixture, ¹Û>N /Û>N @>

Partial pressure of component ), Pa

@~,>

Critical pressure of component ), Pa

R>hh

Reactor tube inner radius, O

¦t

¦O,>

R×¥. R/Û R.

R

Total system pressure, Pa

Reduced pressure of component ), (dimensionless Reactor tube outer radius, O

Average radius of catalytic pore, Z

Reactor tube radius, O

Specific rate of reaction observed, (units depending on the type of kinetics)

rq

Chemical reaction rate, Äi /¹~M. ∙ Å

r¥

Constant of ideal gases, [8.314471 /° _

rst Ò/

Reaction rate of Fischer-Tropsch synthesis kinetics, Äi /¹~M. ∙ Å Total geometric external surface area of catalyst particle, Z

H

Time,

w*

System initial temperature, K

wi wM w~

wZ\]^ wD wO

Absolute temperature of the f phase, K Room temperature, K

Critical temperature of the component ), K Temperature at the surface solid, (K)

Temperature on the side of the wall, K

Reduced temperature of component ), K 71 Environment ACS Paragon Plus

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IJ

i

Superficial velocity of the , phase, Äi! /O ∙ Å

ÕO

Overall coefficient of heat transfer, [, /O ° _

º

Stoichiometric coefficient Phase volume ,, [i! _

ºO

The reactor volume, O!

-i

Flow factor, (dimensionless)

p>q º/ é>

i

§>

i

K

Ñ

Total geometric volume of catalyst particle, Z

Mass fraction of component ) in phase ,, ¹> /¹t

Mole fraction of component ) in the , phase, > /t

Axial reactor coordinate, O

Factor compressibility , (dimensionless)

Greek letters ä

äZ å

Probability of growth of the hydrocarbon chain

}~

Parameter related to the effect of the reactor wall, (dimensionless)

¶D

Parameter related to the Soave-Redlich-Kwong EOS

ΔO

Heat of reaction (enthalpy of reaction) of the Fischer-Tropsch synthesis, /¹>

Δ¦

Pressure drop, Pa/O

¶D

Δst Δ>

Parameter related to the effect of the reactor wall, (dimensionless) Thickness of a hollow cylinder, O

Enthalpy of FTS reaction at temperature w and pressure ¦, /¹>

Heat of vaporization/condensation (or latent heat) of component ), />

kl

! /O! _ Catalyst bed void fraction or catalyst bed porosity, [AW

Fi

External holdup of the , phase, [i! /O! _

k~M. FW,P FW,J

! Catalyst volumetric reaction of the washcoated monolith, [AW /O! _

Dynamic liquid holdup, (dimensionless) Static liquid holdup, (dimensionless)


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F/

i

b

Holdup of , (, =G or L) phase inside catalyst particle, [i! /Z! _

µ , µ

Effectiveness factor of a partially external and internal wetted pellet, (dimensionless)

0ii,MN

Axial effective thermal conductivity, [/O ∙ ∙ ° _

MN

Axial , phase thermal conductivity, [/O ∙ ∙ ° _

Z

Thermal conductivity of solid, [ /O ∙ ∙ ° _

i

>

Are constant Ergun type, (dimensionless)

Thermal conductivity of component ), Ä/[i ∙ ∙ °_Å

i 0ii,OMP Raial effective thermal conductivity, [/O ∙ ∙ ° _ i

OMP i

Radial , phase thermal conductivity, [/O ∙ ∙ °_

D

Thermal conductivity of the wall, [/O ∙ ∙ ° _

¸>

Dynamic viscosity of component ) in the , phase, Ä¹i /[i ∙ _Å

z

Radial coordinate inside spherical catalyst particle, Z

¸0ii i

i

¸Û>N,A nl

ni þ

2>W

2> ΦZ

3~M. 4

é>

Effective viscosity, Ä¹i /[i ∙ _Å

Viscosity of the gaseous mixture, Ä¹i /[i ∙ _Å

! Catalyst bulk (or bed) density, ¹Z /~M.

Density at process conditions of , phase, [¹i /i! _

Tortuosity factor for catalyst, [i /Z _

Liquid phase fugacity coefficient of component ), (dimensionless)

Vapor (gas) phase fugacity coefficient component ), (dimensionless) Thiele modulus of catalyst particle, (dimensionless)

! Catalyst volume fraction, ~M. /O!

Sphericity, (dimensionless)

Acentric factor for Soave-Redlich-Kwong EOS, (dimensionless)


Energy & Fuels

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Subscripts X

Xî

Related to ambient temperature

ÀXH

Cool (cooling medium), related to the heat transfer coefficients by convection

C,,

Dynamic

wÒ

Fischer-Tropsch

ç

Axial

À

Referred to reactor catalytic bed

-

Molecular carbon monoxide

%&

Catalyst

,

Effective

·

Fischer-Tropsch synthesis

Heat

)+

Inner

w

Phase

ℎ

Gas

)

Molecular hydrogen

Ö

Inside

-

Liquid

IH

Number species

)++

Hydraulic

Component index

¢

Interface

)î

Modified

¤

áÒ @ R

Reaction number

Mixture

Outer Particle Referring to the reactor and the reaction 74 Environment ACS Paragon Plus

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Energy & Fuels

RX-

ℎ

IRR

Radial Superficial, holdup of static liquid Shell

Ò

Surroundings

H

Total

Initial condition

Òr w

0

Solid phase Soave-Redlich-Kwong

Tube reactor Wall

Equilibrium condition

Superscripts ,

·

Phase

¤

Gas-solid

·¤

Gas phase

¤,

Liquid phase

·Ò

Gas-liquid

¤Ò

Contact between the liquid and the wall

áÒ Ò

Liquid-solid Number of species Solid phase

Dimensionless groups ç)

·X

·XÛ×P

Biot number, ℎ¤l /

Galilei number, ¹~ ¤!l n /¸

Modified Galilei number, dn/¸ ¹~ Ä

PÎ É ! OÉ

Å g or d

Ë ÏË 5 É Ë ÍË PÎ 6

7Ê ±É Ë


g

Energy & Fuels

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

áI

Nusselt number, [ℎ-/0 /_

rC

Reynolds number, [IJ n i -. /¸ i _

¦C

rCÛ×P

Peclet number, IJ ¤l /L i

\ Modified Reynolds number, d ±É È

¥ ÏÈ PÎ Í 7

g

ÒÀ

Schmith number, [¸/nL _

-A

Modified Lockhart-Martinelli number, S

Òℎ

Sherwood number, [i -/0 /L_

Û

Û

$ T nZ Ï Ï

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