Significance of Pressure Drop, Changing Molar Flow, and Formation of Steam in the Accurate Modeling of a Multi-Tubular Fischer–Tropsch Reactor with Cobalt as Catalyst

Abstract

A Fischer–Tropsch (FT) fixed-bed reactor was simulated with reactor models of different complexities to elucidate the impact of a pressure drop, a change in the total molar volume rate (induced by the reaction) along the tubes, and a change in the axial variation of the external radial heat transfer coefficient (external tube wall to cooling medium, here, boiling water) compared to disregarding these aspects. The reaction kinetics of CO conversion for cobalt as a catalyst were utilized, and the influence of inhibition of syngas (CO, H₂) conversion reaction rate by steam, inevitably formed during FT synthesis, was also investigated. The analysis of the behavior of the reactor (axial/radial temperature profiles, productivity regarding the hydrocarbons formed, and syngas conversion) clearly shows that, for accurate reactor modeling, the decline in the total molar flow from the reaction and the pressure drop should be considered; both effects change the gas velocity along the tubes and, thus, the residence time and syngas conversion compared to disregarding these aspects. Only in rare cases do both opposing effects cancel each other out. The inhibition of the reaction rate by steam should also be considered for cobalt as a catalyst if the final partial pressure of steam in the tubes exceeds about 5 bar. In contrast, the impact of an axially changing heat transfer coefficient is almost negligible compared to disregarding this effect.

1. Introduction

An option for producing liquid fuels, such as diesel oil or jet fuel, which is not based on crude oil, is the Fischer–Tropsch synthesis (FTS). Currently, the synthesis gas for FTS (CO, H₂) is produced from coal or natural gas, e.g., in South Africa, Nigeria, Uzbekistan, and Qatar. In future, other, mainly non-fossil resources may also be considered for FTS: H₂ can be produced via water electrolysis using renewable energy, such as solar and wind. CO₂ is separated from the off-gases from power plants, from the off-gases from production of steel, cement, or industrial chemicals, or, in the future, also from air, as, for example, currently tested in the so-called Orca carbon-capture plant located near Reykjavik in Iceland.

Concentrated CO₂ (after conversion to CO) is then used as a carbon source for FT syngas, a mixture of H₂ and CO in a typical ratio of 2, e.g., via reverse water–gas shift (CO₂ + H₂ → CO + H₂O) or, in future, potentially via the co-electrolysis of CO₂ and H₂O.

In two publications [1,2], we discussed the influence of the distribution of activity of a cobalt catalyst along the tubes on the operation of a cooled multi-tubular FT reactor with and without gas recycle and purge gas. However, the reactor model used so far to simulate the performance of a single tube—and, thus, in principle, of a technical FT reactor with up to 10,000 tubes by simply numbering up—still had drawbacks and simplifications as follows:

(1)

The velocity of the gas u_s in the tubes was assumed as constant, which was an (over)simplification.

(2)

The pressure drop, which affects the reaction rate and gas velocity (and, thus, residence time), was only calculated separately but not considered in the (as yet) isobaric reactor model.

(3)

The decline in the total molar flow in the tubes via FT reaction—about 20% for a CO conversion of 50% (see Section 3.2)—was neglected, although this, accordingly, reduces gas velocity, increases residence time, and, through this—as explained descriptively—the conversion (to be precise, a lower molar flow raises the residual concentrations of CO and H₂ and, thus, the reaction rate and conversion compared to disregarding this effect).

(4)

The heat transfer coefficient α_w,ex (external tube wall to boiling water) was assumed to be constant, but this depends on the pressure and temperature of the boiling water and—even more importantly—on the local value of the radial heat flux. Hence, α_w,ex varies along the tubes, and a maximum is located at the maximum of the axial temperature.

(5)

The effective radial thermal conductivity λ_rad and, also, the heat transfer coefficient on the internal side of the wall α_w,int depend on u_s, but they were calculated simply based on the initial value of u_s, which was assumed to be constant and, hence, fixed in each model calculation. In reality, a change in gas velocity in the axial direction, induced by a pressure drop and/or a decrease in total molar flow from the reaction, changes both λ_rad and α_w,in in the axial direction in the tubes.

(6)

The kinetic equations for the rate of CO conversion neglected any inhibiting influence of steam on the activity of the co-catalyst, which is not true for a high partial pressure of steam, particularly in the rear part of the tubes. Based on our experience, the influence is almost negligible for p_H2O < 5 bar, but it may be relevant if a higher value is reached [3].

We now implemented all these aspects in advanced reactor models and discussed step by step their individual and combined influence on the outcome, evaluated and compared via CO conversion and production of C₂₊-hydrocarbons (HCs), in order to elucidate the significance of each factor in the accuracy of a model of an FT reactor and cooled fixed beds in general beyond FT. All these aspects have not been analyzed to date for a fixed-bed FT reactor.

The kinetics of FTS with Co as a catalyst, the methods of FT reactor modeling, and the data of technical FT reactors were presented in our own recent publication [1] and in the literature, e.g., [4,5,6,7].

2. Objectives and Methodology

2.1. Intrinsic and Effective Reaction Kinetics of Fischer–Tropsch Synthesis (FTS)

The main reaction of FTS is the formation of C₂₊-hydrocarbons:

CO + 2H₂ → (-CH₂-) + H₂O  ∆_RH_CH2⁰₂₉₈ = −152 kJ mol⁻¹

(1)

The term (-CH₂-) represents a methylene group of a paraffinic hydrocarbon. In a (formal) kinetic description of the FTS, the formation of methane is often treated as a separate reaction:

CO + 3H₂ → CH₄ + H₂O  ∆_RH_CH4⁰₂₉₈ = −206 kJ mol⁻¹

(2)

The equations of the chemical rates of CO to CH₄ and C₂₊-HCs in the absence of mass transfer limitations with cobalt as a catalyst (following an approach according to Langmuir–Hinshelwood) were already reported with all kinetic parameters [1]. Here, we only treat important aspects, such as the coefficient C_a (see below) and the impact of pore diffusion, in order to facilitate reading.

The intrinsic chemical rate of CO (without influence of mass transfer) is given by the formation rate of CH₄ and C₂₊-HCs, as CO₂ formation by water–gas shift is negligible for a Co catalyst:

$$r_{m, CO}=-\frac{d{\dot{n}}_{CO}}{d{m}_{cat}}=C_a \left(r_{m,CO, C{H}_4}+r_{m,CO, C_{2+}}\right)$$

(3)

The intrinsic rates r_m,CO,CH4 and r_m,CO,C2+ were experimentally determined with a Pt-promoted (0.03 wt.% Pt to facilitate Co reduction) 10 wt.% Co/γ-Al₂O₃ catalyst [1]. The coefficient of activity C_a in Equation (3) considers the Co content and thus the purely chemical activity. C_a is set to one for 10% Co, and an increase/decline in C_a is considered to be realized by a rise or drop in the Co content. FT catalysts typically contain up to 30% Co (C_a ≈ 3), a value mostly assumed in this study.

In this work, we have also extended the intrinsic rate equations by a term considering the inhibition by steam, which is relevant if a high concentration is reached in the rear part of the tubes. The re-evaluation of our experiments [3] yields the following (rough) approximation:

$$r_{m, CO,H_{2}O}=r_{m, CO} \left(1-\frac{c_{H_{2}O}}{{472 \mathrm{mol} \mathrm{m}}^{-3}} \right)$$

(4)

For example, a partial pressure of steam of 5 bar (120 mol/m³ at 230 °C) leads to a decline in the intrinsic chemical reaction rate by 25%.

Equations (3) and (4) only reflect the intrinsic rate, but pore diffusion limitations decrease the effective rate compared to the intrinsic one for a particle size of several millimeters (here, d_p = 3 mm), relevant for FT fixed bed reactors to avoid an excessive pressure drop; the pores are filled with liquid hydrocarbons, and diffusion of CO and H₂ in liquid HCs is slow. As outlined in [1,2], the effectiveness factor η_pore and the related Thiele modulus ϕ are:

$${\eta }_{pore}=\frac{r_{m,CO,eff}}{r_{m,CO}}=\frac{\tanh \varphi }{\varphi } \approx \frac{1}{\varphi } \mathrm{for} \varphi >2$$

(5)

$$\varphi =\left\{\frac{d_p}{6}\sqrt{\frac{{\rho }_{cat}}{D_{eff,CO,liq} \frac{R T}{ H_{CO}} }}\right\} \sqrt{\frac{r_{m,CO}}{ c_{CO} }}=C_{\varphi } \sqrt{C_a\frac{\left(r_{m,CO, C{H}_4}+r_{m,CO, C_{2+}} \right)}{ c_{CO} }}$$

(6)

For the particle diameter of 3 mm, as assumed here, C_ϕ is 300 kg^0.5 s^0.5 m^−1.5; see previous publication [1]. The effective rate of CO conversion is then given based on Equations (3)–(6) by:

$$r_{m,CO,eff}={\eta }_{pore} r_{m,CO}$$

(7)

For a diameter of the particles of 3 mm, η_pore is lower than 1 above 180 °C and reaches a value of around 0.2 for 240 °C (C_a = 3) [1,2]. The mean molar H₂-to-CO ratio within the particles is then higher compared to the free gas phase with a value of about two. The unwanted formation of CH₄ then rises and lower HCs are formed, as the diffusion coefficient of H₂ in liquid HCs is by a factor of two higher compared to CO. This impact is strong above 240 °C and the CH₄ selectivity (S_CH4) then exceeds 20% by weight compared to 10% in the absence of diffusion limitations [1]. In this study, we therefore limited the temperature to 240 °C and assumed that S_CH4 is constant at 20%, i.e., 80% of CO is converted to C₂₊-HCs. We fixed the H₂-to-CO ratio in the fresh syngas to 2.2, and the H₂ conversion then equals that of CO, which simplifies all mass balances [2].

It should be mentioned that a limitation of the effective reaction rate by external mass transfer does not play a role in Fischer–Tropsch synthesis as FTS is a rather slow reaction; the strong influence of internal mass transfer only occurs if the pores are filled with liquid hydrocarbons. In case of gas-filled pores, i.e., in the initial phase of FTS with a fresh catalyst, even internal diffusion limitations are negligible.

If inhibition of the effective rate by steam is considered, Equations (5) and (6) with r_m,CO,H2O instead of r_m,CO are valid. For a strong limitation by pore diffusion (ϕ > 2: r_m,CO,H2O,eff = η_pore r_m,CO,H2O~r_m,CO,H2O^0.5), as typical for fixed-bed FT synthesis, inhibition of the effective rate by steam is weaker than that of the intrinsic rate, e.g., for a partial pressure of steam of 5 bar, the effective rate drops only by 13% and not by 25%, as stated before for the decline of the intrinsic rate.

2.2. Models (Examined in This Work) of a Multi-Tubular FT Reactor Cooled by Boiling Water

The model to simulate a single tube of a multi-tubular fixed-bed reactor is a pseudo-homogenous two-dimensional model already presented [1,2].

The mass and the heat balance in a differential axial tube section are given by the following Equations (8) and (9):

$$\frac{d\left(c_{i }{u}_s\right)}{dz}=\left({\nu }_{i,R1} r_{m,CO,R1,eff}+{\nu }_{i,R2} r_{m,CO,R2,eff}\right) {\rho }_{bed}$$

(8)

$$c_p {\rho }_g\frac{d\left(T u_s\right)}{dz}={\lambda }_{rad}\frac{1}{r}\frac{dT}{dr}+{\lambda }_{rad}\frac{d^{2}T}{d{r}^2}+\left( r_{m,CO,R1, eff}\left(-{\mathsf{\Delta }}_R{H}_{R1}\right)+r_{m,CO,R2,eff}\left(-{\mathsf{\Delta }}_R{H}_{R2}\right)\right) {\rho }_{bed}$$

(9)

Radial temperature gradients in the bed are taken into account to achieve a reliable calculation of the operation of the FT reactor (temperature profiles in the axial and radial direction, CO conversion, and thermal stability). Each reactor tube has an inner diameter d_t,int of 3 cm. The heat produced by FTS is radially transferred through the pseudo-homogenous phase consisting of catalyst and gas from the fixed bed to the inner tube wall. The radial heat flux within the bed is governed by the radial effective thermal conductivity λ_rad and the internal heat transfer coefficient α_w,int, taking into account the thermal resistance very near the internal side of the wall resulting from the high porosity of the bed at the wall. Finally, heat transfer by conduction in the wall, which only negligibly contributes to the total thermal resistance, and the heat transfer from the external side of the tubes to the boiling water are also considered in the reactor model.

Both the axial dispersion of mass and of heat were deliberately neglected in the reactor model as they are only relevant if very steep axial gradients of concentration or temperature over a length of a few particles are present. This may be different for radial dispersion of mass if the radial difference in temperature in the fixed bed becomes large, e.g., near or during temperature runaway. We then may have a difference of 50 K or more over a length of about 5 particles (radius of tube: 15 mm; particle diameter 3 mm) and not only about 10 K as during “normal” operation (see Figure S14 in the Supporting Information), and the reaction rate near the (cooler) wall is then much lower compared to the center region of the tubes. In the former case, the syngas conversion is relatively low (high concentration) near the wall and high (low concentration) in the center region. Hence, radial dispersion then may lead to a radial “mixing”, i.e., to an adjustment of radial concentration gradients by “dispersive” supply of CO and H₂ from the near-wall region to the tube center. This effect may then, for example, decrease the ignition temperature to a certain extent compared to the case of no radial dispersion of mass. This aspect is, here, not further considered but will be analyzed in future work in more detail.

Table 1. Parameter values used to model the FT reactor with gas recycle and a total CO conversion of 95% (details on heat transfer in Supporting Information).

Constant Parameters (230 °C, 30 bar) Not Varied during Modeling		Value
Length of reactor (single tube) Lt		12 m a
Internal tube diameter dt,int		3 cm
Thickness of tube wall swall		0.3 cm
Content of CO (in fresh syngas) yCO,fresh,SG		0.3125
Content of H2 (in fresh syngas) yH2,fresh,SG = 1 − yCO,fresh,SG		0.6875
Total pressure ptotal		30 bar
Diameter of spherical catalyst particles dp		3 mm
Bulk density of bed/catalyst ρbed		960 kg m−3
Porosity of fixed bed εbed		0.4
Heat capacity of gas mixture cp		29 J mol−1 K−1
Thermal conductivity of gas mixture λg		0.016 W m−1 K−1
Kinematic viscosity of gas mixture νg		2.3 × 10−6 m2 s−1
Thermal conductivity of wall material (steel) λwall		15 W m−1 K−1
Parameters varied during modeling	Comment	Typical value
Content of CO (inlet of reactor) yCO,reactor,in	depends on XCO and corresponding recycle ratio R	0.19 b
Content of H2 (inlet of reactor) yH2,reactor,in		0.42 b
Content of CH4 (inlet of reactor) yCH4,reactor,in c		0.39 a
Pressure drop ∆pbed	depends on us	5.6 bar b
Initial superficial gas velocity us,z=0 (230 °C, 30 bar)	varied	1 m s−1
Heat transfer coefficient (wall to boiling water) αw,ex	depends on us d	1850 W m−2 K−1
Effective radial thermal conductivity λrad		8.4 W m−1 K−1
Heat transfer coefficient (bed to internal tube wall) αw,int		1540 W m−2 K−1

^a The length of the tubes of industrial multi-tubular FT reactors are typically in a range of 12 to 20 m [6,7]. ^b Values according to model M4 for u_s,z=0 = 1 m/s and C_a = 3. ^c The recycle and purge gas is considered to contain only unconverted CO and H₂ and CH₄. It is assumed that H₂O and all C₂₊-HCs are separated as liquids downstream of the reactor (see [2]). ^d λ_rad and α_w,int depend on u_s, which changes along the tubes due to pressure drop and decreasing total molar flow rate. The values listed are initial values at the tube inlet for u_s,z=0 = 1 m/s. α_w,ex mainly depends on the local radial heat flux. The listed value is the one at z = 1.9 m, where the maximum axial temperature of 240 °C is just reached for u_s = 1 m/s and C_a = 3. α_w,ex also includes heat conduction through the wall; see Equation (S18) in the Supporting Information.

shows parameter values utilized to model an FT reactor with gas recycle (unconverted CO and H₂, and CH₄), a purge gas stream, and an assumed total CO conversion of 95%. The values are given for a superficial gas velocity of 1 m/s.

It should be mentioned that the results of modeling of an FT reactor depend on the specific catalyst used, e.g., on the main active metal, here, Co and not Fe, as also industrially used for FTS. Here, we concentrate on cobalt as a catalyst and used our own experimental results of the kinetics.

As mentioned in the introduction, we have now modified and improved the reactor model by considering the pressure drop as well as the decline in the total molar flow, which both influence the gas velocity and residence time. An axially changing gas velocity also influences the radial conductivity λ_rad and the internal heat transfer coefficient α_w,int. All heat transfer parameters were calculated by literature correlations [8,9,10,11,12,13,14]; see Supporting Information.

The influence of pressure of the boiling water and of the radial heat flux was also taken into account for an accurate determination of the local value of the external heat transfer coefficient α_w,ex.

Finally, we also analyzed the impact of the inhibition of the CO reaction rate by steam on the outcome of a model of a cooled multi-tubular FT reactor.

Table 2. Reactor models used here for a multi-tubular fixed-bed FT reactor.

Label of Model	Parameter Considered (+)or Neglected (−)			Comment
	$\mathsf{\Delta }{\dot{\mathit{n}}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$	pbed	Impact of H2O
M0	−	−	−	“Simple” model used in previous publications [1,2] $(\mathrm{constant} u_s, p_{\mathrm{total}}, {\dot{n}}_{total}$, and αw,ex (1 kW m−2 K−1)
M1	−	−	−	As model M0, but accurate calculation of αw,ex (for details see Section 3.1)
M2n	+	−	−	$\mathrm{Used} \mathrm{to} \mathrm{show} \mathrm{only} \mathrm{effect} \mathrm{of} \mathsf{\Delta }{\dot{n}}_{total}$ (and of corresponding axial drop of us), i.e., neglecting pbed
M2p	−	+	−	$\mathrm{Used} \mathrm{to} \mathrm{show} \mathrm{only} \mathrm{effect} \mathrm{of} p_{bed} (\mathrm{and} \mathrm{of} \mathrm{corresponding} \mathrm{axial} \mathrm{rise} \mathrm{of} u_s), \mathrm{i}.\mathrm{e}., \mathrm{neglecting} \mathrm{change} \mathrm{of} {\dot{n}}_{total}$
M3	+	+	−	Accurate model, if inhibition by steam is negligible
M4	+	+	+	As M3, but also considering inhibition by steam

introduces the used models and the respective parameters considered or deliberately neglected to clearly elucidate step by step the influence of all these aspects on the outcome of each model, i.e., the FT reactor performance, mainly “measured” by X_{CO,per pass} and production rate of C₂₊-HCs.

Details on the general performance of an FT reactor including a gas recycle and a purge gas stream were already outlined in our very recent publication [2]. Here, we have always assumed a total syngas conversion (CO, H₂) of 95%, realized by a respective recycle ratio R and a purge gas stream, which is needed as an outlet for all unwanted methane produced as a by-product.

The gas velocity and the syngas composition at the reactor entrance, depending on the total and per pass conversion of CO (X_CO,total, X_{CO,per pass}), were varied, but the geometry of the tubes (internal diameter 3 cm, length 12 m), the initial total pressure (30 bar), and T_max (240 °C) were fixed. The critical cooling (ignition) temperature T_ig, where thermal runaway occurs, was determined in every modeling case. For a safe operation, the maximum of the cooling temperature was fixed to be 5 K below the temperature of ignition (T_ig).

The equations of the mass and heat balances (listed in Supporting Information) were solved by the program Presto, a solver of differential equations (CiT GmbH, Rastede, Germany).

During reactor modeling, the catalytic activity (coefficient C_a) was mostly regarded as constant. Only in two cases an axial activity distribution was considered, a two-zone fixed-bed and a graded distribution with C_a,initial until T_max of 240 °C was reached followed by a continuous increase in C_a to keep the temperature at 240 °C, as presented in the Supporting Information (Table S1).

In this work, we use a value for C_a of three (ideally 30% Co) as the appropriate value for a superficial gas velocity around 1 m/s (and C_a = 2 for 0.5 m/s) to guarantee a safe operation of the reactor; details are given in a former publication [2].

The initial superficial gas velocity at the entrance of the tubes (u_s,z=0) was varied in a broad regime from 0.25 m/s to 1.6 m/s, as this has a strong influence on ∆p_bed, X_{CO,per pass}, production rate of C₂₊-HCs, heat transfer parameters, and on the change in u_s in axial direction:

○

If ∆p_bed is low (low u_s,z=0), u_s drops along the tubes and, in return, the residence time (compared to constant u_s) increases by the then dominating effect of the drop in the total molar flow rate. This, in return, has an influence on the local values (axial direction) of α_w,int and λ_rad. In total, X_{CO,per pass} is then higher compared to a model (as M0 or M1, Table 2) simplified assuming a constant u_s.

○

If ∆p_bed is high (high u_s,z=0), this yields an increase in u_s in axial direction (decrease in residence time compared to constant u_s), as the drop in u_s by the decreasing total molar flow rate is then overcompensated; this again has an impact on the local values of α_w,ex and λ_rad, and, in total, X_{CO,per pass} is then lower compared to a model assuming constant u_s.

○

For a “medium-sized” ∆p_bed (medium-sized u_s,z=0), we may obtain an almost constant gas velocity along the tubes, i.e., the opposite influence of ∆p_bed and drop in total molar flow on u_s cancel each other out. In this rare case, a model with or without considering both aspects coincidentally yields similar results with regard to X_{CO,per pass} or production rate of C₂₊-HCs.

3. Results of Simulation of a Single Tube of a Cooled Multi-Tubular FT Reactor

3.1. Influence of External Heat Transfer Coefficient α_w,ex on Reactor Modeling

In contrast to the “simple” model M0, used in our previous publications [1,2], all other models consider the influence of the pressure of the boiling water and of the local radial heat flux, which changes in axial direction (Figure 1), on the external heat transfer coefficient α_w,ex based on literature correlations (Supporting Information) [12,13,14]. This effect was neglected in model M0, where an estimated constant value of α_w,ex (1 kW m⁻² K⁻¹) was used (Figure 1). The heat transfer parameters λ_rad and α_w,int, both depending via Re_p on the gas velocity, Figure S1, were calculated by literature correlations [4,5,6,7,8] (Supporting Information). For M0 and M1, both assuming constant u_s and p_total, λ_rad and α_w,int are constant (Figure 1 and Figure 2).

For model M1 (and also M2 to M4), the external heat transfer coefficient α_w,ex passes a maximum at z = 2 m (Figure 1), corresponding to the location of the maximum temperature (Figure 3) and the highest radial heat flux. The comparison of M0 and M1, which only differ in value and calculation of α_w,ex (Figure 1), shows that the model data are similar, e.g., X_{CO,per pass} is 44.3% (model M0) and 45.5% (M1),

Table 3. Data of multi-tubular FT reactor according to the models M0 (with constant value of α_w,ex) and M1 (improved calculation of α_w,ex; see text and Figure 1) for constant u_s of 1 m/s (230 °C, 30 bar) and C_a = 3. Conditions: X_CO,total = 95%; S_CH4 = 20%; molar H₂-to-CO ratio = 2.2; T_max = 240 °C; 1825 mol/h syngas per tube at reactor inlet.

Tcool in °C	XCO,per pass in %	yCH4,reactor,in in %	R	Prod. of C2+-HCs per Tube in kgC per h	Reactor Model and Parameter Considered (+) or Neglected (−)
					Model	αw,ex = f(z)	$\mathsf{\Delta }{\dot{\mathit{n}}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$	pbed
219.9	44.3 a	38.7	2.50	1.48	M0	−	−	−
223.0	45.5	38.1	2.38	1.54	M1	+ (see Figure 1)	−	−

^a Throughout this work, we have chosen a precision for the CO conversion (and other parameters) of three significant digits (e.g., here, 44.3%). A higher precision is not justified (also with regard to the insufficient knowledge of the “exact” values of kinetic, heat transfer parameters, etc.). In addition: for a constant maximum axial temperature (here, 240 °C), the value calculated by the model is not exactly 240.00 °C but typically in a range of 239.98 and 240.02 °C to limit the time needed for the variation in the cooling temperature to reach the target value of 240.00 °C. Hence, the normalized rate and thus the conversion are then in a range of 0.9993 and 1.007 of the “true” value of 1.000.

. Nevertheless, the more accurate calculation of α_w,ex, implemented in all models except model M0, should be preferred and was, here thereafter, utilized.

3.2. Influence of Pressure Drop and Change in Total Molar Gas Flow on Reactor Modeling

Table 4. Data of multi-tubular fixed-bed FT reactor according to the reactor models M1, M2_n, M2_p, and M3 for u_s,z=0 = 1 m/s (230 °C, 30 bar) and C_a = 3. Conditions: X_CO,total = 95%, S_CH4 = 20%, molar H₂-to-CO ratio = 2.2, T_max = 240 °C, 1825 mol/h syngas per tube at reactor inlet. Further details are shown in the Figure S4.

Tcool in °C	XCO,per pass in %	yCH4,reactor,in in %	R	Prod. of C2+-HCs per Tube in kgC per h	Reactor Model and Parameters Considered (+) or Neglected (−)
					Model	$\mathsf{\Delta }{\dot{\mathit{n}}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$	pbed	Impact of H2O
223.0	45.5	38.1	2.38	1.54	M1	−	−	−
222.5	49.1	36.2	2.05	1.72	M2n	+	−	−
223.5	42.6	39.6	2.69	1.41	M2p	−	+	−
222.8	45.8	37.9	2.35	1.56	M3	+	+	−

and Figure 3, Figure 4 and Figure 5 show the results of the reactor simulation by models M1, M2_p, M2_n, and M3 for an initial superficial gas velocity u_s,z=0 of 1 m/s (230 °C, 30 bar).

According to the “correct” model 3 (within the four models compared here), which considers ∆p_bed as well as $\mathsf{\Delta }{\dot{n}}_{total}$, the gas velocity u_s is almost constant, as discussed below in detail. Nevertheless, both the effective radial thermal conductivity λ_rad (Figure 2) and the internal heat transfer coefficient α_w,int (Figure 1) decrease to a certain extent along the tubes by the decline in Re_p (=u_s d_p/ν_g), Figure S2, as the gas viscosity (ν_g) rises with decreasing pressure (ν_g~1/p_total). But this effect only leads to minor differences in the axial profiles of temperature (center of tube) and reaction rate for models M1 and M3 (Figure 3 and Figure S4), and the CO conversion per pass and production rate of C₂₊-HCs are very similar, Table 4 (first and fourth row).

If only the decline in ${\dot{n}}_{total}$ (and not ∆p_bed) is implemented in the model (as for M2_n), the conversion is much larger (49.1%) as for the models M1 (45.5%) and M3 (45.8%), see Table 4. For model M2_p, considering only ∆p_bed, this is reversed and X_{CO,per pass} is only 42.6%. Hence, an accurate model should consider both ∆p_bed and $\mathsf{\Delta }{\dot{n}}_{total}$ (as M3) and not only one of these two aspects (M2_p and M2_n), as this even leads to less reliable data than neglecting both aspects (M1).

Figure 5 shows the individual (left) and combined influence (right) of temperature, ∆p_bed, and drop in total molar flow rate on the gas velocity u_s in more detail for model M3. For an initial value of u_s,z=0 of 1 m/s, the influence of $\mathsf{\Delta }{\dot{n}}_{total}$ and ∆p_bed on u_s cancel each other out; the impact of temperature on u_s is negligible, as the (mean) value only varies in a range of 223 to 240 °C.

Table 5. Data of a multi-tubular FT reactor according to models 1, 2_n, 2_p, and 3 for u_s,z=0 = 0.5 m/s (230 °C, 30 bar) and C_a = 2. Conditions: X_CO,total = 95%; S_CH4 = 20%; molar H₂-to-CO ratio = 2.2; T_max = 240 °C; 912.5 mol/h syngas per tube at reactor inlet.

Tcool in °C	XCO,per pass in %	yCH4,reactor,in in %	R	Prod. of C2+-HCs per Tube in kgC per h	Reactor Model and Parameters Considered (+) or Neglected (−)
					Model	$\mathsf{\Delta }{\dot{\mathit{n}}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$	pbed	Impact of H2O
218.0	57.7	31.4	1.41	1.08	M1	−	−	−
216.6	63.4	27.9	1.09	1.25	M2n	+	−	−
218.2	57.1	31.7	1.45	1.07	M2p	−	+
216.9	62.8	28.3	1.12	1.23	M3	+	+	−

and the Figure 6 and Figure 7 show the results of models M1, M2_n, M2_p, and M3 but, now, for a relatively low initial gas velocity u_s of 0.5 m/s (230 °C, 30 bar). For this rather low gas velocity, ∆p_bed is almost negligible, only 1.2 bar compared to 5.6 bar for u_s,z=0 = 1 m/s, and the gas velocity substantially decreases along the tubes by up to about 25% (Figure 7) if models M3 and M2_n are used, both considering the drop of the molar flow rate by the FT reaction. In return, the CO conversion per pass (62.8% and 63.4%, respectively) is then higher compared to the “simple” model 1 (56.9%; Table 5), which assumes (too simplifying) a constant gas velocity. Hence, for a low gas velocity, a model not implementing $\mathsf{\Delta }{\dot{n}}_{total}$ should not be used or only for rough estimations. Model M2_p, only considering ∆p_bed compared to M1, shows that the implementation of (low) ∆p_bed only is of minor importance (Table 5).

Two additional figures are given in the Supporting Information: Figure S4 depicts axial profiles of the effective rate at r = 0 (center of tube) for model 1 and 3 for u_s,z=0 = 1 m/s and Figure S5 for 0.5 m/s. Again, note that effectiveness factor η_pore (center of tube, i.e., at r = 0) is, here, always only around 0.2.

3.3. Influence of Inhibition of Steam on the Performance of an FT Reactor

Table 6. Data for model M3 (no inhibition by steam) and model M4 (inhibition). Conditions: X_CO,total = 95%; S_CH4 = 20%; molar H₂-to-CO ratio = 2.2; T_max = 240 °C; 1825 or 912 mol/h syngas per tube at inlet for u_s,z=0 of 1 or 0.5 m/s (details in Figures S6 and S7).

Tcool in °C	XCO,per pass in %	yCH4,reactor,in in %	R	Production of C2+-HCs per Tube in kgC per h	Reactor Model and Parameters Considered (+) or Neglected (−)
					Model	$\mathsf{\Delta }{\dot{\mathit{n}}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$	pbed	Impact of H2O
us,z=0 = 1 m/s (230 °C, 30 bar); Ca = 3
222.8	45.8	37.9	2.35	1.56	M3	+	+	−
223.1	44.4 a	38.7	2.49	1.49	M4	+	+	+
us,z=0 = 0.5 m/s (230 °C, 30 bar); Ca = 2
216.9	62.8	28.3	1.12	1.23	M3	+	+	−
217.6	59.0 b	30.6	1.34	1.12	M4	+	+	+

^a p_H2O at the end of the tubes (z = 12 m) is 2.5 bar, i.e., the intrinsic rate (Equation (4) is about 13% lower compared to no inhibition by steam; the effective rate then declines by only 7%, Equations (5) and (6). ^b p_H2O at z = 12 m is 4.8 bar, i.e., the intrinsic rate is 25% lower compared to no inhibition; the effective rate then declines by 13%.

compares the influence of considering the inhibition of the reaction rate of CO conversion by steam on the reactor modeling for initial gas velocities (at 230 °C and 30 bar) of 0.5 and 1 m/s. Now, only the “advanced”, most accurate models M3 and M4 are considered, i.e., both $\mathsf{\Delta }{\dot{n}}_{total}$ and ∆p_bed are implemented but either without (M3) or with (M4) steam inhibition.

For model M4, which correctly considers inhibition by steam, the CO conversion per pass is, as expected, in general, lower at 44.4% compared to 45.8% for M3 for u_s,z=0 of 1 m/s, which is still a small deviation, as p_H2O only reaches 2.5 bar at the end of tubes. For a lower initial gas velocity of 0.5 m/s, the deviation of conversion is already quite pronounced at 59% for model M4 compared to 62.8% for M3. Now, a rather high value of p_H2O of 4.8 bar is reached at the reactor outlet. In conclusion, an accurate FT reactor model should include inhibition by steam if the conversion per pass is above 50% and if p_H2O finally approaches 5 bar, respectively.

In order to spotlight the even more pronounced influence of inhibition by steam on the reactor modeling for X_CO >> 50%, we then used “extreme” parameters, a low value for u_s,z=0 of 0.5 m/s, a high value of C_a of 4, and a syngas consisting only of CO and H₂. In addition, the reactor was (contrary to industrial reality) considered as isothermal (∆_RH_i was then just zeroized in the model), and a high temperature of 240 °C was chosen to reach a high CO conversion and, thus, a high partial pressure of steam in the rear part of the tubes (

Table 7. Data of isothermal FT reactor (240 °C) for model M4 (considering ∆p_bed, change in total molar flow, and inhibition by steam) and for M3 (as M4 but without influence of steam) for u_s,z=0 of 0.5 m/s and C_a of 4 (syngas with 31.3% CO and 68.7% H₂).

CO Conversion XCO at Axial Position z			pH2O (yH2O) at z = 12 m	us at z = 12 m	Reactor Model and Parameters Considered (+) or Neglected (−)
					Model	$\mathsf{\Delta }{\dot{\mathit{n}}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$	pbed	Inhibition by H2O
3 m	6 m	12 m
30.8%	57.7%	92.0%	18.4 bar (63%)	0.24 m/s	M3	+	+	−
29.5%	53.1%	81.4%	14.3 bar a (49%)	0.27 m/s	M4	+	+	+

^a The intrinsic rate (Equation (4)) at z = 12 m is 73% lower (M4) compared to no inhibition by steam (M3), and the effective rate declines by 52%, see Figures S6 and S7. ∆p_bed is, in both cases, low (0.9 bar).

, see also Figures S6 and S7 in the Supporting Information). Now, the CO conversion is 81% and 92% with and without inhibition. Thus, model M4 considering inhibition by steam is then clearly needed for a reliable simulation.

3.4. Impact of Gas Velocity on Modeling an FT Reactor if Pressure Drops, Change in Total Molar Flow Rate, and Inhibition of Reaction Rate by Steam Are Correctly Considered

Finally, the initial gas velocity u_s,z=0 was varied for the “best” model M4 and the activity C_a was 3 (Figure 8, Figure 9 and Figure 10,

Table 8. Results of “best” reactor model M4: impact of initial gas velocity u_s,z=0 on the performance of a multi-tubular fixed-bed FT reactor for an axial activity C_a of 3 (X_CO,total = 95%; S_CH4 = 20%; molar H₂-to-CO ratio = 2.2; T_max = 240 °C). The axial profiles of η_pore (r = 0) for u_s (230 °C, 30 bar) of 0.5 and 1 m/s are shown in Figure S19.

us, in m/s		pbed in bar	Tcool in °C	pcool in bar	XCO,per pass in %	yCH4,reactor,in in %	R	C2+-HCs/Tube in kgC per h
z = 0 a	z = 12 m
0.23	0.17	0.3	199.6 b	15.2	66.1	26.5	1.04	0.64
0.48	0.38	1.2	213.3 c	20.1	63.2	28.4	1.10	1.24
0.73	0.66	2.9	219.4	22.9	53.1	34.0	1.73	1.44
0.99	1.04	5.6	223.1	24.6	44.4	38.7	2.49	1.49 (best case)
1.24	1.63	9.8	225.5	25.7	36.8	42.4	3.46	1.45
1.49	3.09	16.7	227.4	26.6	30.0	44.8	4.74	1.36
1.59	5.19	21.5 d	228.0	26.9	26.4	46.8	5.69	1.23
For comparison and illustration: Hypothetic cases for absence of mass transfer resistance by pore diffusion (ηpore = 1)
0.46	0.44	1.4	191 e	12.9	28.9	45.8	5.01	0.43
0.94	1.07	5.7	198 f	14.9	28.4	46.0	5.13	0.84

^a The values of u_s,z=0 for simulation are 0.25, 0.5, 0.75, 1, 1.25, 1.5, and 1.6 m/s and are related to 230 °C and 30 bar; the listed values at the reactor entrance are slightly lower, as T_cool = T_in < 230 °C. ^b In this case, the maximum temperature of 240 °C cannot be realized (runaway): the ignition temperature (T_ig) is 204.6 °C and, for T_cool = 199.6 °C (5 K below T_ig), T_max is only 226 °C. ^c In this case, the maximum of 240 °C can just be realized without risk of thermal runaway; the ignition temperature (T_ig) is 220 °C and, thus, T_cool,max is 215 °C. ^d ∆p_bed and u_s,z=12m increase strongly for u_s,z=0 > 1.6 m/s, see also Figure S15, e.g., for u_s,z=0 = 1.65 m/s (230 °C, 30 bar), we obtain u_s,z=12m = 10.2 m/s (!) and ∆p_bed = 25.5 bar (p_final = 4.5.bar). For such low total pressures, the kinetics were not evaluated and the model is not really reliable anymore. ^e T_max of 240 °C cannot be realized, T_ig is 196 °C, and, for T_cool,max = 191 °C, T_max is only 199 °C. ^f T_max of 240 °C cannot be realized, T_ig is 203 °C, and T_cool,max = 198 °C, and T_max = 208 °C. If C_a is decreased, e.g., to a value of two, T_ig, T_cool,max, and T_max are higher at 208 °C, 203 °C, and 212 °C, respectively, but, nevertheless, the CO conversion per pass is even lower (27.4%).

). As already outlined at the end of Section 2.2, the degree and direction (decline or increase) of the change in gas velocity u_s strongly depends on the initial velocity.

If ∆p_bed is less than 3 bar (u_s,z=0 ≤ 0.75 m/s), u_s decreases along the tubes by the decreasing total molar flow rate (Figure 8), and the residence time rises compared to a constant u_s by the dominating effect of the drop in the total molar flow rate; this, in return, increases X_{CO,per pass}, i.e., the “true” value (model M4) is higher compared to M1 neglecting $\mathsf{\Delta }{\dot{n}}_{total}$ and ∆p_bed (Figure 10).

For ∆p_bed > 9 bar (u_s,z=0 ≥ 1.25 m/s), the effect is reverse; then, u_s increases in axial direction, the residence time decreases compared to a constant u_s, and X_{CO,per pass} (“correct” value according to model 4) is lower compared to model 1, oversimplifying an assumed constant u_s in axial direction.

For a moderate value of ∆p_bed of around 6 bar (u_s,z=0 = 1 m/s), the gas velocity almost remains constant in axial direction (Figure 8), i.e., the influence of $\mathsf{\Delta }{\dot{n}}_{total}$ and ∆p_bed on u_s cancel each other out (see Figure 5). For this specific case, a model with or without considering these two aspects coincidentally leads to similar results of X_{CO,per pass} and production rate of C₂₊-HCs (Figure 10). It should be also noted that, for this superficial gas velocity, the maximum of the rate of production of C₂₊-HCs is reached (Figure 10, Table 8), which is contrary to the too simple model M1.

For completeness, it should be mentioned here that the effectiveness factor (pore diffusion) is below 0.7 for all cases listed in Table 8 (except the last two rows, where any influence of pore diffusion is deliberately neglected), even at the inlet of the fixed bed with the initially low temperature of T_cool, and always around 0.2 at z ≈ 2 m, where the maximum of 240 °C is reached (Figure 9). Hence, the influence of internal mass transfer on the effective reaction rate is always strong for the given conditions, a well-known phenomenon for FT fixed-bed synthesis. For example, in the best case with regard to production of C₂₊-HCs (u_s,z=0 = 1 m/s), η_pore (r = 0) is initially 0.34 at the entrance of the tubes (223 °C), drops to a minimum of 0.18 at z = 2 m, where T_max (240 °C) is reached, and then only slightly increases towards the end of the tubes to a value of 0.20 at z = 12 m (234 °C).

It is interesting that, in the purely hypothetic but technically not at all realistic case of the absence of any pore diffusion limitations in FT synthesis, i.e., if η_pore = 1 is used for the reactor simulation, X_{CO,per pass} and, thus, also the production of C₂₊-HCs per tube would unexpectedly even strongly decrease, e.g., for u_s,z=0 = 1 m/s from 1.49 to 0.84 kg_C h⁻¹ (last row in Table 8). The allowable value of T_cool with regard to thermal runaway of the reactor is then only 198 °C, the maximum temperature only 208 °C, and X_{CO,per pass} drops to 28% compared to 44% in the case correctly considering pore diffusion. For u_s,z=0 = 0.5 m/s, this effect is more pronounced and the output of C₂₊-HCs per tube drops to 0.43 kg_C h⁻¹ compared to 1.24 kg_C h⁻¹ if pore diffusion is (correctly) included in the reactor simulation; see second to last row in Table 8.

The reason for this on first sight really surprising effect is the much higher reactor sensitivity if pore diffusion would not dampen the effective rate and the apparent activation energy (by a factor of about two). The thermal reactor stability without pore diffusion limitation is then only reached at a much lower cooling and maximum temperature; see Table 8 (last two rows). In other words, for FT synthesis, pore diffusion unexpectedly not only “helps” with regard to thermal stability of a multi-tubular reactor but also with regard to reaching a high CO conversion and production of HCs. This important aspect is often disregarded in evaluations of FT fixed-bed synthesis.

Additional instructive figures and a table are given in the Supporting Information:

○

Figure S8 depicts the impact of u_s,z=0 on the axial profile of the total pressure in the tubes.

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Figure S9 shows the influence of u_s on the rate of heat removal from fixed bed to boiling water.

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The influence of u_s on the heat transfer coefficient α_w,ex is shown in Figure S10.

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Figure S11 shows the influence of u_s,z=0 on the axial profile of α_w,int, and Figure S12 the corresponding figure for λ_rad in the bed of the tubes.

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Figure S13 depicts temperature profiles at different radial positions, and Figure S14 presents a selected radial temperature profile for z = 2 m (location of maximum in temperature).

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Figure S15 depicts the influence of u_s,z=0 on pressure drop and final gas velocity u_s,z=12m, indicating a strong rise both in ∆p_bed and u_s,z=12m with increasing u_s,z=0.

○

Axial profiles of the radial heat fluxes in the tubes (heat removal, heat production, and heat flux from/to gas) are given by Figure S16.

○

The parametric sensitivity of the FT reactor with and without influence of pore diffusion is also discussed in the Supporting Information (Figures S17 and S18), which explains in detail that pore diffusion “helps” with regard to thermal stability of a fixed-bed FT reactor.

○

Table S1 compares an FT reactor with constant activity (C_a = 3; simulation by “optimal” model M4) with a two-zone reactor (C_a = 2.5 for z < 6 m and 3.5 for 6 m < z < 12 m) and a reactor with optimal activity distribution (C_a,mean = 3). The data indicate that the output of C₂₊-HCs can be improved by 4% and 8%, respectively.

○

Axial profiles of η_pore (r = 0) for u_s (230 °C, 30 bar) of 0.5 and 1 m/s are shown in Figure S19; selected values at different temperatures are listed in Table S2.

4. Summary

In this work, a cooled multi-tubular FT reactor with a common gas recycle and purge gas stream was simulated by reactor models of different complexity, e.g., with regard to neglect or considering the pressure drop, the change in total molar flow along the tubes, or axial changes in radial heat transfer parameters. The effective reaction kinetics of CO conversion for cobalt as a catalyst were utilized in all reactor models.

An accurate and thus recommendable FT fixed-bed reactor model should consider both the change (decline for FT) in the total molar flow by the reaction and the (general) decrease in total pressure in a fixed-bed reactor by the unavoidable pressure drop. Both effects opposingly change the superficial gas velocity u_s and thus the residence time and syngas conversion, respectively, along the tubes compared to a (too) simple isobaric and isochoric (neglect of change in number of moles during reaction) reactor model presuming constant u_s. Only in rare cases do both effects cancel each other out, such as here coincidentally for u_s of 1 m/s.

A changing gas velocity as well as a drop in the total pressure along the tubes also have an impact on the radial heat transfer, i.e., on the effective thermal conductivity λ_rad and the heat transfer coefficient α_w,int at the internal side of the tube. Hence, these aspects should also be considered for an accurate FT reactor model.

The inhibition of the effective reaction rate by steam should be at least taken into account if a partial pressure of steam at the end of the tubes reaches more than around 5 bar. For typical reaction conditions and a common gas recycle, only a high conversion of CO of more than 50% per pass leads to such a high value of p_H2O at the reactor outlet.