Reactor design Ahmad Rafiee Magne Hillestad Norwegian University of Science and Technology (NTNU), Department of Chemical Engineering, Trondheim, Norway. 1 Research Article Staging of the Fischer-Tropsch Reactor with a Cobalt-Based Catalyst A method for systematic reactor design, described by Hillestad [1], is applied to the Fischer-Tropsch synthesis. The reactor path is sectioned into stages and design functions are optimized to maximize an objective function. Two different objective functions are considered: the yield of wax and a measure of the profitability. With the chosen kinetic model [2] and the path temperature constrained by 240 °C, staging of the Fischer-Tropsch synthesis based on the first criteria will increase the yield of wax. By introducing the cost of heat transfer area in the objective function, the total heat transfer area requirement of a two-stage reactor is significantly less than of a single-stage reactor. Keywords: Fischer-Tropsch synthesis, Path optimization, Plug flow, Reactor design, Staging Received: December 14, 2012; revised: May 29, 2013; accepted: June 07, 2013 DOI: 10.1002/ceat.201200700 1 Introduction In the last decades, the conversion of natural gas through a gas-to-liquids (GTL) process proved to be an alternative for the use of remote natural gas reserves to produce liquid transportation fuels. An increasing world-wide demand for cleanburning fuels has sparked a renewed interest in the study of the Fischer-Tropsch (FT) synthesis. A GTL plant consists of three main sections: (i) synthesis gas (syngas) production: conversion of natural gas to a mixture consisting mainly of H2 and CO, (ii) FT synthesis: conversion of H2 and CO to a wide range of hydrocarbons, (iii) upgrading of products. There are different routes for syngas production: auto-thermal reforming (ATR), steam methane reforming, combined reforming, and heat exchange reforming involving series and parallel arrangements [3]. Over the years, four types of reactors have been utilized for FT synthesis: the fixed-bed tubular reactor known as the ARGE reactor, circulating fluidized-bed reactors known as Synthol reactors, the Sasol Advanced Synthol reactor, and the slurry-bubble column reactor [4, 5]. The upgrading unit involves mainly separation and hydroprocessing. The primary upgrading starts with the removal of light ends and dissolved gases. The same basic technologies used in crude oil refineries have been adapted for FT product refining [3]. – Correspondence: Prof. M. Hillestad (magne.hillestad@chemeng.ntnu. no), Norwegian University of Science and Technology (NTNU), Department of Chemical Engineering, Sem Sælandsvei 4, 7491 Trondheim, Norway. Chem. Eng. Technol. 2013, 36, No. 00, 1–11 The earliest catalysts used for FT synthesis were iron and cobalt. Iron is a highly active catalyst and exhibits water-gas shift (WGS) activity whereas cobalt catalysts do not have this activity, leading to improved hydrocarbon yield. Cobalt catalysts yield mainly straight-chain hydrocarbon products and no oxygenates as with iron catalysts. However, cobalt catalysts are 230 times more expensive than iron, but still a very good alternative to iron catalysts. The reason is that cobalt catalysts demonstrate activity at lower operating pressures [5]. Staging in chemical engineering is not a new idea and has been applied on many processes. Androulakis and Reyes [6] studied oxidative coupling of methane (OCM) and the role of oxygen distribution and product removal on a staged plugflow reactor. Maretto and Krishna [7] introduced staging of an FT slurry-bubble column reactor and optimized the reactor conversion. The advantages of this reactor configuration compared to a single-stage reactor are increased syngas conversion and reactor productivity. In their study, the multi-stage reactor requires additional cooling tubes. Waku et al. [8] performed studies on staged oxygen introduction and selective hydrogen combustion during propane dehydrocyclodimerization reactions on cation-exchanged zeolite. Hwang and Smith [9] investigated the effect of catalyst dilution and distribution of feed to control the reactor temperature of two case studies involving nitrobenzene hydrogenation and ethylene oxidation. Diakov and Varma [10] investigated the effect of feed distribution in a membrane reactor for methanol oxidative dehydrogenation. Guillou et al. [11] examined the influence of hydrogen distribution between stages in a micro-channel FT reactor. Rafiee and Hillestad [12] evaluated staging of an FT reactor with an iron-based catalyst. The design functions, i.e., fluid mixing, hydrogen distribution, heat transfer area distribution, coolant temperature, and catalyst concentration are optimized to max- © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.cet-journal.com 2 A. Rafiee, M. Hillestad liquid phase and the other species are in the gas phase. There are small traces of these components in the gas phase which are neglected here. The average heat capacities of lump 1–5 are: 2.86, 2.56, 2.52, 3.15, and 3.00 kJ kg–1K–1, respectively. The price of products (lump 3–5) are 0.8, 0.9, and 1.1 kg–1, respectively [16]. The mass production rates of lump 2–5 are according to the ASF distribution model: imize the concentration of C11+ at the end of the reactor path. Jess et al. [13, 14] described the effect of particle size and single-tube diameter on the thermal behavior of FT reactors. The potential of sectioning the FT synthesis reactor into stages based on the method proposed by Hillestad [1] is demonstrated. Each reactor stage is represented by design functions. Staging of the reactor path will provide more degrees of freedom for optimization. This study focuses on the FT unit only and investigates the possibilities of cost reduction of the FT reactor. 2 ~ 2 ˆ 2aASF 1 R ~ 3 ˆ 3a2ASF 1 R Fischer-Tropsch Kinetic Model ~ 5 ˆ 11a10 R ASF 1 (1) k1 PH2 PCO 0:05 1 ‡ K1 PCO (2) rCO ˆ k2 PH2 0:6 PCO 0:65 1 ‡ K1 PCO (3) aASF †2 aASF n 1 n ˆ 2; 3; :::; ∞ k1 K1 k2   37 326 8:8 × 10 6 exp RT   68 401:5 1:096 × 10 12 exp RT   37 326 1:6 × 10 5 exp RT 3 Problem Formulation 3.1 Staging of the Reactor Path ‰cI www.cet-journal.com dx ~ x† ‡ uF K
xF uM r~JŠ ˆ ruA R xn x† uH E
x xw † (10) The state vector x is the vector of mass fractions augmented with the temperature (Rafiee and Hillestad [12]). Unit 1:05 3 mreactor s 1 x ˆ ‰x 1 ; x2 ; x3 ; x4 ; x 5 ; xH2 ; xCO ; xH2 O ; hŠT hˆ T Pa (9) A path is a line of production on which basic operations take place which are represented by design functions. The path is sectioned into a number of stages and the design functions (decision variables) are optimized so as to maximize an objective function. The flow model given by Eq. (10) is a concise formulation, representing the change of state variables (mass fractions and temperature) along the path. The derivation of Eq. (10) is described in detail by Hillestad [1]. (4) kmolCH4 Pa (8) The first five reaction rates in Eq. (9) are the rates of lump 1–5 on mass basis. The value of B is determined from Eq. (9). The consumption rate of H2 is calculated according to the stoichiometry of the reactions and the distribution of paraffins and olefins. Table 1. Kinetic and adsorption parameters in Eqs. (2) and (3) [2]. Arrhenius expression
aASF †2 ‡ . . . B
~ CO ‡ R ~2 ‡ R ~3 ‡ R ~4 ‡ R ~5 ‡ R ~H ‡ R ~H O ˆ 0 ~1 ‡ R R 2 2 In order to reduce the number of components, a lumping technique is used for hydrocarbon species: (1) CH4, (2) C2, (3) LPG (C3 and C4), (4) middle distillate (C5–C10), and (5) wax (C11+). At 20 bar and a temperature range of 210–240 °C, vaporliquid equilibrium calculations using the UNISIM DESIGN process simulator indicate that lump 4 and 5 are mainly in the Parameter (6) The conservation of mass requires that the sum of all component reaction rates add up to zero. The kinetic and adsorption parameters are given in Tab. 1. The process considered in this study is a once-through FT reactor (s) and the pressure of the reactor is 20 bar. The weight fractions of FT products heavier than methane (Wn) are assumed to follow the ideal Anderson-Schulz-Flory (ASF) distribution. Wn ˆ n 1 aASF † B (7) The kinetic model applied here is the one given by Iglesia et al. [2] for a cobalt catalyst, where methane production and CO consumption are defined by Eqs. (2) and (3), respectively. This kinetic model is chosen because it takes into account a higher selectivity of methane. However, other available kinetic models could be applied [15]. rCH4 ˆ aASF †2 ‡4a2ASF 1 (5) 2

~ 4 ˆ 5a4ASF 1 aASF †2 ‡6a5ASF 1 aASF †2 ‡ . . . ‡10a9ASF 1 aASF †2 B R In an FT reactor, the synthesis gas (syngas) reacts to form a mixture of hydrocarbons. CO + UH2H2 → (–CH2–) + H2O aASF †2 B Tref †Tref1 11† 1 kmolCO Pa 1:25 3 mreactor s 1 The first five mass fractions in Eq. (11) are the mass fractions of lump 1–5, respectively. ~ x† The dimension of the reaction rate vector R –3 –1 is kg m s . © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Chem. Eng. Technol. 2013, 36, No. 00, 1–11 Reactor design   ~ x† ˆ R ~1; R ~2; R ~3; R ~4; R ~5; R ~H ; R ~ CO ; R ~H O; R ~h T R 2 2 ~ h ˆ D; H †cp 1 Tref1 rCO R 4.2 (13) The total mass flow rate W is independent of pressure change, temperature change, phase change, or mole number change along the reactor path [1]. Mixing: The function uM(n) is a design function representing the fluid mixing. When uM is zero, Eq. (10) represents a plug-flow model and as the  functionincreases proportionally duM ˆ 1 , Eq. (10) becomes the with the length of path dn completely mixed volume. For an intermediate slope   duM 0< < 1 , Eq. (10) represents the plug-flow model with dn recycle [1]. Distributed feed: The design function uM = ar defines extra feed distribution along the path volume. The distribution of extra feed can be continuous or pointwise. In this study, a pointwise distributed feed is applied [1]. Heat transfer: The design function uH = br represents the heat transfer area distribution along the path. Heat transfer area distribution is parameterized by a piecewise constant function [1]. Chemical reactions: The design function uA is the catalyst activity which can vary between 0 and 1. A value of 1 is considered to be the maximum catalyst concentration. Catalyst activity is parameterized by a piecewise constant function. Parameterization of design functions leads to a set of inequality constraints [1]. The model applied here is a pseudoheterogeneous model. The gas and liquid phases flow in the same direction and have the same degree of dispersion. 4 Optimization 4.1 Optimal Control Analogy Objective Functions (12) Heat of reaction (–DrH) at 25  C is 172 kJ per mole of CO consumed in the FT reactor. Simultaneously, the total mass is conserved. dc ˆ uF dn The concentration of wax at the end of the path is a possible objective function, J1 = xwax [1]. The design functions represent costs, such as heat transfer area, extra feed points, and utility streams. A measure of the profit is another possible objective function (J2). Here, it is focused only on the FT synthesis, where the syngas is imported. With a syngas capacity of around 67.45 kg s–1, the price of syngas is 0.23 kg–1. The pressure of the reactor is 20 bar. The costs associated with this study are categorized as: 1. Capital cost: Cost of FT reactor including reactor shell [19], tubes [20], and the price of cobalt catalyst which for the initial load amounts to ∼ 54 kg–1. The investment required for the power production unit is assumed to be 300 kW–1. It is presupposed that process water is free. The selling price of the produced electricity in the power plant is 0.06 kW–1h–1 [19]. 2. Equivalent Annual Operating Cost (EAOC): This cost is the annual operating cost plus the annual capital charge [19]. The annual capital charge is the amortized capital cost over the operating life of the plant to establish an annual cost. For an operating life time of 20 years and an interest rate of 8 %, the annual capital charge is 10 % of the capital cost [19]. The annual profit is equal to the income minus the EAOC. The amount of steam produced in the FT reactor is calculated as the total heat transfer divided by the latent heat of vaporization (assuming 100 % efficient heat transfer): R UA T TW †dn _ steam ˆ m (16) k The latent heat of vaporization (k) is 1889 kJ kg–1. Finally, J2 is formulated as below (tax is neglected): J2 ˆ Income EAOC† EAOC ˆ annual operating cost ‡ 0:1 × capital cost nS nS X X uA;i Di ‡ 1:027 × 104 b 2i Di C2;i ˆ 4:792 × 108 ‡ 8:660 × 105 iˆ1 Chem. Eng. Technol. 2013, 36, No. 00, 1–11 C2,i is the cost of tubes per meter ($ m–1) [20]. C2,i = 2.5 + 98.958(di – 0.02) (14) The objective function (J) is formulated as a function of state variables z ˆ ‰xT ; cŠT at the end of the path. Here, it is required that the temperature along the path does not exceed 240 °C, i.e., T(n) ≤ 240 °C [7, 17, 18]. The path constraints are represented by nonlinear inequality constraints [1]. h(z,u) ≤ 0 iˆ1 (17) The optimal path configuration can be found by solving a problem similar to that of an optimal control problem. max J ‰r; uŠ ∈ U dz s:t: ˆ f z; u†; z 0† ˆ z0 dn 3 (15) (18) All performed approximations have the same effect on all the cases. 4.3 Mathematical Programming The decision variables (design functions) and state vector z = [xT, c]T are discretized. The design functions are discretized by piecewise linear or piecewise constant functions [1]. The system of ordinary differential equations given by Eq. (14) is discretized by orthogonal collocation. On each reactor stage © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.cet-journal.com 4 A. Rafiee, M. Hillestad optimizing the path, no extra feed is beneficial, because it will dilute the wax concentration. The mixing structure for this case is a PFR and there is no catalyst dilution. The reactor temperature hits the maximum to obtain the maximum converIn this study, the optimization algorithm based on SQP, fminsion and then declines. In this case, the concentration of water con in Matlab is applied. The objective functions are maximiis 41.87 wt %. zation of wax production at the end of the path and a measure Case 2: Two stages are selected. The concentration of wax of profitability. The inequality constraints, h(z,u), are path and the average heat transfer area density are higher than in constraints and bounds on decision variables. The equality case 1. There is no cost on heat transfer area. All the design constraints are the discretized state variables at each collocafunctions are free and there are 13 degrees of freedom, i.e., two tion point on each reactor stage. design parameters for each of heat transfer area density, coolant temperature, catalyst activity, and extra feed and five for mixing structure. The results are indicated in Fig. 2. 5 Results Removing liquid wax from the outlet stream of the first stage has no effect on the objective function, because the reacIn all cases (if otherwise stated), the residence time (s) and tion rates are calculated based on the partial pressure of H2 chain growth probability are kept constant at 2.97 m3s kg–1 and and CO in the gas phase. On the other hand, removing water 0.9 [7, 17, 18], respectively. The maximum operating temperafrom the first stage and reoptimizing the path will increase the ture of the reactor is limited by catalyst deactivation. Catalyst concentration of wax to 23.98 %. In this case, CO conversion deactivation is not considered here. and average heat transfer area are 81.90 % and 163.43 m2m–3, Tab. 2 presents the optimization results. In each case, the respectively. input parameters are: objective function criteria, number of Case 2a: In this case, the heat transfer density is required to stages, number of design parameters for coolant temperature, be uniform along the path. The reason may be that similar and number of design parameters for heat transfer area denreactors are wanted. The mass fraction of wax and CO conversity. In Tab. 2, the value of 1 for nuT (or nuH) means that all the sion are increased compared to case 1. The first stage is plug stages have the same coolant temperature or heat transfer area flow with recycle (rP) due to less degrees of freedom compared density. The optimization outputs are: value of objective functo case 2, and rP is there to level out the temperature peak. In tion, mixing structure (PFR, CSTR, or PFR with recycle), averthis case, there are 12 degrees of freedom. age heat transfer area density, coolant temperature, catalyst Case 2b: The same criteria as above are applied and both dilution, and mass fraction of each lump of components. The the coolant temperature and the heat transfer density are refresh syngas and extra feed contain only H2 and CO and H2/ quired to be the same along the path. The reason of having CO to the reactor is 2.1. equal coolant temperatures is that it is intended to use the Case 1: One stage is selected and the target is to maximize same pressure level of boiling water as coolant. The mixing the mass fraction of wax. The profile is illustrated in Fig. 1. By structure for this case is PFR with recycle for the first stage and PFR for the second stage. The number of degrees of freeTable 2. Cases with different optimization criteria, number of stages (ns), number of design padom for this case is 11. rameters for coolant temperature (nuT), and heat transfer area density (nuH). Case 2c: Chain growth probability is varying along the path as a Input Results function of temperature and parnuT a nuH b J uM c ad xWax CO Criteria Case ns tial pressures of H2 and CO based conversion [%] on the correlation proposed by Song et al. [21]. The chain growth 1 1 1 1 21.92 P 123.5 21.92 74.91 probability at the end of the path is 2 2 2 2 22.81 P-P 156.5 22.81 77.97 0.78 and the concentration of wax 2a 2 2 1 22.66 rP-P 151.7 22.66 77.5 is 8.05 wt %. The concentration of J1 lump 4 is increased compared to 2b 2 1 1 22.64 rP-P 151.0 22.64 77.43 case 2 due to a low chain growth 2c 2 2 2 8.05 rP-P 157.2 8.05 75.55 probability. Case 3: Three stages are applied 3 3 3 3 23.02 rP-P-P 155.1 23.02 78.72 here. The optimal mixing structure 4 1 1 1 100.00 % P 117.2 21.79 74.47 is rP-P-P (Fig. 3) and the concenJ2 tration of wax is increased com4a 2 2 2 122.00 % C-P 97.6 22.21 75.97 pared to case 1. All design func4b 2 2 2 125.00 % C-P 96.1 22.31 76.28 tions are free and 20 degrees of a freedom are given. Number of design parameters for coolant temperature; 1 means the same coolant temperature. b Case 4: One stage is selected and Number of design parameters for heat transfer area density; 1 means the same heat transfer area density. cP is PFR, C is CSTR, and rP is a PFR with recycle. dAverage heat transfer area dena measure of the annual profit is sity along the path. maximized. The mixing structure and each collocation point, Eq. (14) is formulated as nonlinear equality constraints [1]. www.cet-journal.com © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Chem. Eng. Technol. 2013, 36, No. 00, 1–11 Reactor design 5 Figure 1. Case 1 and the criteria defined by J1. is a single PFR and the wax concentration of wax is 21.79 wt %. Case 4a: Two stages are applied and there is no distributed feed. The mixing structure is CSTR for the first stage and plug flow for the second stage. The value of objective function is increased by 22 % which is due to the lower heat transfer area requirements compared to the single-stage reactor (case 4). Comparing the heat transfer area density of this case with case 2 demonstrates that by introducing the cost of heat transfer area, the total area is reduced. The results are presented in Fig. 4. If the CSTR in Fig. 4 is set to a PFR, the peak temperature exceeds the maximum temperature. The CSTR serves to level out the temperature peak with less heat transfer area. Case 4b: The same criteria as above are applied but the original feed is distributed between the stages. In this case, 97.48 % of the original syngas is fed to the first reactor and the objective function increases by 25 % compared to case 4. The concentration of wax at the end of the path is 22.31 wt %. Chem. Eng. Technol. 2013, 36, No. 00, 1–11 6 Discussion 6.1 Model Verification The model presented in this study is verified by: (i) UNISIM DESIGN process simulator, and (ii) performing two-phase calculations based on the correlations proposed by Marano and Holder [22] for FT systems. The values of b, coolant temperature, and mixing structure are the same as for case 1. The results presented in Tab. 3 are close to the results given by UNISIM DESIGN and two-phase calculations. Consequently, Table 3. Verification of model. I: this study, II: UNISIM DESIGN, III: two-phase calculations. I II III xWax [wt %] 21.92 21.70 21.63 CO conversion [%] 74.91 74.04 74.47 © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.cet-journal.com 6 A. Rafiee, M. Hillestad Figure 2. Case 2 and the criteria defined by J1. assuming an ideal split between gas and liquid phases is a good approximation. 6.2 Interpretation of the Results This study focuses on the FT unit only and investigates the possibilities of cost reduction of an FT reactor. The kinetic model is the one given by Iglesia et al. [2] which takes into account a higher selectivity of methane. The pressure drop is neglected here. The model applied is a pseudo-heterogeneous model. With the chosen kinetic model, staging of the FT synthesis based on the first criteria will increase the mass fraction of wax and CO conversion. The concentration of wax for a single-, two-, and three-stage FT reactor of the same total volume is 21.92, 22.81, and 23.02 wt %, respectively. By optimizing the path, no extra feed is beneficial because it will dilute the wax concentration. The reactor temperature hits the maximum to obtain the maximum conversion. For a two-stage reactor www.cet-journal.com where all the design functions are free, the number of degrees of freedom is 13. In case 2a, the heat transfer area density is forced to be the same for both stages, and the first stage is a plug flow with recycle to level out the temperature peak. By introducing the cost of heat transfer area in the objective function, the total heat transfer area requirement of a twostage reactor is 16.7 % less than for a single-stage reactor. The optimal mixing structure of a two-stage FT reactor is completely mixed (CSTR) for the first stage and plug flow (PFR) for the second stage. The CSTR is to level out the temperature peak with less heat transfer area requirement. A case study is conducted to investigate the effect of distributing the original syngas feed between the stages. In this case, 97.48 % of the original syngas is fed to the first reactor and the concentration of wax at the end of the path is 22.31 wt %. For a single-stage FT reactor with iron-based catalyst and syngas flow rate of 67.45 kg s–1, the conversion of CO and concentration of wax at the end of the path are 75.23 % and 13.70 %, respectively, [12]. In this case, the water-gas shift reaction is active and part of CO is converted to CO2, i.e., the © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Chem. Eng. Technol. 2013, 36, No. 00, 1–11 Reactor design 7 Figure 3. Case 3 and the criteria defined by J1. concentration of CO2 at the end of the path is 34.81 %. This will increase H2/CO at the end of the reactor path to 5.33. The optimal mixing structure is plug flow [12]. Maretto and Krishna [7] studied the staging of an FT slurrybubble column reactor and optimized the reactor conversion. Staging of the reactor increases syngas conversion and reactor productivity compared to a single-stage reactor. In their study, the catalyst activity is equal to 1 and a temperature difference of 10 °C is considered between the reactor and the coolant. Furthermore, there is no cost on heat transfer area and the multi-stage reactor requires additional cooling tubes. Here, catalyst activity, reactor temperature, and heat transfer area are optimized. 6.3 Realization of the Optimal Design In an industrial GTL plant, the FT reactor will require large volumes and sectioning of the reactor may be necessary in any Chem. Eng. Technol. 2013, 36, No. 00, 1–11 case. Systematic reactor staging will provide more degrees of freedom for reactor optimization. The mixing structure of a two-stage reactor based on the profitability measure (second objective function) is CSTR for the first stage and plug flow for the second stage. The geometry of the slurry reactor, gas and liquid velocities, and the arrangement of the cooling tubes will affect the flow pattern inside the reactor [7]. With the same value of b for fixed-bed and slurrybubble column reactors, more area is needed for a fixed-bed reactor due to a lower heat transfer coefficient. 7 Conclusions The method developed by Hillestad [1] was applied to FT synthesis by means of a kinetic model given by Iglesia et al. [2]. The design functions such as fluid mixing, heat transfer area density, catalyst dilution, and distribution of extra feed are optimized along the path to maximize an objective function. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.cet-journal.com 8 A. Rafiee, M. Hillestad Figure 4. Case 4a and the criteria defined by J2. The objective function is here defined to be the yield of wax (J1) and a measure of profitability (J2). Staging of the FT reactor based on J1 will increase the mass fraction of wax and CO conversion. No distributed feed is beneficial because it will only dilute the wax concentration. By introducing the cost of heat transfer area in the objective function (J2), the total heat transfer area requirement is reduced which will increase the annual profit. The optimal mixing structure for a two-stage FT is completely mixed (CSTR) for the first stage and plug flow (PFR) for the second stage. Having less heat transfer area, the purpose of the CSTR is to level out the temperature peak. If the CSTR is set to a PFR, the peak temperature will exceed the maximum temperature. The syngas feed can be distributed between the stages. In this case, 97.48 % of the main syngas feed is fed to the first reactor and the objective function will increase compared to a single-stage reactor due to less heat transfer area requirements. www.cet-journal.com Staging with optimal distribution of heat transfer area, distribution of syngas, and mixing configuration will increase the profit measure as demonstrated here. The results of this study can serve as initial points for staging of the FT reactor in the overall GTL process with different syngas production configurations [23]. Acknowledgment The authors gratefully acknowledge financial support from the Research Council of Norway through the GASSMAKS program. The authors have declared no conflict of interest. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Chem. Eng. Technol. 2013, 36, No. 00, 1–11 Reactor design Symbols used a A B [m2m–3] [m2] [–] –1 –1 cp,i cp,ref C2,i di E [kJ kg K ] [kJ kg–1K–1] [m–1] [m] [–] h I J1 J2 ~J [–] [–] [wt %] [–] [–] K [–] heat transfer area density heat transfer area scaling parameter to ensure a consistent ASF distribution for all components heavier than methane specific heat capacity of component i reference heat capacity cost of tubes per meter tube diameter diagonal matrix, E = diag (0,0,...., cp,ref cp–1) path constraints identity matrix wax concentration measure of profitability partial derivative of component reactions with respect to x, ~ ~J ˆ ∂R0 x† ‡ diag 0; 0; . . . ; 1† ×    ∂x  a 1 cp;f cp;01 bcp;ref cp;01 diagonal matrix, diag (1,1,...., cp,F cp–1) [kmolCH4Pa–1.05mreactor–3s–1] k1 kinetic parameter k2 [kmolCOPa–1.25mreactor–3s–1] kinetic parameter K1 [Pa–1] adsorption parameter ṁsteam [kg s–1] steam production rate n [–] carbon number nc [–] number of components ns [–] number of stages nuT [–] number of design parameters for coolant temperature number of design parameters for nuH [–] heat transfer area density PCO [Pa] partial pressure of CO PH2 [Pa] partial pressure of H2 rCO [kmol mreactor–3s–1] consumption rate of CO rCH4 [kmol mreactor–3s–1] production rate of CH4 [kmol mreactor–3s–1] component reaction rate on molar Ri basis ~i R [kg mreactor–3s–1] component reaction rate on mass basis T [K] temperature along the path Tref [K] reference temperature Tw [K] temperature of the cooling medium along the path uA [–] catalyst dilution uF [–] feed distribution design function, uF = ra uH [–] heat transfer area distribution design function, uH = rb [–] mixing design function uM u [–] vector of decision variables U [kWm–2K–1] overall heat transfer coefficient UH2 [–] usage ratio of H2 Chem. Eng. Technol. 2013, 36, No. 00, 1–11 V VR W W0 Wn [m3] [m3] [kg s–1] [kg s–1] [–] x [–] xF xW z [–] [K] [–] z(0) [–] 9 residence volume along the path total residence volume of the path total mass flow rate along the path total mass flow rate at the inlet weight fraction of hydrocarbons heavier than methane state vector of mass fractions and temperature, x = [x1,...,xn,h]T state vector of extra feed coolant temperature state vector augmented with mass flow rate, z = [xT,c] z at the inlet of the reactor path Greek symbols a aASF b [kg m–3s–1] [–] [kg m–3s–1] c [–] Di k h [–] [kJ kg–1] [–] n [–] r [m3s kg–1] xi [–] feed distribution chain growth probability heat transfer area design function, b = Ua cp,ref–1 mass flow rate relative to the inlet, W W0–1 volume fraction of stage i, Vi VR–1 latent heat of vaporization of water dimensionless temperature, h = (T–Tref )Tref–1 dimensionless volume of the path (independent variable), V VR–1 space time or residence time, VR W0–1 mass fraction of component i References [1] M. 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Technol. 2012, 35 (5), 870–876. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Chem. Eng. Technol. 2013, 36, No. 00, 1–11 Reactor design Research Article: A cobalt-based catalyst Fischer-Tropsch (FT) reactor is sectioned into stages. Design functions are optimized to maximize the two selected objective functions, namely, the wax concentration at the end of the reactor path and a measure of profitability. Staging of the FT reactor increases the production rate of wax based on the first objective function criterion. Chem. Eng. Technol. 2013, 36, No. 00, 1–11 11 Staging of the Fischer-Tropsch Reactor with a Cobalt-Based Catalyst A. Rafiee, M. Hillestad* Chem. Eng. Technol. 2013, 36 (䊏), XXX … XXX DOI: 10.1002/ceat.201200700 © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.cet-journal.com