An Equation-Oriented Framework for Optimal Synthesis of Integrated

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An Equation-Oriented Framework for Optimal Synthesis of Integrated Reactive Distillation Systems for Fischer-Tropsch Processes Yizu Zhang, Cornelius Mduduzi Masuku, and Lorenz T. Biegler Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b00971 • Publication Date (Web): 14 May 2018 Downloaded from http://pubs.acs.org on May 15, 2018

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An Equation-Oriented Framework for Optimal Synthesis of Integrated Reactive Distillation Systems for Fischer–Tropsch Processes Yizu Zhang,† Cornelius Mduduzi Masuku,†,‡ and Lorenz T. Biegler∗,† †Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania, 15213, United States ‡Department of Civil and Chemical Engineering, University of South Africa, Private Bag X6, Florida, 1710, South Africa E-mail: [email protected] Phone: +1 (412)-268-2232 Abstract To explore reactive distillation (RD) for Fischer–Tropsch synthesis (FTS), we develop a steady-state adiabatic reactive distillation model for this system. Here, the calculation of the vapor–liquid equilibrium (VLE) through cubic equation of state is used to describe the phase behavior. Rate expressions for the FT and the water gas shift reactions are expressed in terms of fugacities and product selectivity of catalysts are based on Anderson–Schulz–Flory (ASF) distribution and experimental data. Next, a mass, equilibrium, heat and summation (MESH) model is extended by considering bypass streams for non-reactive trays and is used to integrate column structure blocks. A step-by-step initialization procedure is proposed to accommodate the complexity of the RD column and the high nonlinearity. This includes case studies of operating variables in a reactive flash model prepared for performance optimization of RD. Finally, a

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typical low-temperature Fischer–Tropsch (LTFT) process, which favors diesel production is implemented in the RD model. Comparing the RD for FTS against conventional slurry reactors, provides results that show that RD has a potential edge in industrial processes.

Keywords Fischer–Tropsch Synthesis, Reactive Distillation, Vapor–Liquid Equilibrium, Kinetic Modeling, Initialization Procedures.

Introduction Commercial scale Fischer–Tropsch (FT) reactors have been installed and operated before World War II and ever since. Typical high-temperature FT (HTFT) process reactors are Circulating Fluidized Bed reactors and Fixed Fluidized Bed reactors, which are suitable for production of gasoline and linear low molecular mass olefins; typical low-temperature FT (LTFT) process reactors are multitubular reactors and slurry three-phase reactors, which are suitable for production of high molecular mass linear waxes.1 Schulz,2 Davis,3 and Sie et al.4 reviewed the developments of reactor design for FT processes. For HTFT processes, higher capacity, better control and online catalyst removal were achieved by replacing circulating fluidized bed reactors with fixed fluidized bed reactors, where there is a wider reaction section and more cooling coils. For LTFT processes, slurry reactors have advantages over multitubular reactors as they are isothermal and have less catalyst loading for the same output. Mathematical modeling of LTFT reactors has been done by Sexena et al.,5 with descriptions of hydrodynamics, kinetics, and FT chemistry. Maretto et al.6 describe the simulation of a commercial size slurry bubble column reactor, and emphasize flow regimes and complex hydrodynamics. Ahón et al.7 proposed a quasi-steady-state model, which also applies to transient simulation of FT slurry reactors. In order to gain larger production 2


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capacity, several studies were performed on the design and scale-up of slurry reactors for the FT process; see Krishna and Sie,8 Krishna et al.,9 Sehabiague et al.10 In multiple reactants or products systems, reactive distillation is a proven reactive separation method that can enhance yields as well as improve product selectivity.11 However, fewer FTS studies have considered reactive distillation, partly because the design and operation issues for reactive distillation systems are considerably more complex than those involved for either conventional reactors or conventional distillation columns.12 York et al.13 describe a catalytic distillation reactor system with the embodiment of connecting separately controlled reaction chambers and separation units. Similar to reactive distillation columns, Maretto and Krishna14 created a simulation model for a multi-stage bubble column for slurry reactor design and optimization, thus reducing the overall backmixing of the slurry phase; however the separation process was not included. Srinivas et al.15–17 presented several simulation cases by using an Aspen RadFrac module, embedded with in-built thermodynamic procedures, and gradually increased complexity in the kinetic model and column structure. Moreover, Srinivas et al.15 show that this RD model predicts performance at par or better than conventional reactors. Masuku et al.18 develop a mathematical model to describe the behavior of an FT reactor, and consider the dynamic interaction between reaction and vapor–liquid equilibrium (VLE). Their results show that the rate of formation of component hydrocarbons is dependent on either the reaction rate or stripping rate, depending on which is rate-limiting. These studies demonstrate that FT-based RD has advantages in managing heat removal and increases conversion for FT reactors. This study extends the above studies by developing efficient equation-oriented optimization models for RD-based LTFT systems. The next section discusses the basic Fischer– Tropsch model and focuses on a compact selectivity model. The third section focuses on phase equilibrium models particularly for vapor–liquid equilibrium (VLE). Reaction kinetics and VLE are combined in the fourth section to develop, implement and validate a reactive flash model. The reactive distillation model is described in the fifth section, with results

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reported in the sixth section. The last section summarizes the key points and conclusions of the paper.

Fischer–Tropsch kinetic model The FTS mechanism is still an issue of active research.19–22 Some studies consider −CH2 − to be a monomer in the FT chain growth.19,20 Recently, M ≡ CH also been indicated as a likely monomer in the FT chain growth.23,24 However, this paper aims to use a general kinetic descriptions that could be used for a number of reported literature product distributions. A detailed kinetic expression is being considered for future work. For the FT polymerization reaction presented, a hydrocarbon chain attached at one end to the FT catalyst grows by sequential addition of a single carbon segments −CH2 −, each with probability α to continue the chain growth process or to terminate. FT polymerization follows the simple scheme:

CO + 2H2 → H2 O + −CH2 −

The chain growth can terminate by hydrogenation to paraffins or by dehydrogenation to olefins which results in the following reactions in our case:

nCO + (2n + 1)H2 → nH2 O + Cn H2n+2

nCO + 2nH2 → nH2 O + Cn H2n Dry1 proposed a number of mechanisms for FT reactions in his literature review and concluded that cobalt-based catalysts favor the production of n−paraffins, while iron-based catalysts also produce unsaturated compounds, primarily 1−olefins, and the water-gas shift reaction readily occurs: CO + H2 O ⇐⇒ CO2 + H2

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Van Der Laan et al.25 provide a critical review of the development in kinetics and selectivity of the Fischer-Tropsch synthesis, and emphasize that process development and reactor design require reliable kinetic expressions and detailed selectivity models. Maretto and Krishna14 describe the syngas consumption rate through the intrinsic kinetic equation developed by Yates and Satterfield.26 Moreover, Wang et al.27 perform experiments in a micro-fixed-bed reactor over an industrial Fe-Cu-K catalyst, and develop a model for overall reaction rate as well as product distribution. Srinivas et al.15 modified the kinetic parameters for FT reactions in the model proposed by Wang et al.27 to conform to the Aspen Plus power-law formulation. Both Krishna et al.9 and Srinivas et al.15 fixed the chain growth parameter α in their simulations. Because of model complexity and accuracy, we assume the production of oxygenates, including alcohols, can be neglected. The related FT polymerization reactions considered in this work include a wide spectrum of hydrocarbon products, including C1 to C56 n−paraffins and C2 to C20 1−olefins; no lumping of hydrocarbons is considered in this study.

Reaction rates The reaction rate for the FT polymerization reaction consists of the sum of reactions with hydrocarbons as products. The rate law is given by Sarup and Wojciechowski28 and is expressed in terms of fugacities:

RF T

1/2 1/2 kF T fˆCO fˆH2 = 1/2 1/2 1 + cfˆ + dfˆ CO

(1)

H2

where c = 0.242, d = 0.185 are constants. Also the reaction rate expression for WGS, an equilibrium-limited reaction, in the form of component fugacity, is taken from Singh and Saraf:29

RW GS = kW GS P 0.75 (fˆCO −

fˆH2 fˆCO2 ) KW GS fˆH O 2

5


(2)

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The equilibrium constant KW GS for the WGS reaction shows a significant temperature dependence, and is taken from Callaghan:30

ln KW GS = (s1 /T + s2 + s3 T + s4 T 2 + s5 lnT )/R

(3)

where s1 = 9998.22, s2 = −10.213, s3 = 2.7465 × 10−3 , s4 = −4.53 × 10−6 and s5 = −0.201 are constants.

Selectivity model The Anderson–Schulz–Flory (ASF) distribution is generally used to describe the molar FT carbon number product distribution:

zn = αzn−1 = αn−1 z1

where n indicates the carbon number and zn is the sum of mole fractions with carbon number n in the total hydrocarbon product. H2 /D2 runs from Ojeda et al.,20 show that carbon number selectivity depends on a single chain growth parameter. In this work, we assume chain growth parameter is a constant on every reactive stage, and is only a function of temperature. Inspired by work by Srinivas et al.,15 they associate the chain growth parameter α with a pre-exponential factor value to express reaction rate in the power-law formulation. The rate expression in this work is given by:

Rn = αRn−1 = αn−1 R1

(4)

M ax C#

X

n × Rn = RF T

(5)

n=1

Since this work aims at modeling RD for the FT process, the formulation and choice of the kinetic model should have the ability to cope with temperature and other variables in the

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RD. We found this to be essential to balance between good conversion and heavy product production for the FT process, both for a reactive flash model and the RD separation process under a temperature profile. A temperature-based correlation of the chain growth parameter α, from Masuku et al.,31 is incorporated in the following selectivity model.

ln(

α2 G0 (T ) )= 1−α RT

(6)

where G0 is the intercept of the Gibbs free energy G of reaction versus carbon number curve. Olefin to paraffin (o/p) ratio is generally catalyst specific. Shi and Davis32 investigated o/p through experimental runs by switching from H2 /CO to D2 /CO syngas feed and showed that reliable o/p values for heavier hydrocarbons can be obtained from H2 /D2 runs. Such o/p selectivities could be easily used in the model developed here for any given catalyst. Based on experimental observations from FT runs, Schulz and Claeys33 postulate that 1−olefins and paraffins are formed in a ratio that depends on carbon number. On a molar basis, ratio of olefin to paraffin o/p for the carbon number is given by the following rate expressions: 1 Rn o/p + 1 o/p = Rn o/p + 1

Rn,p =

(7.1)

Rn,o

(7.2)

where Rn,o and Rn,p is the reaction rate of 1−olefin and n−paraffin with carbon number n respectively. The o/p ratio presented in this work is a discrete function of carbon number but not of operating conditions and catalyst. The following o/p ratio data are obtained from an experimental run from Marano and Holder34 presented in Table 1, the trend is the same as what Schulz and Claeys33 proposed in their work.

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Table 1: o/p ratio based on carbon number (Data from Marano and Holder (1997)34 Component carbon number C2 C3 C4 C5 − C10 C11 − C20 C21+

o/p 1.86 4 3.17 1.70 0.67 0

Phase equilibrium Kinetic and phase equilibrium models are the two major inputs for the RD system. The phase equilibrium model also becomes important when the reaction occurs in the slurry phase, where synthesis gas bubbles through a suspension of fine particles and reactions occur on the surface of catalysts. As Dry,1 Kölbel and Milos,35 and others consider FT reactions only in the liquid phase, we do not consider the existence of a solid phase. Marano and Holder34 reported the characterization of FT liquids in slurry reactors for VLE calculations. Their Equation-of-State (EOS) model was compared with ideal and other models by Ahón et al.7 Srinivas et al.15 used the in-built thermodynamic properties and calculations of Aspen Plus and chose the “PRMHV2” (Peng-Robinson with modified Huron-Vidal mixing rules) option, and applied “free-water” calculations only in the condenser and not on the trays. They also compared two other thermodynamic options, “PSRK” and “UNIFLL”. The condenser model within an FT RD system should have vapor, oil-rich, and waterrich phases. These were accommodated here with a vapor-liquid-liquid-equilibrium (VLLE) calculation in the RD condenser configuration along with a vapor–liquid equilibrium (VLE) calculation in the slurry reactor and stage model. The phase equilibrium model presented in the following paragraph is to illustrate composition relationships for components within vapor–liquid or liquid–liquid phases in an FT system:

y i = K i xi

8


(8)

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oil xaq i = KD,i xi .

(9)

To obtain the equilibrium constants, fugacity of each component in three phases are set equal: fivap = fioil = fiaq

(10)

Vapor Phase Cubic EOS Model Among various EOS models in engineering calculations, the vapor phase fugacity at moderate to high pressure is estimated by a Peng–Robinson (PR) model.36 This model incorporates a proven root selection procedure for specific phases and binary interaction parameter estimation for FT components. The general form of the PR equation of state is:

P =

a(T ) RT − V − b V (V + b) + b(V − b)

where a represents the attractive forces between molecules and b represents repulsive forces, which are both related to the critical properties of a pure substance. The PR equation can be rewritten as:

f (Z) = Z 3 − (1 − B)Z 2 + (A − 3B 2 − 2B)Z − (AB − B 3 − B 2 ) = 0

(11)

where:

Z = P V /RT

(12.1)

A = P a/(RT )2

(12.2)

B = P b/(RT )

(12.3)

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To calculate vapor compressibility factor Zv , Kamath et al.37 propose the root selection based on the first and second order derivative of Eq.(11) as follows:

f 0 (Zv ) = 3Zv2 − 2(1 − B)Zv + (A − 3B 2 − 2B) > 0

(13.1)

f 00 (Zv ) = 6Zv − 2(1 − B) > 0.

(13.2)

Mixing rules with interaction terms are used to improve the prediction of mixture properties and phase equilibria. For two constant cubic equations of state, including the SRK and PR models, Reid et al.38 recommend the following mixing rules:

a= b=

XX i

i0

X

y i bi

√ yi yi0 aii ai0 i0 κii0

(14.1) (14.2)

i

The binary interaction parameters κii0 for all hydrocarbons, carbon dioxide, carbon monoxide and hydrogen, are calculated from the predictive correlation proposed by Nishiumi et al..39 Other binary interaction parameters were taken from Walas.40 Based on the Peng–Robinson model, the vapor phase fugacity coefficient for each component φi,v is given by Eq.(15). Eq. (11) to (16) are used to calculate the vapor phase fugacities. bi A 2ai bi Zv + ε1 B lnφi,v = (Zv − 1)( ) − ln(Zv − B) − ( )( − )ln( ) b Bε3 a b Zv + ε2 B

(15)

fi,v = φi,v yi P

(16)

where ε1 = 2.41421, ε2 = −0.414214 and ε3 = 2.282843.

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Oil phase Experimental results reported by Lu et al.41 showed that water constitutes a small fraction in the liquid phase, and Srinivas et al.17 show column performance is insensitive to liquid mole fraction of water. For our equilibrium models, we determine compositions of all components including water from the liquid fugacities in Eq.(17) and Eq.(18). We estimate liquid fugacities in the oil phase from Marano and Holder,42 based on asymptotic behavior correlations of n−paraffins and 1−olefins. This correlation was developed from VLE models of several Mobil bubblecolumn slurry reactors. For supercritical components, we use Henry’s constants for Eq.(17) and we use activity coefficients for Eq.(18). That is, we calculate liquid phase fugacity fi,l only for CO2 , CO, H2 , and C1 to C3 hydrocarbons (Henry’s components) by Eq.(17), and for the other components, water and C4 + hydrocarbons by Eq.(18). sat fi,l = Hi∞ xi exp[V¯i∞ (P − Psol )/RT ]

(17)

sat fi,l = γi∞ Pisat xi exp[V¯i∞ (P − Psol )/RT ]

(18)

Also, solubility of water in hydrocarbons has been studied by Black et al.,43 Englin et al.,44 who show that solubility of water in hydrocarbons has a negligible carbon number effect, which allows us to represent the oil phase with one hydrocarbon. In our model, we use the correlation proposed by Heidman et al.45 to predict solubility data of water in n−octane over a wide range of temperature:

ln(xoil aq ) = A + B(

1 ) + C(1 − Tr )1/3 + D(1 − Tr ) Tr

(19)

where A = −0.66037, B = −7.1130, C = −0.67885, D = −1.43381. Therefore, we obtain

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the activity coefficient of water in the oil phase: 1 xoil i

γioil =

(20)

Aqueous phase According to Eq.(10), (17) and (18), liquid-liquid distribution coefficient KD,i for Henry’s components, i.e. CO2 , CO, H2 , and C1 to C3 hydrocarbons, are obtained from:

KD,i

Hi∞,oil = ∞,aq Hi

(21)

γi∞,oil γi∞,aq

(22)

and for C4 + components and water:

KD,i =

Table 2: Parameters for calculation of Henry’s law constants in water solution (Data from Sander46 ) Components unit CO2 CO H2 CH4 C2 H6 C2 H4 C3 H8 C3 H6

◦ kH (at 298.15K) mol/(kg · bar) 0.035 0.00099 0.00078 0.0014 0.0019 0.0047 0.0015 0.0048

−∆soln H R

K 2400 1300 500 1600 2300 1800 2700 3400

From Sander,46 the Van’t Hoff equation is applied to calculate Henry’s law constant in water under condenser temperature:

◦ kH (T ) = kH × exp[

−∆soln H 1 1 ( − ◦ )] R T T

12


(23)

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Related parameters for calculation of Henry’s constants are listed in Table 2. For C4 + hydrocarbons, correlations for aqueous solubility xaq,sat of n−paraffins and 1−olefins, as well i as the correlation between solubility and molar volume are from McAuliffe.47 Water from both vapor oil = fwater liquid phases is considered in equilibrium with the vapor phase by equating fwater

in the above oil and vapor phase calculations.

Reactive flash model development The continuous well-mixed FT slurry reactor is developed as a reactive flash model, shown in Figure 1. Aside from syngas feed, a second feed stream of C30 n−paraffin solvent is fed continuously to ensure slurry conditions in the reactor. Two streams exit the reactor—a vapor product and a homogeneous liquid product stream containing water.

Reactive flash model formulation

1. Syngas; 2. Solvent; 3. Reactive Flash (CSTR) shown as 2-phase 4. Vapor product stream; 5. Liquid product stream

Figure 1: Illustration of Reactive flash stream flow

At steady state, the mass balance (Eq.(24)) for the reactive flash model is expressed for each component separately. The summation equation (Eq.(25)) and enthalpy calculation (Eq.(27)) are applied for all streams. The energy balance equation and equilibrium equation

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are Eq.(26) and Eq.(8) respectively. X

zi,s Fs − V yi − Lxi +

s

X

νi,j ξj = 0

(24)

j

X

xi =

i

X

yi = 1

(25)

i

Q = ∆Hv,out + ∆Hl,out − ∆Hv,in − ∆Hl,in ∆H =

X

∆Hi

(26) (27)

i

As described in Biegler et al.,48 the enthalpy of liquid and vapor mixtures are computed based on standard heat of formation and heat of vaporization:

∆Hv,i =

0 Hf,i

Z

T

+

0 ∆Hl,i = Hf,i +

T0 Z T

0 Cp,i (T )dT

(28.1)

0 Cp,i (T )dT − ∆Hvap,i (T )

(28.2)

T0 0 Cp,i (T ) = Ai T + Bi T 2 + Ci T 3 + Di T 4

(28.3)

Instead of using liquid heat capacities, we use heat of vaporization ∆Hvap,i (T ) to calculate enthalpies for liquid mixtures. The Watson49 correlation is used to calculate ∆Hvap,i (T ) which has a monotonic decrease with increasing temperature:

∆Hvap,i (T ) = ∆Hvap,i (Tb )[(Tc,i − T )/(Tc,i − Tb )]0.38

(29)

where Ai , Bi , Ci and Di , critical volume Vc , acentric factor ω, heat of formation at standard 0 conditions Hf,i , boiling point Tb,i and critical point Tc,i are taken from Reid.38

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The model formulation for reactive flash: min z(T,W ) = 1

x,y,L,V

s.t. VLLE :

Eq.(8) − (23),

Kinetics :

Eq.(1) − (7),

Reactive flash module : Eq.(24) − (29),

(Initialization :

Eq.(30))

Simplification and Initialization The above flash kinetic and VLE model comprises nonlinear, nonconvex constraints, is formulated as the stage model to apply over a wide range of operating conditions, and is a crucial component for the subsequent RD model. As part of the solution strategy for the flash, we simplify the kinetic model as (Eq.(30)): P

RF T,n = 1 × 10 RW GS

Fs1

−Ea,F T −Ea,F T − ) Fref RT RTref P F s1 −Ea,W GS −Ea,W GS −4 exp( − ) = 1 × 10 × s1 Fref RT RTref −6

×

s1

αn (T ) exp(

(30.1) (30.2)

The fractional part accommodates feed conditions and the exponential term is derived from Arrhenius equations. The αn (T ) term is set to accommodate the increasing carbon number and temperature. The initialization procedure is used to solve a sequence of models ordered from the lowest complexity to the highest, the result of the previous model is used as the initial point of the next: 1. Solve the reactive flash model first incorporated with simplified kinetics (30) and simplified VLE calculation (all fugacity coefficients set to 1).

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2. Update the above model with detailed kinetics (1)-(7) and solve, using the previous solution for initialization. 3. Update the above model with detailed VLE calculations and solve, using the previous solution for initialization. 4. Add Energy Balance calculations to obtain heating condition and solve, using the previous solution for initialization.

Figure 2: Liquid, vapor and total product distribution over C# for reactive flash

Model validation A base case run for reactive flash is conducted where Tref , Fref , solvent feed, catalyst loading, H2 /CO ratio and pressure are 495 K, 2.78×10−1 kmol/sec (1000kmol/hr) syngas, 2.78×10−5 kmol/sec, 200 kg, 2.03 and 21 bars respectively. We validate the reactive flash model by matching experimental and simulation results. Product distribution of mole flow rate in 16


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logarithmic scale for both phases is obtained from our base case run shown in Fig.2, which is comparable with experimental runs reported from Marano and Holder.34 The nonsmooth trends of the hydrocarbons are a result of the actual vapor pressure data used in the calculation. The product spectrum generated under various temperatures in Fig.3 is compared to that reported by Dry.50 Results from each of these simulations are used to calculate conversion for FT reactions based on carbon number as well as chain growth probability α, shown in Fig.4. With the increase in temperature, α shows a monotonic decrease, while the conversion of FT reactions increases between 190 ◦C to 260 ◦C. This shows that our combination of the ASF distribution and determination of α captures basic FT kinetic characteristics, and provides a better sense of the stage performance for later implementation of the RD model.

Figure 3: Relationship between Chain Growth Parameter and hydrocarbon selectivity

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Figure 4: Reactive flash performance under different temperatures

Reactive distillation model development The model for conventional equilibrium stages is given by the the MESH (material and energy balance, phase equilibrium, and summation) equations, with the extension accompanied by chemical reactions, as reviewed in Taylor and Krishna.51 In this section, a novel MESH model with tray bypass for optimization of column size and utilities is introduced and a framework of setting up an adiabatic RD for FT process is presented. The framework includes an initialization procedure and related mathematical model formulations. A general depiction of RD structure and stream flow is established in Fig. 5, where we assume fixed concatenated reactive stages, non-reactive stages in a rectifying section above the reactive section, with a non-reactive stripping section below. Only reactive stages receive liquid solvent feed and the vapor feed is on the last stage of the reactive section. Moreover, tray bypass is only considered for non-reactive stages, while the condenser and reboiler are present with temperature of the reboiler freed to allow overall convergence. 18


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Figure 5: Illustration of a general Reactive Distillation Column (RD) structure and Block connection

19


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RD model formulation For distillation optimization the optimum number of trays is usually determined through use of integer variables. Viswanathan and Grossmann52 developed a rigorous mixed integer nonlinear programming (MINLP) model for distillation column optimization and Ciric and Gu53 extended this MINLP model to synthesizing RD columns. Nevertheless, rigorous VLE and kinetics models add significant nonlinearity and nonconvex constraints to the problem. In this study we formulate the problem into a pure NLP, where some of the difficulties of the mixed integer problem can be avoided. As noted above, the MESH model is extended by considering bypass streams for nonreactive trays. Here, a continuous variable, ζk ∈ [0, 1] serves as a tray efficiency factor to indicate the existence of tray k and the extent of bypass for this stage; ζk = 0 leads to total bypass for tray k, while ζk = 1 determines no bypass. For detailed description of the bypass model, see Dowling and Biegler.54 Another variable γk ∈ [0, 1], the stage connection factor, links streams between adjacent stages or extracts intermediate streams as product; γk = 0 denotes total extraction and γk = 1 denotes total connection.

Figure 6: Stream flow illustration for non-reactive stages

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The non-reactive stage model is presented in Fig.6. Stage k receives inlet liquid stream γl,k−1 ζl,k Lk−1 from the upper stage and vapor stream γv,k+1 ζv,k Vk+1 from the lower stage. Also, we label the condenser to be the first stage, the reboiler as the last stage, and number the other stages from the top down. To initialize stage k, say to connect stage k with stage k + 1, the portion of inlet vapor streams entering stage k, γv,k+1 , increases from 0 to 1, with liquid condensing on stage k. Further γl,k also increases from 0 to 1; this allows the liquid stream from stage k to return to stage k + 1. This incremental procedure builds the model with growing complexity of column structure, and can be adapted to alternative column structures or configurations. Mass balance is achieved among inlet streams that are not by-passed γl,k−1 ζl,k Lk−1 , γl,k+1 ζl,k Vk+1 and outlet streams L∗k , Vk∗ for non-reactive stages in Eq.(31). ∗ ζk γv,k+1 Vk+1 yi,k+1 + ζk γl,k−1 Lk−1 xi,k−1 − Vk∗ yi,k − L∗k x∗i,k = 0

(31.1)

Lk xi,k = L∗k x∗i,k + (1 − ζk )Lk−1 xi,k−1

(31.2)

∗ Vk yi,k = Vk∗ yi,k + (1 − ζk )Vk+1 yi,k+1

(31.3)

Also, among inlet feed Fs,k , liquid and vapor inlet Lk−1 , Vk+1 , and outlet Lk , Vk for reactive stages mass balance is defined by Eq.(32)) on a component basis. X

Fs,k xsi,s + γv,k+1 Vk+1 yi,k+1 + γl,k−1 Lk−1 xi,k−1 − Vk yi,k − Lk xi,k +

s

X

νi,j ξi,k = 0

(32)

j

An illustration of the stream flow relationship between a partial condenser, the 2nd stage and decanter is shown in Fig.7. Mass balance for condenser and reboiler are achieved through Eq.(33) and Eq.(34) respectively.

γv,2 V2 yi,2 − Vcond yi,cond − Lcond xi,cond = 0

(33)

γl,N T −1 LN T −1 yi,N T −1 − VN T yi,N T − LN T xi,N T = 0

(34)

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Lcond xi,cond = LHC xi,HC + Lwat xi,wat

Page 22 of 38

(35)

Figure 7: Illustration of stream flow for Decanter A decanter is set on the top of the column to exclude most of the water from liquid stream coming out of condenser. Recent work shows that excess water may deactivate FT catalyst in certain circumstances, but it also acts as a cooling medium. Thus water returning to the system could be controlled by a “free-water reflux ratio”. Light product extraction are allowed and controlled by γl,1 . The outlet stream for decanter are a 1st liquid RichHydrocarbon (RHC) subscripted by HC and a 2nd liquid Rich-water (Rwat) subscripted by wat. The mass balance is achieved for decanter through Eq.(35). The summation equations (Eq.(25)) are applied for each stream. The energy balance equation are different for non-reactive stages (Eq.(36)), reactive stages (Eq.(37)), condenser (Eq.(38)) and reboiler (Eq.(39)).

∗ ∗ Qk = ∆Hv,k,out + ∆Hl,k,out − ζk γv,k+1 ∆Hv,k+1,in − ζk γl,k−1 ∆Hl,k−1,in

(36)

Qk = ∆Hv,k,out +∆Hl,k,out −∆Hv,f eed,in −∆Hl,f eed,in −γv,k+1 ∆Hv,k+1,in −γl,k+1 ∆Hl,k−1,in (37) Qcond = ∆Hv,cond,out + ∆Hl,cond,out − γv,2 ∆Hv,2,out

(38)

Qrebo = ∆Hv,N T,out + ∆Hl,N T,out − γl,N T −1 ∆Hl,N T −1,out

(39)

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Initialization framework A base case RD is defined in which the column structure, number of stages and connection condition are specified, and operating conditions of the column are fixed. Here we choose 4 reactive stages, a total of 6 stages in the column, with 0.278 kmol/s (1000 kmol/hr) syngas fed into the bottom reactive stage. The H2 /CO feed ratio is fixed to be 2.03, and a relatively small amount of 2.78×10−5 kmol/s C30 n−paraffin solvent is fed across the 4 reactive stages; the effect of feed temperature is neglected for all cases. Catalyst loading and temperature profiles are fixed but temperature is later freed to achieve design specifications, the results of which are shown in Table 3. The column is operated at 21 bars and a condenser temperature of 35 ◦C, as in York et al..13 We can see that the overall operating condition is the same as with the reactive flash base case. The RD model can be initialized in each stage with gradual increase of connecting flows, as in Dowling and Biegler.54 The initialization framework for RD is given by: 1. Model four individual reactive flashes simultaneously (no reciprocal connections) with distributed feed under simplified kinetics and VLE calculation (see initialization procedure for reactive flash). 2. Improve the above model with detailed kinetics. 3. Connect one reactive flash at a time into a cascade to formulate the reactive section. In the mean time, transfer the syngas feed from each individual reactive flash to the bottom flash. 4. Incorporate non-reactive stages into column configuration (optional). 5. Incorporate condenser, decanter and reboiler into column configuration. 6. Provide initial guess of Zk = 1 and φV,i,k = 1, calculate RD model with detailed VLE. 7. Add Energy Balance equations to the model and solve. 23


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Up until this point, we could obtain solution under fixed operating conditions under the model: min

xk ,yk ,Lk ,Vk

z(Tk ,Wk ) = 1

s.t. Eq.(8) − (23)

VLLE :

∀k

Eq.(1) − (7) ∀kr

Kinetics :

Eq.(28) − (29),

RD module :

(31) − (39), (Initialization :

Eq.(30))

Each stage may have a heat duty, which makes it a non-adiabatic RD base case. To achieve adiabatic behavior (or near adiabatic behavior), we minimize the heat duty for intermediate stages (stages except condenser and reboiler) and transform the model into an optimization problem. Temperature of reactive stages Tkr are freed in the range of 190 ◦C to 260 ◦C (typical LTFT) to accommodate heat released. Note that water from the decanter is excluded from the system, and light product extraction is freed (reflux ratio changes) to achieve convergence for all cases. The model formulation to achieve minimization of intermediate stage heat duty is presented, which corresponds to an adiabatic RD base case:

min

Tk , reflux ratio

s.t.

zβ =

X

pk + qk

Q k = pk − q k , ¯ k ≤ pk + qk ≤ Q ¯k, βQ

VLLE : Kinetics : RD module : (Initialization :

Eq.(8) − (23),

∀k

Eq.(1) − (7),

∀kr

Eq.(28) − (29), (31) − (39), Eq.(30))

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where positive slack variables pk and qk are introduced when the sign of Qk is unknown. β ¯ k is the original heat duty before optiis the steering parameter that goes from 1 to 0 and Q mization. Thus the adiabatic RD base case problem transforms into a series of optimization problems in which the freedom for Qk is larger each time. The purpose of this procedure is to achieve an adiabatic point, and test whether the degrees of freedom or column specifications are sufficient to achieve adiabatic behavior or not. In our findings, adiabatic behavior of RD is hard to achieve and intermediate cooling must be available. From results in Table 3, the first reactive stage receives direct cooling from the reflux and cooling is also applied on the 2nd reactive stage. With design specifications declared and heat for intermediate stages set, an optimization model with a user-defined objective could be established: min

Tk ,reflux ratio

s.t. VLE : Kinetics : RD module : (Initialization :

z = User defined Obj. Qk = pk − qk = Self defined value, Eq.(8) − (24),

∀k

Eq.(1) − (7),

∀kr

Eq.(28) − (29), (31) − (39),

∀k

Eq.(30))

The base case model, adiabatic model and optimization model are solved in sequence. All of the simulations were performed using GAMS with the CONOPT solver with the components, kinetics, and the VLE model mentioned in the earlier sections.

Results and discussion Operating conditions and results for an RD base case (operating condition fixed) are shown in Table 3, where results for a comparable reactive flash are also listed. While both configuration have similar FT conversion, RD seems to have the ability to inhibit WGS reactions because 25


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of the water removal in the decanter. In comparison to the flash result in Fig.2, product distribution for RD is shown in Fig.8. Here the ‘total HC component’ line shows a deficiency in the carbon number range from around C5 to C12 ; this is due to a light liquid product extraction from the decanter which is not included in this plot. This product stream allows clean naphtha to be obtained from the top. The steep decline of vapor product flow rate with increasing carbon number shows RD’s capacity to deliver liquid products.

Figure 8: Liquid, vapor and total product distribution over C# for RD To provide an intuitive understanding of the range of carbon number in hydrocarbon product from RD, a comparison of bottom liquid product flow rate for both configurations is presented in Fig.9. Results show that RD has the ability to preserve more HC product from around C8 to C32 , which is the typical range for gasoline and diesel. We also see a visible difference for HC with carbon number larger than C40 . This is caused by the stage temperature profile shown in Table 3; higher FT conversion is achieved on low temperature stages where there is more catalyst, resulting in α holding a larger value that favors production

26


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Energy & Fuels

Figure 9: Distribution of liquid product flow rate by Carbon Number comparison between reactive flash and RD

of high-carbon-number hydrocarbons. As a result, under similar typical LTFT operating conditions and similar conversion, RD has an edge over normal slurry reactors in preserving liquid production and production selectivity toward wax. Table 3: Operating conditions and basic results for reactive flash and base case RD Configuration Units (Base case) RD (stage No.)

1 (cond.) 2 (reac.) 3 (reac.) 4 (reac.) 5 (reac.) 6 (rebo.) Reactive flash

Temperature (stage) [K] 495 (avg.) 308.15 485 490 500 505 505 495

Catalyst loading (stage) [kg] 30 (sum) 0 100 67 20 13 0 200

27

FT Conversion (stage) %C 41.7% (sum) 0 15.8% 14.1% 6.6% 5.3% 0 40.6%


WGS Conversion (stage) %C 17.7% (sum) 0 5.2% 6.3% 3.4% 2.8% 0 36.4%

Heat Duty (stage) kW -468.20(sum) -88.19 -135.61 -135.58 -62.10 -46.72 0 -7232.25

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Reflux rectifier case study We consider two RD models based on the adiabatic RD model formulation; the first has the same base case configuration, while the second has non-reactive stages to see if separation extent increases. Because of high exothermicity and pressure, we have found that the reboiler does not really strip light products from bottom liquid in the base case simulations. Instead, Srinivas et al.16 mention that a “Reflux Rectifier” (RR), rather than a “Reactive Distillation” column is a preferred configuration. In fact, the syngas feed to the FTS provides the vapor load to the column when introduced below the bottom-most stage of the column. Hence a reboiler is not required as the reaction exotherm is sufficient to provide the heat required for distillation. Table 4: Operating conditions and basic results for Adiabatic 6-stage RD and 6-stage RR Configuration Units RD (stage No.) 1 (cond.) 2 (reac.) 3 (reac.) 4 (reac.) 5 (reac.) 6 (rebo.) RR (stage No.) 1(cond.) 2(non-reac.) 3(reac.) 4(reac.) 5(reac.) 6(reac.)

Temperature (stage) [K] 482.33 (avg.) 308.15 483.99 504.96 477.22 463.15 463.15 491.02(avg.) 308.15 452.11 482.71 464.97 494.75 521.65

Catalyst loading (stage) [kg] 200 (sum) 0 100 67 20 13 0 200(sum) 0 0 100 67 20 13

FT Conversion (stage) %C 40.93% (sum) 0 13.95% 23.40% 2.63% 0.95% 0 32.64%(sum) 0% 0% 13.69% 4.79% 4.97% 9.19%

WGS Conversion (stage) %C 13.39% (sum) 0 4.32% 7.24% 1.38% 0.46% 0 13.36%(sum) 0% 0% 5.33% 1.66% 2.16% 4.21%

Heat Duty (stage) kW -452.51(sum) -350.83 0 -100.00 0 0 0 -396.48(sum) -266.48 0 0 -130.00 0 0

The RR configuration contains one non-reactive rectifying stage, 4 reactive stages and a condenser (6 stages total). The MESH equations for the reboiler remains, but duty for the reboiler is set to zero. No design specifications are defined, although duty of intermediate stages is fixed at zero except for the 2nd reactive stage, after its adiabatic base case is achieved. The results in Table 4 are obtained with the sum of top product naphtha, bottom product diesel and set wax liquid fractions as the objective to be maximized. 28


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Energy & Fuels

The addition of rectifying stage 2 accounts for the difference of FT conversion in the RD and RR configurations, as liquid from the top stage acts as a cooling medium to balance the exothermic FT reaction. This leads to a lower temperature on the 4th stage (second reactive stage) of RR, which results in lower conversion. The stage-wise liquid composition for RD and RR of naphtha (C5 -C7 ), fuel gas (C1 -C4 ), gasoline (C8 -C12 ), diesel (C13 to C18 ), wax (C19 -C56 ) and water is shown in Fig.10 and Fig.11. Results show that, we could obtain a clean naphtha stream from the first or second stages, and a mixture containing naphtha, diesel and wax from the bottom. Since the hydrocarbon product is predominately linear paraffin/olefin in LTFT processes, along with a good chance of long-chain heavy product yield, it seems a good idea to produce high cetane number diesel fuels from the process. According to Dry,55 the naphtha produced from LTFT process would be a feed stock for thermal cracking to ethylene better than crude oil-derived naphtha. The straight-run distillate diesel fraction should have a cetane number up to 75 after hydrotreatment, and under mild conditions and recycling the product heavier than diesel, the wax can be cracked to extinction.

Figure 10: Liquid product distribution stage profile (6-stage RD Configuration)

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Figure 11: Liquid product distribution stage profile (6-stage RR Configuration) Table 5: Product yield comparison for reactive flash, RD and RR Configuration Units Reactive flash RD RR

(bottom) Liquid product yield [kg/sec] 0.107 0.295 0.289

Vapor product yield [kg/sec] 2.868 2.475 2.620

At this point, we have comparable results from the three configurations: reactive flash, RD and RR. Yields from both phases for these configurations are shown in Table 5. Adding rectifying stages should be carefully treated when the unit favors wax production. We could also see that column configuration would play a vital role for heat exchange.

Conclusions This study describes the modeling simulation and optimization of a slurry reactor, a 6-stage RD and a 6-stage RR system. Aspects including separation, conversion, product yield, and liquid product distribution profile are compared. We observe that the RD configuration 30


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Energy & Fuels

preserves more liquid product and has an edge in selectivity toward wax compared to the slurry reactor. The RR system (without reboiler) has advantages over the RD configuration. However, adding the “rectifying section” should be carefully treated, as the feasible region for operating variables is narrow and could result in poor performance in conversion and production toward wax. These results were obtained from an equation-oriented optimization modeling framework, where specific modules were developed and integrated in the simulation and optimization of RD for FTS. The general framework of how these modules are incorporated and developed is presented and a tailored step-by-step initialization procedure is proposed. Simplified modules are first incorporated to initialize more rigorous ones, which guide the success of the stage model. A generalized sequence to connect stage models and other column structures was then proposed to formulate a reactive distillation model. This approach enables the decompostion of the original problem, which in turn decreases computational time and increases robustness. The results of this study lead us to extend this framework for more general RD systems for FTS in the future. We intend to formulate larger, more challenging optimization problems and more detailed kinetic models for a broader slate of FT-based chemistries and product requirements.

Acknowledgement We gratefully acknowledge the partial support for this work from the China Scholarship Council, and the National Research Foundation of South Africa (Grant Number: 113652). The opinions, findings and conclusions or recommendations expressed in this publication are those of the authors. These funding bodies accept no liability whatsoever in this regard.

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Nomenclature Notation ◦

Referenced state

∞

Infinte dilution

sat

Saturation properties

i ,i0

Component in system

j

Reaction in system

k

Stage number

l

Liquid phase

s

Feed streams

v

Vapor phase

aq

Aqueous phase

cond

Condenser

FT

Fischer-Tropsch reactions

in

Inlet streams

NT

Number of trays

oil

Oil phase

out

Outlet streams

rebo

Reboiler

s1

Syngas feed streams 32


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Energy & Fuels

sol

Solvent

vap

Vapor phase or vaporization

W GS

Water-gas-shift reactions

n

Hydrocarbon carbon number

r

Reactive (stages)

Parameters −∆soln H R

Enthalpy of dissolution temperature dependence constant, K

κii0

Binary intereaction parameters

ν

Stoichiometric coefficient

Ea

Activation energy, J mol−1

G0,F T Intercept of Gibbs free energy of FT reaction versus carbon number curve, kJ mol−1 GF T

Gibbs free energy of FT reaction, kJ mol−1

R

Gas constant, J K −1 mol−1

Variables α

Chain growth parameter

V¯

Partial molar volume, cm3 /g mole

∆H

Enthalpy of stream flow, kW

γ

Activity coefficient or Stage connection factor

φv

Vapor phase fugacity coefficient

ξ

Extent of reaction, kmol/sec 33


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

ζ

Tray efficiency factor

aii , ajj Mixing rule variables F

Feed mole flow rate, kmol/sec

f

Component fugacity, bar

H

Henry’s constant, bar

K

Phase equilibrium constant

KD

Liquid-liquid distribution coefficient

kF T

FT reaction rate constant, kmol kg −1 s−1 bar−1

kH

Henry’s law constant for component solubility in water, mol kg −1 bar−1

kW GS WGS reaction rate constant, kmol kg −1 s−1 bar−1.75 L

Liquid stream mole flow rate, kmol/sec

P

Pressure, bar

Q

Heat duty, kW

Rj

Reaction rate of reaction j, kmol kg −1 s−1

T

Temperature, K

Tr

Reference temperature for calculation of water solubility in oil phase, Tr = T /T3c , T3c = 539.1 K

V

Vapor stream mole flow rate, kmol/sec

x

Mole fraction in liquid phase

y

Mole fraction in vapor phase 34


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Energy & Fuels

Z

Compressibility factor

z

Molar fraction in total hydrocarbon product or mole composition in feed stram or model objective

References (1) Dry, M. E. Catalysis today 2002, 71, 227–241. (2) Schulz, H. Applied Catalysis A: General 1999, 186, 3–12. (3) Davis, B. H. Catalysis Today 2002, 71, 249–300. (4) Sie, S.; Krishna, R. Applied Catalysis A: General 1999, 186, 55–70. (5) Saxena, S.; Rosen, M.; Smith, D.; Ruether, J. Chemical Engineering Communications 1986, 40, 97–151. (6) Maretto, C.; Krishna, R. Catalysis today 1999, 52, 279–289. (7) Ahón, V. R.; Costa Jr, E. F.; Monteagudo, J. E.; Fontes, C. E.; Biscaia Jr, E. C.; Lage, P. L. Chemical Engineering Science 2005, 60, 677–694. (8) Krishna, R.; Sie, S. Fuel Processing Technology 2000, 64, 73–105. (9) Krishna, R.; Van Baten, J.; Urseanu, M.; Ellenberger, J. Chemical Engineering Science 2001, 56, 537–545. (10) Sehabiague, L.; Morsi, B. I. International Journal of Chemical Reactor Engineering 2013, 11, 309–330. (11) Glasser, D.; Hildebrandt, D.; Liu, X.; Lu, X.; Masuku, C. M. Current Opinion in Chemical Engineering 2012, 1, 296–302. (12) Hauan, S.; Ciric, A. R.; Westerberg, A. W.; Lien, K. M. Chemical Engineering Science 2000, 55, 3145–3159.

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(13) York, K. M.; Keller, A. E.; Wright, H. A.; Harkins, T. H. Catalytic Distillation Reactor. Patent Application, AU 2001/017698 A (Australia), 2001. (14) Maretto, C.; Krishna, R. Catalysis Today 2001, 66, 241–248. (15) Srinivas, S.; Malik, R. K.; Mahajani, S. M. Industrial & Engineering Chemistry Research 2008, 47, 889–899. (16) Srinivas, S.; Malik, R. K.; Mahajani, S. M. Industrial & engineering chemistry research 2009, 48, 4710–4718. (17) Srinivas, S.; Malik, R. K.; Mahajani, S. M. Industrial & Engineering Chemistry Research 2009, 48, 4719–4730. (18) Masuku, C. M.; Lu, X.; Hildebrandt, D.; Glasser, D. Fuel Processing Technology 2015, 130, 54–61. (19) Ojeda, M.; Nabar, R.; Nilekar, A. U.; Ishikawa, A.; Mavrikakis, M.; Iglesia, E. Journal of Catalysis 2010, 272, 287–297. (20) Ojeda, M.; Li, A.; Nabar, R.; Nilekar, A. U.; Mavrikakis, M.; Iglesia, E. The Journal of Physical Chemistry C 2010, 114, 19761–19770. (21) Inderwildi, O. R.; Jenkins, S. J.; King, D. A. The Journal of Physical Chemistry C 2008, 112, 1305–1307. (22) Van Santen, R.; Markvoort, A.; Filot, I.; Ghouri, M.; Hensen, E. Physical Chemistry Chemical Physics 2013, 15, 17038–17063. (23) Shi, B.; Jin, C. Applied Catalysis A: General 2011, 393, 178–183. (24) Shi, B.; Liao, Y.; Naumovitz, J. L. Applied Catalysis A: General 2015, 490, 201–206. (25) Van Der Laan, G. P.; Beenackers, A. Catalysis Reviews 1999, 41, 255–318. (26) Yates, I. C.; Satterfield, C. N. Energy & Fuels 1991, 5, 168–173. (27) Wang, Y.-N.; Ma, W.-P.; Lu, Y.-J.; Yang, J.; Xu, Y.-Y.; Xiang, H.-W.; Li, Y.-W.; Zhao, Y.-L.; Zhang, B.-J. Fuel 2003, 82, 195–213. 36


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