The world energy infrastructure currently uses mainly fossil fuels such as hydrocarbons and various types of coal with steadily growing contribution from renewable energy sources as the energy economy based on hydrogen as energy carrier is still in its infancy. One of the key problems associated with carbonaceous fuels is unfavorable deposition of solid carbon from gaseous carbon-hydrogen-oxygen (C-H-O) mixtures met in a range of industrial processes (Rostrup-Nielsen 1997). The main sources of depositing carbon were described in the literature as (i) molecules of elemental carbon created in thermal cracking reactions in gas phase (Jankhah et al. 2008) and (ii) C-H-O compounds of various proportions of the elements, decomposed on catalyst surfaces (Chen et al. 2011, Korup et al. 2012). Carbon deposition in (i) consists in direct-phase transformation of the element molecules from gaseous state into the solid form.
Decomposition of several hydrocarbons was often used to produce the so-called pyrolytic carbon (pyrocarbon), which found significant applications initially in the space and aircraft industries. Nowadays, carbon nanomaterials are being manufactured mainly in the form of single- or multi-walled nanotubes (Dasgupta et al. 2011, Prasek et al. 2011). In contrast, carbon deposition process (coking, fouling) is highly undesirable in petrochemistry (Benzinger et al. 1996) at cracking and dehydrogenation processes and also in the solid oxide fuel cell technology. Likewise, combustion of carbon containing fuels is frequently accompanied by soot formation due to incomplete oxidation (Wijayanta et al. 2012).
However, this study is focused on conditions of carbon deposition from reformed fuels that is detrimental mainly to catalyst surfaces, depending on the morphology of the surface carbon (Pakhare and Spivey 2014). Therefore, the phenomenon of solid carbon formation both in syngas production and in preprocessing of fuels for solid oxide fuel cells (SOFCs) is of profound importance. The presence of C1 to C4 hydrocarbons and CO in reformate gases was found to be a major precursor of carbon formation in SOFCs (Bae et al. 2010). On the other hand, the nucleation of graphite over a catalyst is extremely sensitive to its nanostructure (Bengaard et al. 2002). The deposition usually occurs on solid surfaces, but nuclei of solid carbon may also form in the fluid phase (Guo et al. 2013, Lemaire et al. 2013).
This review starts from a general description of literature data on deposition of various forms of solid carbon from gaseous reformate mixtures. Based on the literature experimental data, an original three-phase carbon diagram is proposed for a wide P-T range followed by validated mathematical expressions for equilibrium carbon vapor pressure over carbon solid forms. In the following chapter, quantitative description of thermodynamic equilibrium is included for a range of allotropic forms of solid carbon, including two filamentous forms. Then the published models of deposition equilibrium are briefly presented. Based on minimization of the Gibbs energy of the considered C-H-O systems, it was possible to compute equilibrium phase concentrations and detect the depositing carbon form. Series of such simulations for varied temperature and pressure allowed to construct ternary C-H-O plots showing boundaries of both graphitic and filamentous carbon deposition zones.
2 Carbon deposits
Three major types of industrial reforming of hydrocarbon/alcohol fuels can be distinguished. Those are steam reforming (SR), catalytic partial oxidation (CPOX), and dry reforming (DR). While SR and DR are endothermic and CPOX is exothermic, a combination of SR and CPOX is named oxidative SR and its specific energy-neutral version is called autothermal reforming (Diaz Alvarado and Gracia 2012). Depending on the process conditions, all those reforming processes can result in formation of carbonic solid. Thermal decomposition of hydrocarbons and alcohols proceeds along complex paths, and the mechanism details are still searched for (Alberton et al. 2007, Dufour et al. 2009, Wang et al. 2009). Most of the studies of carbon deposition indicate formation of elemental carbon in the reforming reactions as the main source of carbon deposition, however, without supplying a validated mechanism. Two models were considered for pyrolytic carbon (Zheng et al. 2013): a deposition droplet model proposed by Shi et al. (1997) and the combined mechanism of nucleation on graphene and of carbon growth at the edge of graphene (Hu and Hüttinger 2002). Several carbon containing intermediate species were detected during methane decomposition at catalyst surfaces, and associated mechanisms of filamentous deposit growth were proposed (de Bokx et al. 1985, Snoeck et al. 1997a, Wagg et al. 2005).
A range of carbonaceous deposit types, both amorphous and crystalline, was found in broad studies of carbon deposition. The deposits widely differ in morphology and reactivity (Bartholomew 2001, Wagg et al. 2005, Diaz Alvarado and Gracia 2012). Carbon deposition can result in chemical reactions over a catalyst surface, where several forms including atomic carbon can appear (Kee et al. 2005), or in gas phase due to free-radical reactions ultimately forming polyaromatic compounds (Sheng and Dean 2004). McCarty and Wise (1997) observed four main types of carbon on a nickel catalyst. Other early classifications (Rostrup-Nielsen 1979, 1984) proposed three types of solid carbon formed at reforming of fuels: (i) encapsulating film, (ii) whisker-like (filamentous), and (iii) pyrolytic carbon. Those carbon forms were also observed by Sehested (2006). The encapsulating tar-like carbon film (i), according to Rostrup-Nielsen (1979, 1984), is formed by slow polymerization of higher hydrocarbon radicals at temperatures below 773 K. According to Sehested (2006), the encapsulating carbon (gum) deposits consist of a thin film of hydrocarbon or a few layers of graphite. Formation of the filamentous, graphitic type in the form of tubular filaments (whiskers, fibers ii) prevails in the temperature range from 720 K up to 870 K (Rostrup-Nielsen, 1979, 1984), whereas at the latter temperature, the formation of pyrolytic carbon (soot, iii) begins to predominate as result of thermal cracking of hydrocarbons. The mechanism of pyrolytic carbon deposition in the range of 1073–1273 K was studied by Zheng et al. (2013). Alstrup et al. (1998) found that carbon filaments are formed on nickel catalysts when hydrogen is present in the gas phase while encapsulating carbon dominated in pure CO reactions.
Filamentous carbon deposits were detected more than 120 years ago by Hughes and Chambers (1889) during their search for carbon filaments for electric lighting. The finding was confirmed by Davis et al. (1953) by means of electron micrography. Sehested (2006) underlines the detrimental role of whisker carbon for catalysts and links development of pyrolytic carbon with the appearance of higher hydrocarbons at elevated temperature. Generally similar results were reported by Alberton et al. (2007) and He and Hill (2007) indicating the importance of catalyst type and the highly destructive role of carbon dissolved within the catalyst structure above 873 K. During catalytic and thermal cracking of ethanol in the temperature range from 523 K to 1123 K, Jankhah et al. (2008) found that carbon deposits can be formed as filaments, both rectilinear and helicoidal. Essentially, carbon nanotubes (CNTs) are layers of coaxially assembled graphene planes, and their continuous catalytic production was reviewed by Ying et al. (2011). A variety of CNTs can exist ranging from single-walled to multi-walled tubes with Young’s modulus over 1 TPa and tensile strength about 150 GPa (Paradise and Goswami 2007). According to a thermodynamic analysis carried out by Gozzi et al. (2007), the transformation of solid carbon from graphite to multi-walled carbon nanotubes (MWCNTs) becomes spontaneous at the temperature, T, of 704±13 K. The authors also found that thermal decomposition of methane to form MWCNTs is thermodynamically favored at T>809 K. In a following paper (Gozzi et al. 2008), they concluded that carbon nucleation from gas phase usually results in a mixture of various nanotubes and also amorphous or/and graphitic forms. Proportions of those carbon forms depend mainly on temperature and the catalyst type and its granularity. Similar conclusions about coexistence of different forms of solid carbon were drawn by Snoeck and Froment (2002) and Makama et al. (2014). Thermodynamic data for single-walled CNTs in bundles were also obtained using an indirect solid galvanic cell method (Gozzi et al. 2009).
A specific two-dimensional form of solid carbon, called graphene, can also be formed on metal surfaces, e.g. Ni (Arkatova 2010) and Cu. The properties of graphene were extensively described by Soldano et al. (2010). Three methods of graphene preparation were employed: (i) segregation of carbon from the bulk of the metal, (ii) chemical vapor deposition of hydrocarbons, and (iii) carbon vapor deposition (Xu et al. 2013). Despite intensive investigations, many fundamental aspects of the graphene nucleation are still not fully understood (Mehdipour and Ostrikov 2012, Zhang et al. 2012, Kim et al. 2013).
A purely chemical kinetic analysis of pyrocarbon deposition was performed by the team led by Hüttinger (Becker and Hüttinger 1998a,b,c,d, Hu and Hüttinger 2002). Kazakov et al. (1995) proposed a model for soot formation in laminar flames, and Appel et al. (2000) presented a new kinetic model of soot formation in flames. The kinetics of nucleation and growth of soot particles in turbulent combustion flames was also numerically analyzed (Zucca et al. 2006, Chernov et al. 2012). A kinetic expression for the coke formation rate on a nickel catalyst was derived by Ginsburg et al. (2005) and Asai et al. (2008), and evidence of a prior formation of unstable carbides was presented by Kock et al. (1985). Based on an elementary kinetic description presented by Bessler et al. (2007) for SOFCs, a modeling study for predicting carbon deposition from reformates was also carried out (Yurkiv et al. 2013). A review and a thorough discussion of the present knowledge of the factors controlling coke formation in SOFCs were presented by Wang et al. (2013).
It can be summarized that carbon deposition rate strongly depends on temperature, gas phase composition, presence of catalysts, and the type of applied catalyst (Rostrup-Nielsen 1972, de Bokx et al. 1985, Kee et al. 2005, Mallon and Kendall 2005, Ke et al. 2006, Kim et al. 2006, Chen et al. 2011, Yurkiv et al. 2013). Several authors (Rostrup-Nielsen 1972, 1984, Jankhah et al. 2008, Cimenti and Hill 2009, Essmann et al. 2014) highlighted that both the thermodynamic equilibrium conditions and the kinetic rate of all reactions and phase transitions should be considered in predictions of carbon deposits. However, Rostrup-Nielsen and Christiansen (2011) showed that the equilibrium constants for methane decomposition decisively depend on the metal catalyst type and stated that “[t]he important question is whether or not carbon is formed and not the rate at which it may be formed.” Therefore, the following chapters are focused on the analysis of current literature information for temperature-pressure conditions for stable phases, which were first presented in the form of a gas-liquid-solid diagram for pure carbon and then in ternary diagrams for C-H-O reformates in thermodynamic equilibrium.
3 Carbon phase diagram
Carbon is the lightest element of Group IV and can form allotropes of three different covalent bond formations in condensed phases: sp3 hybridization (diamond, liquid), sp2 (graphite, nanotubes, graphene, fullerenes, liquid), and metastable sp (carbynes) (Ghiringhelli et al. 2008). Thermal functions for gaseous carbon species C1 to C7 were published (Lee and Sanborn 1973) over 40 years ago.
The overall graphical form of equilibrium phases is usually presented as a pressure vs. temperature, P-T, phase diagram indicating areas of existence of thermodynamically stable forms. Two main points are characteristic for such diagrams: the triple phase point for the coexistence of solid, liquid, and gas phases and the critical liquid-gas point. The two points for carbon appear at high temperatures, above 4000 K. Therefore, experimental investigations of the carbon equilibrium conditions required applications of special techniques. The published papers refer to either gas-solid (graphite) equilibrium (Marshall and Norton 1950, Clarke and Fox 1969, Lee and Sanborn 1973, Leider et al. 1973, Baker 1982, Lee and Choi 1998, Havstad and Ferencz 2002, Joseph et al. 2002, Miller et al. 2008) or equilibrium for liquid-graphite-diamond (Fried and Howard 2000, Ghiringhelli et al. 2005, Savvatimskiy 2005, Wang et al. 2005, Ghiringhelli et al. 2008, Yang and Li 2008, Umantsev and Akkerman 2010). The liquid-gas phase boundary was given little attention with the exception of Bundy et al. (1996). However, research papers on carbon equilibria were mostly devoted to theoretical studies rather than to the experimental ones.
3.1 Carbon solid – liquid and liquid – gas equilibria
van Thiel and Ree (1989) proposed a phase diagram of graphite-diamond-liquid for bulk carbon. The P-T diagram was enhanced for bulk carbon by Bundy et al. (1996) up to T=6000 K and extended for diamond melting by Wang et al. (2005) above 8000 K and also refined to account for carbon crystal size by Savvatimskiy (2005), who presented experimental thermodynamic data for the graphite melting points. The diagram was further adjusted for carbon nanocrystals of graphite and diamond by Yang and Li (2008). However, the entire phase diagram should cover the regions of thermodynamically stable three phases: (i) solid phases of graphite (G), diamond (D), and CNTs; (ii) gas phase (vapors, V); and (iii) a single liquid carbon (L) zone since Ghiringhelli et al. (2008) rejected existence of two phases of liquid carbon, as suggested elsewhere. The border line between the G and V areas ends at the triple point G-L-V, which was assumed according to Leider et al. (1973) for TGLV=4100 K and PGLV=1.12 bar. It should also be mentioned that the NIST-JANAF tables (Chase 1998) suggest that TGLV=4765 K and PGLV=26.95 bar, and the same temperature value is suggested for 1 bar by the CRC Handbook (Lide 2005). Ghiringhelli et al. (2005) established the location of the second triple point for graphite-diamond-liquid carbon at TGDL=4250K and PGDL=164,000 bar. It was also found that the P-T coordinates of the triple point, TGDL, shift to lower P and T with decreasing carbon nanocrystal size (Yang and Li 2008).
According to Togaya (1997), the line between the two triple points, GLV-GDL, has a maximum at the temperature of 4790 K and P=5.6 GPa. The coordinates of the critical point, CR, where differences between the carbon liquid and gas phases vanish was adopted based on Leider et al.’s (1973) approximation at TCR=6810 K and PCR=2300 bar. The line dividing the gas (V) and liquid (L) carbon phases was assumed linear between CR and GLV, and the line between the diamond (D) and liquid (L) areas was based on Savvatimskiy (2005) and Wang et al. (2005) phase diagrams. The complete carbon phase diagram shown in Figure 1 was based on Bundy et al.’s (1996) proposal; however, it was widened to include important gas range and upgraded using a wide collection of literature data described in the next section. In order to better show details of the carbon diagram, the axis of equilibrium carbon pressure, in Figure 1 is presented in the logarithmic scale.
An important issue results from the considerations in section 3.1 that no stable liquid carbon phase is expected within the temperature range of T<3000 K. It follows from the graph that the lines between G-D and D-L and L-V areas are of little importance to the deposition equilibrium and only the (de)sublimation lines between carbon vapors V and either solid graphite G or nanotubes CNT play significant role.
3.2 Carbon solid – gas equilibria
The graphite-vapor equilibrium was empirically studied by measuring evaporation rates at temperatures above 2000 K (Marshall and Norton 1950, Lee and Sanborn 1973, Joseph et al. 2002). Carbon vapors were assumed to contain several polyatomic species (Clarke and Fox 1969, Leider et al. 1973) ranging from C1 to C7; however, C6 was later ruled out by Joseph et al. (2002). Careful selection, tuning, and generalization of available experimental data for carbon led to establishing approved thermochemical tables, such as NIST-JANAF data (Chase 1998). Therefore, the equilibrium line for carbon sublimation that divides the area of carbonic vapors (V) and graphitic solid (G) in Figure 1 was based on the NIST-JANAF data presented in details in the next section.
Typical temperature of reformate gases ranges from ambient up to 1500 K; therefore, this range is adopted in the thermodynamic analysis for carbon solid depositions of this present study. Thus the following thermodynamic analysis for reformates refers to relevant gas-solid equilibria in that range, first for graphite and then for other carbon deposits.
3.2.1 Carbon vapors over graphite
Required conditions for carbon deposition usually refer to the state of thermodynamic equilibrium of the species containing carbon in the solid and gas phases. It is therefore essential to apply proper tools of modeling of the equilibrium for C-H-O species and use current data of thermodynamic properties of different forms of elemental carbon both in the gas and solid phases.
The standard thermodynamic approach that assumes equilibrium of a closed system is generally (e.g. Smith et al. 2005) characterized by two conditions: a minimum of the total Gibbs energy at constant temperature and pressure, G, of the considered system, which consists of a set of nj moles of species “j”:
and also by equality of the partial molar Gibbs energy, μj, i.e. chemical potential of each species in all phases (Smith et al. 2005), either gas (g), liquid (l), or solid (s).
For a physical phase transfer or a chemical reaction between species Aj linked by the stoichiometric equation and in the state of their thermodynamic equilibrium, denoted by “*”, the species system undergoing phase transfer or/and reaction obeys Equation (3).
The chemical potential of a pure species in any phase is usually presented as a sum of its standard-state value, and the contribution resulting from a ratio of the species activities in a current state, aj, and in the reference state, normally the standard state,
Equation (4) is valid both for pure species and for species in a single-phase mixture. The aj activity depends on the actual temperature, T; pressure, P; and composition of the considered phase. As accepted by convention (e.g. Atkins 1998), the standard activity of any pure element in its condensed state is equal to the unit for its stable form in the standard conditions (pure species under P0=0.1 MPa) using various definitions and units of the activity. In a similar way, the actual activity can be expressed by means of various measures of species concentration, such as pressure, molar ratio, etc. The species activity, aj, for gas phases is usually termed “fugacity”, fj, and is defined as a product of the non-dimensional fugacity coefficient, ϕj, and a species pressure, Pj:
In that case, the standard activity of a species, is also related to the standard pressure, P0. On the other hand, for gas species under pressures greatly lower than its critical pressure and for temperature very much higher than the species critical temperature, both the species compressibility factor and fugacity coefficient are close to 1 (Atkins 1998). Therefore, for the saturated carbon vapor pressures, below 10−19 bar and the analyzed temperatures up to 1500 K (cf. Figure 1), one can safely assume a0 [Ci(g)]=P0 and also a[Ci(g)]=P[Ci(g)]. In addition, the solid-state activities of graphite as the stable form of solid carbon, both the standard, a0[Cg(s)], and actual, a[Cg(s)], ones are also very close to 1 at total pressure close to P0.
However, one should distinguish between various allotropic forms of carbon, appearing both in the gaseous, Ci(g), and solid, Ck(s), states. Most published gaseous forms of carbon arising from reformates are Ci(g) for i=1 up to 8. Solid carbon, Ck(s), can be limited here to four basic classes: graphitic, k=g, two filamentous k=f, and amorphous k=a. When necessary, two filamentous forms will be distinguished, the multi-wall, k=MW, and single-wall, k=SW.
Applying Equations (3–5) and the conventions for carbon in the gas (j=Ci(g)) or solid (j=Ck(s)) phases in the state of thermodynamic equilibrium (*), we get:
and as in the considered cases of carbon deposition below 1500 K, ϕCi≅1, one can express the equilibrium (saturation) pressure over the graphitic carbon solid phase, shown by the “/Cg(s)” index as
That form is convenient for analyzing the relationship P*[Ci(g)] using the standard values of the chemical potential for various gaseous carbon molecules, Ci(g), available either directly or indirectly in the subject literature (Yaws 1995, 2012, Chase 1998, Scientific Group 1999, Lide 2005) and also in the open source Cantera (Goodwin 2009) or the commercial software HSC (HSC Chemistry 2014). The chemical potential of formation of gas carbon molecules from graphite is equal to the numerator, μ0[Ci(g)]–μ0[Cg(s)] of Equation (7). Due to the convention used in physical chemistry, it is often assumed μ0[Cg(s)]=0 for the basic solid form of atomic species such as graphite in the reference standard conditions. However, this must be coherent with parallel assumptions for the standard partial molar enthalpy, and partial molar entropy, to comply with the general relationship G0=H0–TS0. One should also remember that a direct deposition process of carbon vapors is only possible when the local pressure of carbon vapors, P[Ci(g)], exceeds the corresponding equilibrium (saturation) counterpart, P*[Ci(g)], resulting from Equation (7).
The temperature influence on the equilibrium pressure of gaseous carbon allotropes, C1 to C8, over solid graphite is depicted in Figure 2, as computed with the HSC software (HSC Chemistry 2014). It shows that the first molecule, C1, exhibits principal contribution to the total equilibrium pressure of gaseous carbon and the contribution of C4 to C8 molecules can be safely neglected.
The total pressure of carbon vapors over graphite (/Cg(s)),
The Miller et al. (2008) correlation for such equilibrium pressures, P* in Pa, against temperature, T in K, recalculated in SI units can be presented as
The dependence is graphically shown for comparison in Figure 2 as the thin solid line. The relationship of Equation (9) is especially useful for a quick comparison with the equilibrium pressure of gaseous carbon resulting from selected reactions of C-H-O reformates. It was also concluded that the three sources (Chase 1998, Miller et al. 2008, HSC Chemistry 2014) deliver very similar parameters of equilibrium conditions for carbon gases over solid graphite. In addition, a similarly good agreement in the individual equilibrium pressures for C1 up to C5 over graphite was found when data from Chase (1998) and (HSC Chemistry 2014) were compared (not shown here).
Two additional P vs. T lines for CNTs, SWCNT, and MWCNT are also shown in the figure, and their experimental origin is presented in the next section. The equilibrium carbon vapor pressures over the nanotubes are not largely different from the total pressure over graphite within the investigated temperature ranges, as can be seen in Figure 2.
In summary, it may be stated that gaseous carbonic species accompany any solid carbon, although at very low partial pressures. In addition, surpassing the carbon equilibrium pressures described in Equation (7) creates conditions of supersaturation of carbon vapors and tendency to solid carbon depositions.
3.2.2 Equilibrium conditions for non-graphitic deposits
As early as 1905, Schenck and Heller (1905a,b) found that the equilibrium of the Boudouard reaction was influenced by the catalysts used and the obtained reactant compositions differed from those without presence of a catalyst. A similar conclusion was drawn by Pring and Fairlie (1912) in respect to the equilibrium of methane pyrolysis. The discrepancies were later ascribed to formation of solid filamentous carbon (Cf(s)), which has the standard chemical potential, μ0[Cf(s)] different from that of graphite. The values of μ0[Cf(s)] were first estimated by Rostrup-Nielsen (1972) and de Bokx et al. (1985) resulting in their equality to that for graphite, μ0[Cf(s)]=μ0[Cg(s)], roughly at 950 K for the selected size of catalyst grains.
Three methods of estimation of the standard chemical potential were published for solid carbon deposits, (Ck(s)), other than graphite (Cg(s)). The most accurate type was based on measuring the standard electromotive force, E0, of high temperature, solid-state galvanic cells (Jacob and Seetharaman 1995, Gozzi et al. 2007, 2009) with single-crystal CaF2 as a solid electrolyte. The virtual, resultant cell reactions between graphite and purified solid forms of filamentous or amorphous carbon were Cg(s)=Ck(s). Then the standard chemical potential change of the “i” reaction between solid carbon species, Δiμ0[C(s)] was calculated as a temperature function from the measured standard electromotive force, using Equation (10):
where n is the number of electrons transferred per carbon atom in the cells and F denotes the Faraday constant. The investigated chemical potential of the non-graphitic solid carbon, μ0[Ck(s)], can be then derived from
However, the reference value of μ0[Cg(s)] in Equation (11) was dependent on the convention used.
Another method of determination of the chemical potential value for single-walled CNTs, μ0[Cf(s)], is based on precise oxidation calorimetry (Levchenko et al. 2011) with entropy changes calculated using heat capacities of graphite and single-walled CNT.
Yet most published estimates of μ0[Ck(s)] relied on measuring gas reactant partial pressures at equilibrium conditions, and they were determined either at the beginning of coking (Bernardo et al. 1985) or more typically at the coking threshold obtained for both growing and decreasing pressures. Carbon depositions accompanying the Boudouard reaction (21) and methane pyrolysis (23) were investigated and reported in a wide temperature range, in most cases, with dispersed nickel as a catalyst. Usually, the relevant equilibrium quotients (constants), Kp,i, used in the subject literature, like those cited in Equations (22), (24), and (26), respectively, for reactions (21), (23), and (25), were originally calculated as follows:
A number of experimental data for such Kp,i quotients are available in the subject literature (Bromley and Strickland-Constable 1960, Rostrup-Nielsen 1972, Bernardo et al. 1985, de Bokx et al. 1985, Tavares et al. 1994, Rostrup-Nielsen et al. 1997, Snoeck et al. 1997a,b, Snoeck and Froment 2002). Formally, each of the partial pressures in Equations (12) to (14) should be divided by the reference pressure, P0=1 bar. Therefore, those published quotient values depend on the pressure units used, and this must be accounted for when calculating the chemical potential deviation by means of Equation (15) or more general Equation (19). The “i” subscript denotes a considered transformation reaction between different allotropic forms of solid carbon, Cg(s) and Ck(s).
That energy difference can be used to determine the standard chemical potential for deposited non-graphitic carbon, μ0[Ck(s)], from Equation (11), applying literature thermodynamic data for graphite, μ0[Cg(s)]. Such method was already used in 1960 by Bromley and Strickland-Constable (1960) and also by the Rostrup-Nielsen group (Rostrup-Nielsen 1984, Bernardo et al. 1985). In principle, it should be possible to derive temperature dependences of μ0[Cf(s)](T) for filamentous deposits (Ck=Cf) based on the equations for threshold constants cited by Snoeck and co-authors (Snoeck et al. 1997a,b, Snoeck and Froment 2002) for methane cracking and wet reforming of methane. However, those constants resulted in different standard enthalpy and entropy values for varied methane partial pressures (Snoeck et al. 1997a) and also the computed μ0[Cf(s)] values significantly differed from those published earlier. In addition, one has to conclude that majority of the papers quoting quantitative data for Δiμ0[Ck(s)] did not clarify how the μ0[Cg(s)] values for graphite were calculated for determining Kp,i[Cg(s)]. Nevertheless, it is informative to compare the published temperature functions of the chemical potential deviation, Δiμ0[Ck(s)]=f(T), and a collection of such plots of black (for CNT) and grey solid lines is graphically presented in Figure 3 along with information on data sources. The dotted reference line of Δiμ0[Cg(s)]=0, for graphite is also shown in Figure 3.
Preferred depositing conditions for filamentous carbon rather than graphite are those where Δiμ0[Ck(s)]<0, and most of the lines in Figure 3 go below the chemical potential level for graphite (dotted line) at the temperature ranging from 430°C to 680°C. It can also be concluded from Figure 3 that a relatively wide scatter exists between the direct or calculated data for which were published by different authors, and it may be useful to identify possible sources of such discrepancies. For instance, even from data in a single paper (Snoeck and Froment 2002), five different Δiμ0[Ck(s)] values can be derived (1a to 1e in Figure 3). In the authors’ opinions, two key reasons for the presented scatter can be indicated: (i) lack of crystallographic purity of the investigated carbon deposits and (ii) use of different values for the standard chemical potential of graphite, which led to differences in the Kp,i[Cg(s)] equilibrium quotients. The latter may have stemmed from different selections of the standard and reference states used in the published papers. For instance, the standard entropy value for graphite at 298.15 K is assumed either about 5.7 J mol−1 K−1, e.g. in Rostrup-Nielsen and Bak Hansen (1993), Lide (2005), and HSC Chemistry (2014), or 0.0 J mol−1 K−1, e.g. in Chase (1998) and Yaws (1995, 2012). Moreover, some authors apparently assumed μ0[Cg(s)]=0, independent of temperature for the graphitic solid carbon, which is in clear disagreement with the dependence of both standard enthalpy and entropy on temperature for every species.
Based on the evidence delivered by Gozzi et al. (2007), one may expect that usually more than one crystallographic form of solid carbon appears in carbon deposits. The photographic SEM evidence of presence of filamentous carbon, which was often shown in relevant papers, cannot be regarded as a sufficient proof of the deposit purity. Consequently, it was accepted in this study that thermodynamic data only for pure forms of filamentous carbon or graphite should be used in predicting equilibrium conditions of their deposition. Therefore, reliable experimental data for the purified multi-wall (Gozzi et al. 2007) and bundled single-walled (Gozzi et al. 2009) CNTs, shown in Figure 3 as black solid lines, are applied with thermodynamic fundamentals in the following sections to derive conditions of carbon deposition for various reformate compositions. Relevant relationships reported for purified CNTs were published by Gozzi et al. (2007) for Ck(s)=MWCNT in temperature ranging from 820 K to 920 K:
and by Gozzi et al. (2009) for bundled SWCNT in the temperature range from 750 K to 1015 K in a polynomial form:
where F denotes the Faraday constant.
Latini et al. (2014) reported in 2014 that there were no articles other than the two by Gozzi et al. (2007, 2009) that dealt with thermodynamic measurements at high temperature on carbon nanostructures. In the study of Wagg et al. (2005), an upper bound of the chemical potential of SWCNT growth threshold was estimated for T>700°C as
which indeed results in significantly higher values than those calculated from Equation (17).
The necessity and importance of purifying the investigated samples of solid carbon deposits were clearly explained by Gozzi et al. (2009). Moreover, the authors in a further study obtained significant differences in the standard enthalpy and entropy even for separated and single-wall-in-bundle carbon nanofibers. A similar rationale can possibly explain substantial differences in Δiμ0[Ck(s)]=f(T) for amorphous carbon samples (Jacob and Seetharaman 1994, Terry and Yu 1995). It should also be mentioned that in another study (Lee et al. 2013), the standard partial molar enthalpy and entropy of the CNT formation were estimated as ΔfH0=54.46 kJ mol−1 and ΔfS0=68.9 J mol−1 K−1 based on literature data of de Bokx et al. (1985) and the NASA data (McBride et al. 2002) for graphite. The values were then used to derive C-H-O concentration boundaries for depositing CNT at varied temperature and resulted in the deposition boundary lines in a ternary diagram largely different (Lee et al. 2013) from the corresponding one shown in Figure 4B.
Perhaps Diaz Alvarado and Gracia (2012) were the only authors who concurrently considered published equilibria of various solid carbon forms: graphite, MWCNT, amorphous, and polymeric for glycerol reforming systems. The authors used data for multi-walled nanotubes based on Gozzi et al. (2007) and found existence of two regions: the most favorable carbonaceous solid type was graphite below 450°C, whereas CNTs were preferred to form above that threshold temperature. However, the deposition threshold temperatures determined in this study, shown by the vertical broken lines in Figure 3, were 431°C for the graphite-to-MWCNT change (as in Gozzi et al. 2007) and about 590°C for the MWCNT-to-SWCNT transformation. In conclusion, it should be stressed that carbon vapors are always expected in equilibrium over solid graphite or CNT, however, with a very small partial pressure.
4 Published models of deposition equilibrium
Several proposals for quantitative description of equilibrium conditions for carbon deposition from gaseous reformates have already been published, and an early communication in 1972 was delivered by Rostrup-Nielsen (1972). In the paper, formation of whisker carbon was evidenced, and the equilibrium constants for actual reaction with carbon deposition, Kp,observed, were compared to the theoretical equilibrium constant computed for graphite, Kp,graphite. Higher CH4 or CO concentrations were detected (Rostrup-Nielsen 1979) before carbon formation on catalysts than for non-catalyzed graphite deposition, which implies smaller constants, Kp,observed, than those for graphite, Kp,graphite, at a given temperature (Rostrup-Nielsen 1984). The excess chemical potential, Δμ[Ck(s)], of the deposited carbon compared to that based on graphite was estimated (Rostrup-Nielsen 1972) on the basis of the two constants, Equation (19).
That approach was further developed by the introduction of the so-called principle of equilibrated gas (Rostrup-Nielsen 1984), which states “[c]arbon formation is to be expected if the gas shows affinity for carbon after the establishment of the methane reforming and the shift equilibria.” In that case, Kp,observed>Kp,equilibrium and Δμ[Ck(s)]<0, and it is true except for noble metals as catalysts (Rostrup-Nielsen and Christiansen 2011). The steady-state activity of solid carbon, a[Ck(s)], was also defined based on a kinetic analysis (Rostrup-Nielsen 1984). The principle of actual gas was presented by Rostrup-Nielsen and Christiansen (2011) and has been often used to assess the risk of carbon formation in a non-equilibrium, steady-state process. In that case, the quotient, Kp,observed, should be calculated for actual, non-equilibrium concentrations.
In an early study, de Bokx et al. (1985) concluded that the deposited filamentous carbon in the presence of catalysts is a metastable carbide intermediate of a higher chemical potential than that of graphite but is not decomposed to graphite due to a high activation barrier for the homogeneous nucleation of graphite, which causes a very low rate of the nucleation. It also implies that almost filament-free operation is possible for more carbon bearing gases than it results from graphite equilibria.
A thermodynamic analysis of carbon deposition accompanying methane SR (Xu and Froment 1989) was carried out using a ratio, Vi, of the pressure quotient of the actual partial pressures of species “j”, pj, to the quotient for equilibrium conditions of the “i” reaction, the latter being the equilibrium constant, Kp,i, for that reaction.
That approach was also applied by Hou and Hughes (2001) for methane reforming. Magnitude of the ratio indicates whether a reaction goes to the right (Vi<1) or to the left (Vi>1) or remains at equilibrium (Vi=1). A related presentation of the conditions of carbon deposition was used in an analysis for catalyzed reforming of methane (Ginsburg et al. 2005), in a SOFC fueled by sewage biogas (van Herle et al. 2004), and also when the fuel cell was fed with methane (Klein et al. 2007).
One of the similar approaches (Tsiakaras and Demin 2001) was based on consideration of the Boudouard reaction of carbon disproportionation, Equation (21).
The relevant equilibrium constant of Equation (21), Kp,21, for species pressures, pi, was used (Tsiakaras and Demin 2001) to define the carbon activity, a[Ck(s)],21, which is a reciprocal of Vi.
It was commonly assumed that the carbon activity equals 1 at the thermodynamic equilibrium, and solid carbon formation is possible only when a[Ck(s)],21>1 (Assabumrungrat et al. 2004, Tsiakaras and Demin 2001). That analysis was also extended by considering carbon activities resulting from other reactions (Assabumrungrat et al. 2005, Sangtongkitcharoen et al. 2005, Assabumrungrat et al. 2006): methane pyrolysis (23) and reverse gasification (25)
In seven further studies (Sangtongkitcharoen et al. 2005, Klein et al. 2007, Nikooyeh et al. 2008, da Silva et al. 2009, Vakouftsi et al. 2011, Nematollahi et al. 2012, Trabold et al. 2012), the same or slightly transformed definitions, as those in Equations (22), (24), and (26), were applied to determine values of the solid carbon activity, a[Ck(s)]. The calculations were based on minimization of the Gibbs energy using either commercial software or thermodynamic data from literature.
Other types of carbon deposition analyses were based on the C-H-O diagrams, and they were relatively often quoted in the subject literature. Bartholomew (1982) was perhaps the first to publish equilibrium isotherms at 450°C and 1.4 atm both for graphite and amorphous carbon plotted in a C-H-O ternary diagram. The isotherm lines were based on literature experimental data and clearly showed the deposition region for amorphous carbon was smaller than that for graphite, i.e. its boundary was shifted towards higher C content of C-H-O mixtures. Eguchi et al. (2002) possibly pioneered systematic application of that approach supported by commercial software for computing the equilibrium phase contents with assumed temperature, pressure, and input moles. The boundaries were also presented in a ternary C-H-O diagram. A comprehensive study by Sasaki and Teraoka (2003b) led to detail boundaries of graphitic carbon deposition for equilibrium temperature ranging from 100°C to 1000°C. Essentially the same approach was reported in several other papers (Koh et al. 2001, 2002, Sasaki and Teraoka 2003a, Sasaki et al. 2004, Kee et al. 2008, Aravind et al. 2009, Chen et al. 2011, Lee et al. 2013, Liu et al. 2013) using various software codes. Determination of the carbon deposition boundary was based on assumption of a very small but non-zero amount of solid carbon in equilibrium (Koh et al. 2002) or a fraction of 10–6 of the total carbon amount (Sasaki and Teraoka 2003a). A similar criterion of the minimum carbon fraction, xC,min, for carbon deposition to occur was proposed by Heddrich et al. (2013) as xC,min=10–4, which resulted from the thermodynamic equilibrium concentrations computed by minimizing the Gibbs energy of reacting systems.
In another study, Jankhah et al. (2008) performed a thermodynamic analysis for DR of ethanol and also for its thermal cracking based on a commercial software that uses the Gibbs energy criterion, Equation (1). Minimum temperatures for thermodynamic exclusion of carbon formation at various oxidant/ethanol ratios were determined. Later on, Wang and Cao (2010) assumed, in their analysis of the conditions for solid carbon formation, that the partial molar Gibbs energies both of gas carbon and solid carbon, μ[C(g,s)], have zero value independent of temperature. This assumption may have caused significant impact on the authors’ assessment of critical temperatures above which coke is not formed. A two-case limit model for the driving force of graphene deposition from hydrocarbons was proposed by Lewis et al. (2013) for a hot wall reactor. Case 1 corresponded to thermodynamic equilibrium of chemical reactions of hydrocarbons in the gas phase, and a separate Case 2 considered the solid-gas phase equilibrium composition over a growing graphene film.
Thermodynamic properties of metallurgical coke formation were also investigated and discussed (Terry and Yu 1991, Jacob and Seetharaman 1994, Jacob and Seetharaman 1995, Terry and Yu 1995). A decrease of solid carbon amount resulting from application of the amorphous carbon properties instead of those for graphite was verified by Cimenti and Hill (2009) in their thermodynamic calculations. They also found that due to a higher energy of the amorphous form than that for graphite, a joint use of properties of the two carbon forms results in the same equilibrium composition as that for graphite.
It follows from the literature presented in this section that the thermodynamic equilibrium is one of the key factors controlling the formation of carbon deposits during reforming of C-H-O mixtures. The thermodynamic analyses carried out for a range of oxidized fuels indicated the predominant role of the mixture composition and temperature in solid carbon formation. The boundaries of no-deposition operational regimes were determined using different thermodynamic criteria and properties. In the majority of the published thermodynamic approaches, different authors considered solid carbon and typically neglected existence of carbon vapors in their thermodynamic calculations. However, the carbon element is known of having different allotropic forms both in the gas and solid phases and their thermodynamic equilibria may be critical in understanding the conditions of carbon deposition. Those aspects are considered in the following chapter.
5 Thermodynamic equilibrium for C-H-O mixtures
5.1 Temperature dependence of carbon deposition equilibrium
Carbonaceous depositions can occur as a result of side reactions in reforming popular hydrocarbons such as methane from natural/bio gas, propane with butane from LPG, or higher hydrocarbons from diesel fuel. The preceding section on gas-solid carbon equilibrium clearly quantified the equilibrium partial pressures of carbon vapors, which exist over different crystallographic forms of solid carbon. However, most authors who analyzed thermodynamic conditions of carbon deposition neglected the presence of carbon vapors in Equations (22), (24), and (26), assuming perhaps that all atomic carbon was completely transferred to the solid phase and therefore did not influence the reaction equilibrium. The importance of carbon gas pressure was explicitly found in only two papers (Bromley and Strickland-Constable 1960, da Silva et al. 2009); however, the solid carbon activity was used there in calculations of the reaction constants. Critical compositions of C-H-O mixtures for carbon deposition to occur were usually studied with the help of commercial software based on principles expressed in Equations (1) to (7).
Several papers (Bartholomew 1982, Koh et al. 2001, Eguchi et al. 2002, Koh et al. 2002, Sasaki and Teraoka 2003a,b, Sasaki et al. 2004, Kee et al. 2008, Aravind et al. 2009, Chen et al. 2011, Lee et al. 2013, Liu et al. 2013) reported C-H-O equilibrium calculations and their comprehensive presentation in ternary diagrams. Most of the published studies considered only the graphitic form of solid carbon. However, the thermodynamic data for CNTs by Gozzi et al. (2007, 2009), respectively, for the multi-wall and single-wall in bundles are also considered in this section. As already mentioned, the authors of this study decided that only the galvanic cell data (Gozzi et al. 2007, 2009) for filamentous carbon can be regarded as trustworthy and useful in the thermodynamic analysis. By analogy to approaches applied by other authors (Diaz Alvarado and Gracia, 2012), the chemical potential dependence on temperature by Gozzi et al. (2007) was assumed valid down to 600 K (inclined dotted line in Figure 3). In addition to graphite data as a reference, table data for diamond and experimental data for the chemical potential of formation of amorphous carbon (Jacob and Seetharaman 1994) were also added and applied in the HSC software (HSC 2014). However, the amorphous data along with that for diamond did not turn out significant within the ranges considered in this study, i.e. temperature between 200°C and 1000°C and pressure of 1 and 10 bar.
5.2 Numerical predictions of C-H-O equilibria
A wide series of computations of equilibrium phase concentrations of C-H-O reformates were performed with finding of the depositing carbon form, for varied temperature between 200°C and 1000°C with 100°C increments and two pressure levels of 1 bar and 10 bar. Based on those results, boundary lines between the deposition and no deposition zones were drawn in ternary diagrams and their selection is presented in Figures 4–7. In order to verify the simulation procedure, such boundary plots for 1 bar and graphite only in the solid phase (Figure 4A) were first compared with the original ternary plots published by Sasaki and Teraoka (2003b) (not shown here) and the two line sets turned out practically identical within the temperature range from 200°C to 1000°C.
The effect of adding Δiμ0[Cf(s)] data for filamentous carbon, either MWCNT [Equation (16)] or SWCNT [Equation (17)], as a thermodynamic equilibrium alternative to graphite is visually presented in Figure 4. In the performed calculations, the most stable forms of solid carbon were generally graphite for temperatures from 200°C to 431°C; MWCNT from 431°C up to about 577°C; and above it, the SWCNT prevailed. Therefore, corresponding boundary lines for a given temperature are identical up to 400°C, as only graphite was computed in the solid phase, and differ significantly for higher temperatures where filamentous carbon was predicted. Based on the presented results, it may be stated that by the time new reliable thermodynamic data for the chemical potential of filamentous carbon are published, the boundary lines shown in Figure 4B should be used as reference in reforming processes with nickel catalyst where filamentous carbon deposits may be formed.
For clearer viewing of the differences in the boundary lines, two diagrams for the total pressure of 1 bar are shown in Figure 5. Each of them presents three pairs of corresponding temperature lines either for graphite only (broken lines) or for all considered carbon solid forms (solid lines). Extension of the carbon deposition regions towards lower C concentration, with reference to those for graphite, is due to accounting also for the filamentous carbon forms and is illustrated in Figure 5 by grey shading. One can conclude that the deposition boundary lines for P=1 bar differ mainly for the reformate temperature close to 800°C and the line differences decrease towards the limits of the studied range from 500°C to 1000°C.
5.3 Pressure effect on carbon deposition equilibrium
The course of deposition boundary lines was found dependent also on the total pressure, P, of the C-H-O mixtures. A visual comparison of ternary plots with the equilibrium boundary lines computed for 10 bar, separately for graphite and for all considered solid carbon allotropes, is shown in Figures 6A and B, respectively. The equilibrium simulations were done for the same temperature range from 200°C to 1000°C as before for 1 bar. For that elevated pressure, the conditions of carbon deposition also significantly depended on the solid carbon allotropes considered. Identical to the threshold temperatures at 1 bar, graphite was in solid phase for temperatures from 200°C to 431°C, while MWCNT formed solid phase from 431°C up to about 577°C, and above it, the SWCNT prevailed.
A graphical presentation of the effect of considering either graphite only or filamentous carbon allotropes in the equilibrium simulations is shown in Figure 7 for the total pressure of 10 bar, which is qualitatively similar to that in Figure 5 for 1 bar. One can notice that both at the total pressure of 1 bar and 10 bar of the C-H-O mixtures, carbon deposition can occur when C>0.5, independent of the H and O atomic ratios. When considering all solid carbon forms, the maximum shift of the equilibrium boundary line relative to that for graphite was found for the reformate temperature of about 900°C, i.e. somewhat higher than for the total pressure of 1 bar.
The character of temperature influence on the course of the deposition boundary lines depends on the hydrogen concentration; for low H values, an increase in temperature lowers the critical C value, whereas for high hydrogen concentration (lower left part of diagrams), the temperature effect is opposite. Those effects are qualitatively similar for the two considered pressure values of 1 and 10 bar and either carbon deposit.
The effect of total pressure on the studied equilibrium conditions can also be assessed by comparing locations of the corresponding pairs of boundary lines in Figures 4A and 6A for graphite only and those in Figures 4B and 6B for all solid carbon forms. Two other ternary graphs were prepared (not shown here) for all carbon solid forms with boundary line pairs for 1 and 10 bar and for three temperature levels in each diagram, similarly to those in Figures 5 and 7. From their analysis, it was concluded that the maximum divergence in the lines, i.e. the strongest pressure effect, was in the temperature range from 700°C to 800°C when graphite was the only solid form allowed. However, the temperature range of the maximum influence of pressure was shifted up by about 100°C when all solid carbon allotropes were considered in equilibrium simulations for the total pressure of 10 bar, compared to those for 1 bar.
6 Concluding remarks
The Gibbs and activation energies of chemical reactions and physical phase transformations are dependent on temperature; therefore, both the equilibrium conditions and kinetics of the relevant carbon phase changes are also temperature sensitive, which was confirmed by several researchers. Carbon can deposit in variety of crystallographic forms on different catalysts, including single-walled and multi-walled nanotubes (Jankhah et al. 2008, Gozzi et al. 2008, Makama et al. 2014). Although thermodynamic criteria indicate either graphite or fibrous carbon as stable solid forms at T<1500 K, the kinetics of their mutual direct transformation can be extremely slow and difficult to detect.
Wide literature information on both theoretical and experimental studies on carbon deposition from C-H-O reformates was examined in this review. Based on the analysis of published models, it was concluded that the thermodynamic conditions for deposition of various carbon allotropes require stringent specification. In the first step, the three-phase carbon diagram was updated for only graphite and diamond in the solid phase. It was confirmed that within the temperature and pressure ranges up to 1500 K and 10 bar, respectively, the carbon equilibrium should be considered between only the gas and solid phases.
Physical chemistry fundamentals along with relevant experimental data for filamentous carbon were reviewed in Section 3. Significant differences were found in the standard chemical potentials estimated from the literature equilibrium concentrations for carbon deposition reactions. The lack of crystallographic purity of the carbonaceous deposits was suggested as the major cause of the discrepancy. Therefore, only the electrochemical measurement data for purified both single-walled an multi-walled CNTs (Gozzi et al. 2007, 2009) were accepted for filamentous carbon in the thermodynamic analysis presented in Sections 3 and 4. Verified table data for the chemical potential of other solid forms, such as graphite, diamond, and amorphous carbon, were also used in the computations.
C-H-O atomic concentration diagrams with equilibrium ternary plots for deposition boundaries of the carbon form, characterized by a minimum of its Gibbs energy (Equation (1)), were constructed for varied temperature and pressure. The obtained diagram for graphite as the single solid carbon was practically identical with that published by Sasaki and Teraoka (2003b). Within the analyzed ranges of T-P, only either graphite and or one of the two nanotube types turned out as the depositing carbon form. The following threshold temperatures were estimated for carbon allotropes in solid phase: 431°C for the transition of graphite to multi-walled nanotubes, which is similar to conclusions of Diaz Alvarado and Gracia (2012). A higher threshold at about 580°C was found for SWCNT replacing MWCNT, independent of pressure at 1 and 10 bar. The thresholds are shown by dotted lines in Figure 3. Consideration of the chemical potential data for filamentous carbon in addition to that for graphite resulted in shifting the deposition boundary lines towards lower carbon ratio in C-H-O reformates for temperatures above 431°C. This was valid independent of pressure, P, at constant temperature and of temperature for constant pressure. The strongest effect of the presence of filamentous carbon on lowering the critical carbon atomic ratio, C, was found for reformates without hydrogen, i.e. for H=0. The maximum shift in the boundary molar C ratio was estimated as ΔC1,G-CNT=−0.042 for nearly 800°C at 1 bar and as ΔC10,G-CNT=−0.052 for nearly 900°C at P=10 bar. The two shift values corresponded to an increase in the critical oxygen-to-carbon atomic ratio, O/C, by about 17% or 28% for 1 bar and 10 bar, respectively.
It is also envisaged that similar analyses focused on practical fuel reformates from different reforming methods will be presented as a follow-up in a separate paper.
The research program leading to these results received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) for the Fuel Cells and Hydrogen Joint Undertaking (FCH JU) under grant agreement no.  with the STAGE-SOFC acronym. Information contained in the paper reflects only the view of the authors. The FCH JU and the Union are not liable for any use that may be made of the information contained therein. The work was also financed from the Polish research funds awarded for project no. 3126/7.PR/2014/2 of international cooperation within STAGE-SOFC in the years 2014–2017.
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About the article
Zdzisław Jaworski, DSc, PhD, has been employed as an academic teacher since 1980 and, from 2004, has held a professor position of Chemical Engineering at the West Pomeranian University of Technology, Szczecin. His research interest was focused on numerical modeling of transport processes, mixing and reaction technology, heat transfer, multiphase fluid flow, rheology, thermodynamics, multi-scale modeling, and fuel cell technology. He has led several research projects, both national and European, and published over 150 refereed papers.
Barbara Zakrzewska has been employed as an academic teacher since 1996 at the West Pomeranian University of Technology, Szczecin. She was awarded a PhD with distinction in 2003. Her research interest was focused on CFD modeling of transport processes in mixing technology and application of artificial neural network in reactor modeling, and currently, she is involved in multi-scale modeling in chemical and process engineering and fuel cell technology. She has participated in several national and three European research projects.
Paulina Pianko-Oprych was awarded her PhD with distinction in 2005, and in 2016, she received the DSc degree. She has been a lecturer since 2008 at the West Pomeranian University of Technology, Szczecin. Her research results were published in 70 papers. With over 16 years of experience in CFD applications, she is involved in fuel cell modeling and the creation of BoP system models using process simulator tools. She is now a research-team leader and project manager in two EU projects.
Published Online: 2016-08-26
Published in Print: 2017-05-24
Citation Information: Reviews in Chemical Engineering, Volume 33, Issue 3, Pages 217–235, ISSN (Online) 2191-0235, ISSN (Print) 0167-8299, DOI: https://doi.org/10.1515/revce-2016-0022.
©2016, Zdzisław Jaworski et al., published by De Gruyter.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0