Parametric Studies of Steam Methane Reforming Using a Multiscale

Subscriber access provided by READING UNIV

Article

Parametric studies of steam methane reforming using a multiscale reactor model Flavio Eduardo da Cruz, Seçgin Karagöz, and Vasilios Manousiouthakis Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03253 • Publication Date (Web): 30 Oct 2017 Downloaded from http://pubs.acs.org on November 6, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Industrial & Engineering Chemistry Research is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 46

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

Industrial & Engineering Chemistry Research

Parametric studies of steam methane reforming using a multiscale reactor model Flavio Eduardo da Cruz, Seçgin Karagöz, and Vasilios I. Manousiouthakis* Dept. of Chemical and Biomolecular Engineering, University of California, Los Angeles, Los Angeles CA, 90095

*[email protected]

Abstract This work investigates the influence of porous catalyst structural parameters on a packed bed reactor’s performance, through the application of a multiscale reactor model. Constitutive equations, at the catalytic pellet and packed bed reactor length scales, are derived using the Reynolds transport theorem. Diffusive fluxes in the micro-scale (catalytic pellet) and macro-scale (reactor) domains are calculated using the Dusty-Gas-Model (DGM) and Stefan-Maxwell-Model (SMM) equations respectively, while Chapman-Enskog theory is applied to estimate diffusion and viscosity coefficients. Simulations are carried out for a case study on hydrogen production through steam methane reforming, using a finite element numerical scheme. The employed multiscale model enables the computation of catalyst effectiveness factors throughout the reactor, thus quantifying the effect on reactor performance of various catalyst structural characteristics, such as volumetric fraction, tortuosity, thermal conductivity, and mean pore diameter.

ACS Paragon Plus Environment


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

1. Introduction

In industrial applications, a common route for hydrogen production is through steam methane reforming (SMR) with natural gas as feedstock. The reforming process is conducted by feeding, at high temperature and pressure, a steam/methane mixture in metal tubes filled with catalyst pellets. A large furnace, generally containing hundreds of tubes, supplies the energy required to run the endothermic reforming process. The SMR process/furnace represents a significant portion of the capital and operating cost, including energy consumption,1 of the overall hydrogen production process.

To maximize reactivity inside the SMR tubes, the porous catalyst pellets are engineered to have a high internal surface area per volume ratio. Nevertheless, pellet performance is not dependent only on the catalyst surface density, but also the effectiveness of surface area utilization. To this end it is essential that the transport of the multicomponent reacting mixtures within the catalyst pores be quantified. The complex nature of reaction-diffusion in porous media necessitates that both macroscopic and microscopic effects be considered, thus resulting in a large number of parameters influencing overall system performance.2 Prominent such parameters are the pellet’s volumetric fraction, tortuosity, thermal conductivity, and mean pore diameter.3–5 The DGM is often employed to assess the effect of these parameters, by rigorously describing the transport of gases through the catalyst pores. Although early propositions for this theory were made by James Clerk Maxwell6 in the second half of the 19th century, the DGM formulation has been the object of continuous review and improvement.7,8 The accuracy of the DGM approach has been investigated and compared with other diffusion models, such as the Fickian model and the Wilke-Bosenquet model.9–11 These studies highlighted considerable differences in the species partial pressure profiles within the pellet. In addition, at high steam to methane ratios, the implementation of DGM was shown to be essential for accurate predictions. Nevertheless, even though the DGM is considered a more precise model at industrial conditions,12 its complexity leads to the study of reaction-diffusion effects in steam methane reforming catalyst pellets through the use of Fickian diffusion models.13,14


Page 2 of 46

Page 3 of 46

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


A useful approach to deal with diffusion limitations on reactive systems using porous catalysts is the effectiveness factor concept.15 Due to its simplicity and faster processing times, the use of the effectiveness factor became very popular. Models that use the effectiveness factor concept to account for diffusion limitations in the reacting system are classified as pseudo-homogeneous, while models simultaneously solving constitutive equations at different length scales, i.e., for the catalytic pellet and for the reactor, are classified as heterogeneous models.16 Several studies use the SMR reaction to compare heterogeneous with pseudo-homogeneous approaches,2,17 aiming to quantify differences since the pseudohomogeneous models require less computational power.

The importance of multiscale modeling in developing the insights needed to guide parametric studies and experimental design, and to enhance our comprehension of the impact of fundamental mechanisms, has been laid out in a number of visionary works by several researchers.18–29 Models for multiphase systems, such as packed bed catalytic reactors, have also been considered using averaging methods in the derivation process.19–23 In this work, a multiscale model is obtained from the application of the Reynolds transport theory on each component-phase subsystem of the main system, developing constitutive species, energy, and momentum equations, at the catalytic pellet and packed bed reactor length scales. Transport at these scales is quantified using the DGM, and SMM respectively, while Chapman-Enskog theory is used to estimate diffusion and viscosity coefficients. In the next section, the developed multiscale model is used to carry out a case study on hydrogen production using steam methane reforming. This study quantifies the effect of volumetric fraction, tortuosity, thermal conductivity, and mean pore diameter on reactor performance. Finally, the obtained results are discussed and conclusions are drawn.



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

2.

Page 4 of 46

Mathematical formulation

The catalyst pellet and the packed-bed reactor (PBR) are both multiphase systems containing solid and fluid phase material randomly distributed over the respective system’s domain. Even though the length scales of these systems are significantly different from one another, conservation laws must hold. Thus, the derivation of the pellet’s and PBR’s constitutive equations can be obtained through the repeated application of the Reynolds transport theorem (RTT)30,31 to each phase present within a control volume contained in the pellet and reactor domains, which are exclusive and complementary to each other. Throughout this work, the pellet and reactor domains are identified by the superscripts p and r , each composed by fluid and solid phases, identified by the subscripts f and s respectively. Considering B an extensive property of the subsystem defined by both phase    f , s and 

 domain    p, r , and b the corresponding intensive property, Eq.(1) shows RTT’s application to

phase  , and domain  .

dB  t  dt



           b    ,V dV    b  v  n   , A dA t  CV  CS

(1)

In the above equation, the LHS is a source term representing the rate of change of the extensive 

property B . The first term of the RHS is an accumulation term that accounts for the rate of change with  respect to time of B , within phase  in the fixed (static) control volume CV defined over the multiphase  domain  . The extensive property B is calculated through the volumetric integration over CV of the    specific intensive property b times the mass density  of phase  , times the volumetric fraction   ,V

of phase  , all within the differential control volume dV of domain  . The second term of the RHS is a


Page 5 of 46

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


flux term that accounts for changes in the property B within phase  in the CV defined over the domain  , due to the movement of matter across the control surface CS representing the boundary of

CV . The flux of B is calculated through the surface integration of the inner product of the CS normal vector n with the vector obtained by the scalar multiplication of the mass average velocity v of phase

 at the CS with the product of the intensive property b times the mass density  of phase  , times the surface fraction  ,A of phase  , all evaluated at the differential control surface dAt of domain  .

The constitutive laws for the multiscale model are developed by applying Eq.(1) on the pellet domain and on the reactor domain separately, for each phase within the respective domain, which generates one conservation equation for each domain-phase pair presented in the packed-bed reactor   system. Equation 1 is applied. Considering B and b to be the mass of phase  in the domain  and 1

respectively, and assuming constant mass in each individual phase, i.e., no solid formation or depletion, the total mass conservation equation is obtained as shown in equations (2) to (5).

Pellet-Fluid:

Pellet-Solid:

0





(2)

 p  p  p         s , A   sp,i vsp,i  s ,V  s ,i    t  i 1 i 1   

(3)

 p  p   p         f , A   fp,i v fp,i  f ,V  f ,i    t  i 1 i 1   

0









(4)





(5)

Reactor-Fluid:

0

 r r  f ,V  f      rf , A  rf vrf t

Reactor-Solid:

0

 r r  s,V s      sr, A sr vsr t

  Setting B and b to be equal to the species’ i mass in phase  within the domain  and its

mass fraction respectively, the species mass conservation is obtained as shown in equations (6) to (9).



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



 p p p  f ,V w f ,i  f      fp, A n fp,i  t

Pellet-Fluid:

M i R fp,i 

Pellet-Solid:

M i Rsp,i  M i R rf ,i 

Reactor-Fluid:



; ;

i  1,

(6)

i  1,



(7)



 r r r  f ,V w f ,i  f      rf , A wrf ,i  rf v rf  j rf ,i     t

M i Rsr,i 

Reactor-Solid:



 p p p  s ,V ws ,i s      sp, A nsp,i  t



Page 6 of 46





 r r r  s ,V ws ,i  s      sr, A wsr,i sr vsr  jsr,i     t

i  1,

;

(8)

i  1,

;

(9)

Although equations (2) to (5) are linear combinations of Eq.(6) to (9) in their respective phase-domain subsystem, one can decide to solve either the component balances for all species, or the total balance plus the component balances for all species but one. In the pellet-fluid subsystem’s species mass conservation, the combined mass flux np,i is defined through the DGM equation, Eq.(10), presented below:

 x fp, j p p x fp,i p p   1  p p  n   f ,A nf , j       f , A n f ,i  f , A f ,i Mj  M i   M i DiK   1 1 BO  p   c fp,i  x fp,i c fp,tot   p  c f ,i  p p ; i  1, p  p  DiK  f   

1  eff j 1 Dij













(10)

The DGM considers one stagnant solid species, steady-state operation, and negligible viscous effects. Even though the DGM considers thermal diffusion and thermal transpiration, these effects are neglected in this study. On the other hand, pressure gradient effects are accounted for in this study, although they are typically neglected.32,33 Despite the success of the DGM in describing gas transport in the transition region, the Binary Friction Model (BFM)34 establishes inaccuracies in the DGM approach, which manifest themselves in low-pressure applications (< 2 bar), although they are not substantive for the high-pressure SMR process presented in this work.

In the reactor-fluid subsystem’s species mass conservation, the diffusive mass flux j rf ,i is defined through the SMM equation, Eq.(11), presented below:


Page 7 of 46

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


   x rf ,i x rf , j  1 1 r  r    r j f , j  r j f ,i     f ,i  j 1  Dij   f , j     x rf ,i  wrf ,i  r   x rf ,i x rf , j  r x f , i     p    r r   pr j 1   f Dij    

    ;  Dir D rj   1 r  r  r   r T f   w f ,i w f , j   T f 

i  1,

(11)

In the pellet-fluid subsystem’s species mass conservation, the species i volumetric generation rate is quantified by the underlying reaction schemes kinetic model. Eq.(12) reflects the relation between the volumetric and the specific generation rates for the pellet domain.

R fp,i   sp,V  sp R fp,i

(12)

In the reactor-fluid subsystem’s species mass conservation, the species i volumetric generation rate is quantified by Eq.(13), which enforces its equality to the product of a surface to volume adjustment ratio times the overall ith species combined diffusion-convection mass flux, Eq.(10), at the control surface

CS p of the pellet domain.

R

r f ,i



r s ,V



Ap  p p  n  sp,V V p  f , A i



CS p

  

(13)

  Setting B and b to be equal to the total momentum and respective mass average velocity v of

phase  within the domain  , momentum conservation is obtained as shown in equations (14) to (17).

Pellet-Fluid:

Pellet-Solid:

  p  p p p      f ,V  w f ,i  f v f ,i    i 1   t   p Sf      p p p  p p        p  f , A  w f ,i  f v f ,i   w f ,i v f ,i     i 1  i 1       p  p p p    s ,V  ws ,i  s vs ,i   i 1   t  S sp             sp, A  wsp,i  sp vsp,i   wsp,i vsp,i    i 1  i 1   


      

(14)

(15)


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

S rf 

Reactor-Fluid:











 r r r  f ,V  f v f     rf , A  rf v rf v rf t

Ssr 

Reactor-Solid:



Page 8 of 46

 r r r  s ,V  s vs     sr, A  sr vsr vsr t





(16)

(17)

The source term S in Eq.(14) through Eq.(17) typically consists of Reynold stresses, body forces, momentum exchange, mass exchange, equilibration and nonequilibrium pressure, and average stresses per unit of volume.35 For the reactor-fluid subsystem, Eq.(16), the RHS is zero when negligible variation on both density and viscosity is assumed, and the source term in the LHS is then described by the rate of change in the molecular momentum tensor, S rf   p r  rf . The Ergun equation,36,37 Eq.(18), is a phenomenological model applied in this work as a closure model to compute this source term. r r  v rf 1   rf ,V  150 1   f ,V   f r    p  p  1.75 v f  0 d    r 3   dp   f ,V   r

(18)

  Setting B and b to be equal to the total energy (composed of the sum of internal, kinetic, and

potential energies) and respective specific energy of phase  within the domain  , energy conservation is obtained as shown in equation Eq.(19).





              S        , A Q      , A    v       ,V U   K              t              1 h j         , A  H   K     v      ,A  ,i  ,i       Mi  i 1 

(19)

The energy equation for both the catalyst pellet and the PBR is developed from Eq.(19). In this development, the solid phase is considered stagnant, and the potential and kinetic energies, the work done by the fluid onto the solid, and the effect of interphase viscous friction are all considered negligible. Fourier’s law is used to account for heat conduction, and the fluid is considered ideal. Then, the energy equations in the fluid and solid phases within the pellet domain are presented in Eq.(20) and (21).


Page 9 of 46

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




Pellet-Fluid:



q fp s     fp, A k fp T fp  p      p p p p T f         f ,V  f  c f ,i CP ,i  t   1 p  p      fp, A     i 1 CP ,i  j f ,i T fp  M  i 1   i          fp,V p p       t    h fp,i R fp,i   h fp,i   fp,V c fp,i    t  i 1   i 1  p p p p  p p         f , A  f  c f ,i CP ,i   v f T f  i 1     h fp,i    fp, Ac fp,i v fp   i 1  





q

Pellet-Solid:



p f s



      (20)    

p p p p p p p Ts       s , A ks Ts    s ,V  s CV t

(21)

The proximity of the solid and fluid phases in the pellet domain induces a high heat transfer coefficient between the two phases, which in turn leads to a common temperature Tc p for the pellet’s solid-fluid composite system. The subscript c , in this case, does not indicate a phase per se, but a composite system formed by both solid and fluid phases within the pellet. The resulting energy equation for the composite system in the pellet domain is given by Eq.(22).

    fp, A k fp   sp, A ksp  Tc p          p    1      fp, A  CPp,i  j fp,i Tc p    p  p C p   p  p  c p C p   Tc  s ,V s V , s f ,V f f ,i P ,i     M  i 1    i  i 1  t              h fp,i R fp,i   h fp,i   fp,V c fp,i       p P p    p  p c p C p v p  T p   f ,V f   f , A f  f ,i P,i  f  c  t    i 1 i 1 t  i 1          h fp,i    fp, Ac fp,i v fp i 1  



(22)



For the reactor domain’s fluid phase, the RTT derived energy balance is shown in Eq.(23).

r     r r   r r  T f   c C    f , V f f , i V , i   q r     r k r T r  t i 1   f s f ,A f f                 rf ,V p rf   1 r  r   r r  CP ,i  j f ,i T f     t     f , A    Mi     i 1           rf , A  rf   c rf ,i CPr ,i  v rf T fr      1 hr    r jr      i 1    f ,i f , A f ,i      i 1  M i    



Reactor-Fluid:








(23)


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

Page 10 of 46

In this work, the heat exchanged between the fluid and solid phases within the reactor domain is composed by three terms, Eq.(24): the enthalpy carried by the species’ mass flux between the reactor and the pellet domains, Eq.(25); the convective heat flux exchanged between the fluid in the reactor and the pellet solid phase, Eq.(26); and a source term qer that accounts for any external heat flux crossing the considered reactor control surface CS r for one dimensional models, Eq.(27).

q rf  s  qmr  qcr  qer

 Ap qmr   sr,V  p p  V  s ,V

(24)

CS     1 r p p    h  n  n    f ,i f , A i    M i  1 i      p

 Ap qcr   sr,V  p p  V  s ,V

CS p    p rp r p   s , A  T f  Tc    

 4  wr r  w  t   T f  T  if 1D  q   d   0 if 2D or 3D   r e

(25)

(26)

(27)

For the case study presented in this work, the heat transfer coefficients in Eq.(26) and (27) are calculated according to equations (28)38 and (29)39, as functions of the Reynolds and Prandtl numbers, defined in equations (30) and (31) respectively.



tw

T , p ,c  r f



rp

r

r i

 k rf  p d 

T , p ,c  r f

r

r i

  dp   1  1.5  t d  

 k rf  p d 

  r 13 p 0.6     2  1.1  Pr   Re       

Re  p

Pr  r

  r 1 3 p 0.59     Pr   Re   

v rf  rf d p

 rf CPr , f  rf k rf


(28)

(29)

(30)

(31)

Page 11 of 46

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


Case study: Steam Methane Reforming (SMR)

The model capabilities are demonstrated on a case study involving steam methane reforming (SMR) process. SMR is an important industrial process, widely used for hydrogen production. The goal is to identify the effect of catalytic pellet structural parameters, such as volumetric fraction, tortuosity, thermal conductivity, and mean pore radius, on SMR process performance. The parametric investigation carried out in this work considers steady-state, one dimensional, reactive transport in both the reactor and pellet domains, i.e., a spherical symmetric catalyst pellet and a tubular reactor with a flat velocity profile are considered at steady-state conditions, where spatial variations only occur in the radial and axial directions respectively. A schematic representation of the model is presented in Fig.1. The pellet domain is considered as a collection of pellets distributed along the reactor axis z , so that information is exchanged between the two domains downstream the process. The solution of the differential equations describing the system behavior in the pellet interior requires information from the pellet boundary. On the other hand, solution of the differential equations describing the system behavior in the reactor interior requires source term information related to the pellet boundary. Thus, solution of the overall differential equation system must be carried out in a simultaneous matter. At the end of this process, temperature, pressure, and species profiles are known at each axial location z within the reactor domain, and at each radial location

r within each pellet domain, at each axial location z within the reactor domain.



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

Figure 1 - Packed-bed reactor multiscale model scheme

In order to obtain equivalent 1-dimensional conservation equations for this case study, the control volume for the pellet domain is considered to be an infinitesimally small spherical annular shell, which allows the transformation of the integro-differential in Eq.(1) into an ODE in the pellet’s radial direction. Correspondingly, the species conservation is carried out over the fluid phase as the solid phase is considered rigid, static, and impenetrable by the fluid species. In the reactor domain, the considered control volume is an infinitesimally small cylindrical slice, which allows the transformation of the integro-differential in Eq.(1) into an ODE in the reactor’s axial direction. Correspondingly, the species conservation is carried out over the fluid phase, while species communication with the pellet domain is quantified through appropriately defined source terms in the species balance equations. For the pellet-fluid subsystem, a source term for the momentum balance is obtained by adding the DGM for all species.8 In addition, for the reactor-fluid subsystem, fluid phase heat conduction and energy


Page 12 of 46

Page 13 of 46

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


transport through diffusion are considered negligible in comparison with convective transport. Under these conditions, the above multiscale model presented in Eq.(2) to (27) can be simplified as follows: For a spherical, isentropic, steady-state pellet, component mass balance in the fluid phase is given by Eq.(32), the momentum balance in the fluid phase is given by Eq.(33), the DGM equation is given by Eq.(34), and the fluid phase pressure is evaluated through Eq.(35).

M i R fp,i  sp,V  sp r 2 





d  2 p r  f , A j fp,i  w fp,i  fp v fp   dr 

;

i  1,

p   BO   x f ,i 1 p p  p p  1   n   c  p    f , A f ,i  f ,tot  p p    p   f  j 1 DiK  i 1 M i DiK 





 c fp, j p p c fp,i p p 1  f , A n f ,i   f ,A nf , j  p eff  Mj j 1 c f ,tot Dij   Mi







d d    c fp,i  x fp,i c fp,tot dr  dr





eff

(32)

    

(33)

  1  p p    f , A n f ,i    M i DiK 





 1 1 BO  p d p   p  c p ; i  1, p  p  f ,i dr  D  iK f    p p  c fp,tot RTc p

(34)

(35)

Equation (22) shows the energy balance in the composite system, i.e., considering both fluid and solid phases within the pellet domain. The corresponding balance for a spherical, isentropic, steady-state pellet domain is given by Eq.(36).



p f ,A

k fp   sp, A ksp 

1 r2

 p  d  2 d p   H 0  T C p (T )dT   R p  p p  r T      f ,i   c s , V s i   0  dr    T P,i    i 1    dr

(36)

For a unidimensional, isentropic, steady-state axial reactor, the total mass balance in the fluid phase is given by Eq.(37), the component mass balance in the fluid phase by Eq.(38), the Stefan-Maxwell diffusion model by Eq.(39), pressure drop is evaluated by Ergun Eq.(40), and energy balance in the fluid phase is given by Eq.(41).

0



d r r r  f ,A f v f dz

    v   constant r f

r f


(37)


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



r s ,V



Ap  p p  n  sp,V V p  f , A f ,i



z ,r p

 

 d r   r r d r wrf ,i   j f ,i  ;     f ,A  f v f dz dz   

 x rf ,i  wrf ,i x rf ,i x rf , j  1 r 1 r  d r j f , j  r j f ,i     x f ,i         dz Dijr   rf , j  f ,i pr j 1    

Page 14 of 46

i  1,

 d r   p  ;  dz 

i  1,

(38)

(39)

 v rf   sr,V  150 sr,V  rf d r   p    1.75   rf v rf     3 p p dz d    r    d   f ,V 

(40)

 r  A p   p rp r z ,r p   p     T  T  s ,V  p p   s , A  f c        s ,V V    p  z ,r  r  A p    1 r p p    r r   r r   r d r   h f ,i f , A ni     f , A  f   c f ,i CP ,i   v f Tf      s ,V  p p     i 1   dz        s ,V V   i 1 M i      4t   wr T fr  T w     d   

(41)

For the boundary between pellet and reactor domains, interphase mass-transfer resistances are assumed to be negligible in this work. Thus, the generally applied type 3 boundary condition (Robin) is reduced to a type 1 boundary condition (Dirichlet) at the pellet surface. Therefore, at the external surface of the catalytic pellet ( r  r p ) the considered boundary conditions are shown in in Eq.(42) to (44), while at the center of the pellet ( r  0 ) symmetry boundary conditions are applied, as shown by Eq.(45) to (47).

cip  r p , z   cir  z 

i  1,

;

(42)

p p  r p , z   pr  z 

(43)

q p  r p , z     r p   rp Tcp  r p , z   Tfr  z 

(44)

2

d p ci  0, z   0 dr

;

i  1,

(45)

d p p  0, z   0 dr

(46)

d p Tc  0, z   0 dr

(47)


Page 15 of 46

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


For the reactor domain, boundary conditions are considered at the entrance of the packed-bed reactor (

z  0 ), as shown by Eq.(48) to (50).

c rf ,i  0   c 0f ,i

;

i  1,

(48)

T fr  0   T 0

(49)

pr  0  p0

(50)

Along the axial direction, an external heat profile is assumed to hold the reactive tube’s inner wall w w temperature constant throughout the reactor’s length, i.e., T  z   T .

The reversible reactions R1, R2 and R3 of the SMR processes and their respective enthalpy of formation are shown below:

 R1  R2   R3

  CO + 3H 2 CH 4  H 2O  

H 1 : 206.1 kJ/mol

  CO 2 + H 2 CO + H 2O  

H 2 :  41.15 kJ/mol

  CO 2 + 4H 2 CH 4 + 2H 2O  

H 3 : 164.9 kJ/mol

The intrinsic kinetics of the steam reforming reaction are implemented according to Xu and Froment’s model,40 as shown in Eq. (51) to (54) in terms of the species partial pressures in the pellet domain, which is where the reactions actually take place. Rate coefficients and adsorption parameters, as well as their respective units for the applied SMR kinetic model are found on Tables S1 and S2, in the Supporting Information.

R1 

 

k1

p  p H2

2.5

3 p   pHp2 pCO 1 p p p p   CH 4 H2O    DEN 2 K1  

p  pHp2 pCO k2  p p 1 2 R2  p  pCO pH2O   pH2  K 2   DEN 2


(51)

(52)


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

R3 

k3

p  p H2

3.5

  pp pp  CH4 H2O 





2

p   p H2

p  pCO 1 2    DEN 2 K3 

Page 16 of 46

4

 pHp2O  p p DEN  1  KCO pCO  K H2 pHp2  KCH4 pCH  K H 2O  4  pHp   2 

(53)

(54)

The production rate inside the pellet domain is connected with the reactor domain by calculating the species combined diffusion-convection mass fluxes calculated at the surface of the pellet, and considering the relation between pellet volume, pellet area, and the volumetric fractions of the pellet and the reactor, as indicated in Eq. (13) during the model derivation. Volumetric and surface fractions employed in the above multiscale model, can be related to one another using the tortuosity concept that is commonly employed in the literature.41 The correlation between p p volumetric fraction  f ,V , area fraction  f , A , and tortuosity  , for the pellet-fluid subsystem is given by

Eq.(55) and its derivation can be found on the Supporting Information.



 fp,V  fp, A

(55)

Fluid viscosity and thermal conductivity are calculated for gas mixtures,42 from each individual species’ properties by using the Eq.(56) to (58).

   x  T     f    i i  i 1    x j  ij   j 1 

(56)

   x k T    k f    i i  i 1    x j  ij   j 1 

(57)






Page 17 of 46

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


1  Mi ij  1  8  M j

  

1 2

   T  1 2  M j 1   i       j T    M i 

14

  

   

2

(58)

For ideal gas mixtures, Stefan-Maxwell diffusivities and binary diffusivities are nearly identical34. For the case study in this work, binary diffusion coefficients for polar and non-polar gas mixtures are estimated using the model presented by Poling, Praunitz and O’Connell’s,43 which was developed from the Chapman-Enskog theory. The full set of equations used in the simulation to estimate properties for polar and non-polar gas-mixtures are shown in Tables S3 and S4 in the Supporting Information. Other applicable relations used in the model and implemented in the simulation mass transport definitions are presented in the Table S5. The simulations performed in the case study consider the reactor and pellet domain property dependencies shown in Table S6.

The above multiscale model was realized and numerically solved using the COMSOL Multiphysics® software. The final multiscale implemented model contains Eq.(12), (13), (28) to (58) plus the supplementary relations given in Tables S2 to S5, i.e., Eq.(S5) to (S37) presented in the Supporting Information. The pellet sub-model was verified by considering a reaction A  B with linear kinetics, taking place under isothermal, isobaric conditions, in a pellet within which Knudsen diffusion can be ignored, species diffusion is described by constant effective diffusivity coefficients. The obtained model solution was consistent with the classic, analytical, spherical catalyst pellet, diffusion-reaction model solution involving hyperbolic sines.42 The reactor sub-model was also verified by first considering that the binary diffusion and Knudsen diffusion coefficients for all species, and the wall heat transfer and thermal conductivity coefficients are all very large, and that the numerical values of 150 and 1.75 in the Ergun equation are 5 orders of magnitude smaller. The obtained simulation results from our model are then consistent to an isothermal, isobaric, plug-flow reactor model featuring the same intrinsic kinetics.



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

Page 18 of 46

The parameters and boundary conditions used in the simulations for the case study were obtained from the literature13,32,44 and are presented in Table 1. Parameter values for the calculation of diffusion and viscosity coefficients were also obtained from the literature.42,43 Table 1: Boundary values and simulation parameters Description Parameter

Value

Reforming tube radius

dr

100.0 mm

Reforming tube length

Lr

20.0 m

Reforming tube void fraction

 rf ,V

0.6

Catalyst pellet diameter

dp

1.0 cm

Catalyst pellet density

 sp

2,355.2 kg/m3

Inlet velocity

v 0f

1.80 m/s

Inlet pressure

p0

29.35 bar

Inlet temperature

T0

800 K

Tube inner wall temperature

Tw

1,100 K

Carbon monoxide to methane ratio

c 0f ,CO c 0f ,CH 4

4.021E4

Carbon dioxide to methane ratio

c 0f ,CO2 c 0f ,CH 4

0.04705

Hydrogen to methane ratio

c 0f , H 2 c 0f ,CH 4

0.00724

Water vapor to methane ratio

c 0f , H 2O c 0f ,CH 4

2.96622

Inlet molar concentration ratios

For the specified conditions, the equilibrium conversion for the SMR multi-scale PBR model were validated by comparing with an equilibrium reactor featuring Froment’s kinetics model. In addition,


Page 19 of 46

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


different feed conditions, pressures, and temperature presented in the literature, as well as their respective conversion values,13,44,45 were also used in the model to validate CH4 equilibrium conversion.

3. Results and discussion

The local influence of four catalytic pellet structural parameters were investigated: volumetric fraction, tortuosity, solid phase thermal conductivity, and mean pore diameter. Typical values for these parameters vary within the following ranges: volumetric fraction from 0.4 to 0.7; tortuosity ranging from 2 to 8.13,41 Solid phase thermal conductivity from 2.5 W/(m.K) to 250 W/(m.K); and mean pore diameter from 2 nm to 100 nm. The effects of these parameters on the molar volumetric generation rate of H2, at the position z/Lr=0.05 in the reactor domain, are as follows: For constant  f ,V , Fig. 2(a), 2(b), 2(c) illustrate the rate’s dependence on  (with constant d pore , k s ), p

p

p

p p d pore (with constant  , ksp ), and ksp (with constant  , d pore ), respectively.

p p p For constant  , Fig. 3(a), 3(b), 3(c) illustrate the rate’s dependence on  f ,V (with constant d pore , k s ), p p p p p d pore (with constant  f ,V , k s ), and ksp (with constant  f ,V , d pore ), respectively.

p p For constant d pore , Fig. 4(a), 4(b), 4(c) illustrate the rate’s dependence on  f ,V (with constant  , k sp ), 

p p p (with constant  f ,V , k s ), and ksp (with constant  f ,V , ), respectively.



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

(a)

(b)

Figure 2 - Variation of H2 generation rate, at z

(a)

(c)

p p p (b), and k s (c) for constant  f ,V Lr  0.05 , with  (a), d pore

(b)


Page 20 of 46

(c)

p p (b), and k s (c) for constant  Lr  0.05 , with  fp,V (a), d pore


Page 21 of 46

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


(a)

(b)


(c)

p Lr  0.05 , with  fp,V (a),  (b), and ksp (c) for constant d pore

The system shows monotonic behavior on the studied catalyst parameters, within their considered ranges. Increases in volumetric fraction, and mean pore diameter, and decreases in tortuosity, lead to increases in the hydrogen generation rate. In addition, from figures 1(c), 2(c), and 3(c), one can observe that different values of solid phase thermal conductivity have little influence on hydrogen production. Based on this assessment, results are shown in the figures below for two sets of parameters that represent





p =100 nm the two extremes within the considered parameter ranges: Combination 1  =2,  fp,V =0.7, d pore





p =2 nm is the is the most favorable for hydrogen production, while Combination 2  =8,  fp,V =0.4, d pore

least favorable for hydrogen production. In all cases, the solid phase thermal conductivity is

k sp  25 W  m  K  .



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

Page 22 of 46

Reactor species molar fractions, at different normalized tube lengths, for combinations 1 (straight lines) and 2 (dotted lines), are shown in Fig.5. As expected, the hydrogen (methane) mole fraction increases (decreases) earlier in the reactor for parameter combination 1.

r

Figure 5 – Reactor species molar fractions at different z L , for combinations 1 (straight lines) and 2 (dotted lines)

This macro-scale behavior is better understood by the micro-scale results, shown in Fig. 6, which illustrate the species molar fraction pellet radial profiles, at different normalized tube lengths, for combinations 1 (straight lines) and 2 (dotted lines). It is readily apparent that diffusion in the pellet interior is the process controling step, as the species radial profiles for both combinations are relatively flat. Further, it can be seen, that as tortuosity increases and volumetric fraction decreases, the diffusive transport of all species in the pellet is reduced. Consequently, parameter combination 1 (solid lines), which corresponds to lower tortuosity, higher volumetric fraction, and larger pore diameter, is more favorable to hydrogen production than combination 2 (dotted lines).


Page 23 of 46

z/Ltube = 0.1

z/Ltube = 0.2

z/Ltube = 0.5

H2O

H2

CO2

CO

CH4

z/Ltube = 0.01

2O

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


Figure 6 – Species molar fraction pellet radial profiles at different z for combinations 1 (straight lines) and 2 (dotted lines)


Lr ,


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

Pressure variation inside the pellet domain can also be assessed through the multi-scale model. p Reactor-Pellet pressure difference radial profiles for different mean pore diameters d pore and normalized

tube lengths z Lr , are shown in Fig.7.

z = 0.01

z = 0.05

Figure 7 – Reactor-Pellet pressure difference  p p  r p , z   p p  r , z   radial profiles   p

at different d pore and

z Lr , for  =4,  fp,V =0.6, k sp  25 Wm 1 K 1 


Page 24 of 46

Page 25 of 46

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


Although pressure variation within the pellet domain is mild, its value changes along the reactor’s axial direction. In the initial portion of the PBR, the local pressure inside the pellet is lower than the respective pressure at the pellet surface (reactor’s local pressure). At 3.38% of the reactor length, pellet and reactor have the same pressure, and from this point up to the reactor’s exit, the pressure inside the pellet is higher than the pressure at the pellet surface. At about 20% of the reactor length, the reactor-pellet pressure difference reaches a minimum (as it is negative) and, after that, pressure inside the pellet decreases until approaching zero at the reactor outlet. A possible interpretation of this phenomenon is that at short reactor lengths the reforming reactions are vigorous. Although, as a result, the total molar flow at the pellet surface is outward, the molar flow of all species except hydrogen is vigorously inward. The high diffusivity coefficient of hydrogen implies that its diffusive flux component in the DGM is higher than its pressure driven component. This is not the case for the other species, which have significantly lower diffusivity coefficients. Therefore, their pressure driven component is a significant contributor to the overall flux. Thus, at short lengths, the inward pressure component of all species except hydrogen dominates the pressure component of hydrogen leading to an overall inward pressure gradient. The above pressure effects become more intense for reduced pore diameters, reaching a maximum variation between reactor operating pressure and pellet internal pressure of about 0.1 bar. Although this difference is not insignificant, this suggests that the assumption of constant pressure within the catalyst pellet for steam methane reforming can be considered adequate. The variation of the reactor fluid temperature along the reactor length is shown in Fig.8. At short (long) reactor lengths, parameter combination 1 leads to lower (higher) fluid temperatures than combination 2. This is understandable, as combination 1 facilitates the endothermic reforming reactions, thus leading to lower temperatures than combination 2, at short reactor lengths.



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

Figure 8 - Reactor fluid temperature axial profile, for combinations 1 (straight lines) and 2 (dotted lines)

The variation of the Reactor fluid - Pellet surface temperature difference along the reactor length is shown in Fig.9. At short reactor lengths, parameter combination 1 leads to reactor fluid - pellet surface temperature differences that are higher than the corresponding differences for combination 2. This is again understandable, as combination 1 facilitates more than combination 2 the endothermic reforming reactions, which are more vigorous at short reactor lengths.


Page 26 of 46

Page 27 of 46

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


Figure 9 - Reactor fluid-Pellet surface temperature difference axial profile, for combinations 1 (straight lines) and 2 (dotted lines)

Temperature variation inside the pellet, i.e., the difference between the temperature at the pellet’s surface and its center, has also been assessed along the length of the reactor and is shown in Fig.10. The most intense temperature changes inside the pellet occur for combination 1 rather than combination 2. Nevertheless, the highest temperature difference between the surface and the center of the pellet is only 0.048% (0.0015%) for combination 1 (combination 2). It is important to point out that although temperature has a strong influence on reaction rates, and the residence time inside the pores is quite large, the solid phase thermal conductivity has little influence on hydrogen production, since the reaction rate in most of the catalytic pellet interior is small, as also supported by the small values of the effectiveness factors shown in Fig.13.



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

Figure 10 - Pellet surface – Pellet center temperature difference axial profile, for combinations 1 (straight lines) and 2 (dotted lines)

The endothermic needs of the SMR process are provided by heat influx at the reactor wall. Figure 11 presents the axial profile of the reactor wall heat influx for combinations 1, and 2. At 5.73% of the reactor length, the SMR heat flux varies from 335.6 kW/m2 for combination 1, to 250.2 kW/m2 for combination 2. These results suggest that the pellet structural parameters have a significant influence on the axial profile of the external heat flux, and thus their influence cannot be neglected in SMR studies aiming to quantify temperature peaks, and localize hot spots.


Page 28 of 46

Page 29 of 46

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


Figure 11 – Reactor wall heat flux axial profile, for combinations 1 (straight lines) and 2 (dotted lines)

Methane conversion along the reactor length is shown in Fig.12. Combination 1 requires a shorter reactor length than combination 2, to attain the same methane conversion. Indeed, 60% conversion is attained at 25.96% of the reactor’s length for combination 1, while the same conversion is attained at 42.28% of the reactor’s length for combination 2. This indicates that the pellet’s structural parameters can play an important role in the design of intensified SMR reactors.



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

Page 30 of 46

Figure 12 – Methane conversion axial profile, for combinations 1 (straight lines) and 2 (dotted lines)

Effectiveness factors are used in pseudo-homogeneous simulations to capture the combined effects of reaction-diffusion transport. The species i effectiveness factor, Eq.(59), is given by the ratio of the net mass flow of i at the pellet’s surface, over the mass generation rate of the same species if all catalyst material was completely exposed to the conditions at the pellet’s surface.46

i ˆ

 

p f ,A



nip  n dAt

CS

 M 

p i s ,V

cat Ri p CS dV

; i  1,

(59)

CV

The above definition of effectiveness factor suggests that it can assume positive or negative (and even zero or infinite) values, as demonstrated in the literature for pseudo-homogeneous SMR reactors.13,14 The same behavior is observed in this study, as shown in Figures 13(a) and (b), which illustrate the species’ effectiveness factor axial profile for combinations 1, and 2, respectively.


Page 31 of 46

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


Figure 13 – Species’ effectiveness factor axial profile, for combinations 1 (a) and 2 (b)

The pellet structure influence on the effectiveness factor was found to be significant, with values for combinations 1 and 2 differing by about an order of magnitude. The methane, and hydrogen effectiveness factors are almost constant and equal to each other throughout the reactor, for both combinations 1, and 2. The carbon monoxide, and water effectiveness factors show some increased axial variability, while the carbon dioxide effectiveness factor exhibits a singularity and sign reversal for both combinations 1, and 2, indicating that the carbon dioxide combined reaction-transport and intrinsic reaction generation rates have the same sign (positive) for short reactor lengths, opposite sign at long reactor lengths, while the intrinsic



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

Page 32 of 46

rate becomes zero at 8.84% of the reactor’s length for combination 1, and at 24.14% of the reactor’s length for combination 2. This behavior complicates the application of the effectiveness factor concept to pseudo-homogeneous SMR reactor models. To overcome this problem, an average effectiveness factor concept is proposed for 1-dimensional catalytic reactor models. Considering that both the numerator and denominator of Eq.(59) are functions of the reactor axial coordinate z , one can define an average effectiveness factor as the ratio of the cumulative transport-reaction rate along the reactor, over the cumulative intrinsic reaction rate, as shown in Eq.(60).

 Acr  sr,V    fp, A nip  n dAt   z  dz 0  V p sp,V     CS  iave  z  ˆ r r   Lr  A  c s ,V M i sp,V cat Ri p dV   z  dz 0  V p sp,V   CS   CV  Lr





; i  1,

(60)

The evolution of the species i average effectiveness factor along the reactor length is shown in Figures 14(a) and (b), for combinations 1 and 2 respectively. Both figures demonstrate that for all species (with the exception of CO for combination 2) the average effectiveness factor is a monotonically increasing function of the reactor length. Indeed, at short reactor lengths, the vigorous intrinsic species generation rates make transport limitation effects within the pellet more pronounced, while at longer reactor lengths, the intrinsic species generation rates slow down thus making the transport limitations less apparent. The other important feature shown in figures 14 (a) and (b) is that the carbon dioxide average effectiveness factor no longer exhibits a discontinuity feature. This will be better understood following the close examination of the numerator and denominator terms of Eq.(60) presented in figures 15 and 16 below. Another interesting feature is that the average effectiveness factor for all species evolve closely to each other, until 11% (23%) of the reactor length for combination 1 (combination 2). At these reactor lengths methane conversion stands at 46.6% and 53.3% for combinations 1 and 2 respectively. In turn, this


Page 33 of 46

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


potentially explains the reasonable predictive capabilities of pseudo-homogeneous SMR models using the same effectiveness factor for all species.14

Figure 14 - Average species’ effectiveness factor axial profile, for combinations 1 (a) and 2 (b)

Additional insight into the average effectiveness factor behavior described above can be gained through close examination of the numerator and denominator terms of Eq.(60) for combinations 1 and 2. As stated earlier, the carbon dioxide average effectiveness factor no longer exhibits a discontinuity feature. As



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

shown in figures 15 and 16 below, the signs or the numerator and denominator of Eq.(60) are the same throughout the reactor, and for both combinations 1 and 2. Since the denominator never crosses zero for both combinations 1 and 2, no average effectiveness factor singularity is observed. Figures 15 and 16 also demonstrate that for all species (with the exception of CO2) the numerator and denominator of the average effectiveness factor are monotonic functions of the reactor length.

Figure 15 – Cumulative transport-reaction rate axial profile, for combinations 1 (a) and 2 (b)


Page 34 of 46

Page 35 of 46

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


Figure 16 - Cumulative intrinsic reaction rate axial profile, for combinations 1 (a) and 2 (b)

Another important feature of figure 16 is that the species average effectiveness factor denominators for combinations 1 and 2 exhibit similar axial profiles, with the combination 2 denominator assuming higher values. This is to be expected, given that both denominators represent intrinsic reaction rates, evaluated at



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

the prevailing reactor conditions (species concentrations, temperature) which are higher at shorter reactor lengths for combination 2 versus combination 1.

4. Conclusions

The influence of catalyst pellet structural parameters on reactor performance was investigated using an RTT derived multiscale model that employs the DGM and SMM transport models. An SMR packedbed reactor case study was carried out, quantifying the effects of mean pore diameter, tortuosity, volumetric fraction, and thermal conductivity, on hydrogen production. The employed model relaxes the commonly employed constant pressure assumption inside the pellet, and establishes that the pellet’s tortuosity has a considerable influence on the overall process. The effect of the mean pore diameter is pronounced in the smaller part of its range (between 2nm and 10nm), where Knudsen diffusion dominates. Changes in the volumetric fraction within the range used in this study are important, although less impactful to the system in comparison with the aforementioned parameter effects. The pressure and temperature variation inside the pellet were found to be negligible throughout the reactor. The pellet structure influence on the species effectiveness factor was found to be significant, with values for combinations 1 and 2 differing about an order of magnitude. In addition, singularities and sign reversals were found for the carbon dioxide effectiveness factor, indicating that the carbon dioxide combined reaction-transport and intrinsic reaction generation rates have the same sign (positive) for short reactor lengths, and an opposite sign at long reactor lengths. To overcome this problem, an average effectiveness factor concept is proposed for 1-dimensional catalytic reactor models as the ratio of the cumulative transport-reaction rate along the reactor, over the cumulative intrinsic reaction rate. It is shown to be a monotonically increasing function of the reactor length, that additionally exhibits no singularity features. Improved insight into its behavior is gained through examination of its numerator and denominator terms. The average effectiveness factors for all species are shown to evolve closely to each other for short reactor lengths, thus potentially explaining the reasonable predictive capabilities of pseudo-homogeneous SMR models which use the same effectiveness factor for all species.


Page 36 of 46

Page 37 of 46

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


Acknowledgements Financial support from CAPES – Brazilian Federal Agency for Support and Evaluation of Graduate Education within the Ministry of Education of Brazil – through grant BEX 18691/12-9, and the U.S. Department of Energy through DOE grant DE-EE0005763 “Industrial Scale Demonstration of Smart Manufacturing Achieving Transformational Energy Productivity Gains” are gratefully acknowledged. Insightful discussions with Dr. Mina Sierou of COMSOL are also gratefully acknowledged.

Supporting Information The Supporting Information is available free of charge on the ACS Publications website. Derivation of the relation between volumetric fraction, area fraction and tortuosity; and tables with supplementary equations used in the simulations.

Notation: English Symbols

A p  m2 of domain p  Area of the control surface of domain p Acr  m 2  Cross section area of the tubular reactor

BO  m 2  Viscous flow parameter of domain p

 mol of species i in phase  within domain   c,i   Species’ i molar density of phase  within m3 of phase  within domain    domain 

 mol of phase  within domain   c,tot  3  Total molar density of phase  within domain   m of phase  within domain  



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

  J of species i in phase  within domain  CP,i   Species’ i molar specific heat at th mol  K of the i component of phase  within domain      constant pressure of phase f within domain 

  J phase s within domain  CV   Species’ i molar specific heat at constant volume of   mol  K  of phase s within domain   phase s within domain 

CS   m 2 of domain  

Control surface of domain 

CV   m3 of domain  

Control volume of domain 

 m2 of phase f within of domain   Dij   Species’ i and j binary diffusion coefficient of phase  s   within domain 

 m2 of domain p  D   Species’ i and j effective binary diffusion coefficient in domain p s   eff ij

 total m2 of domain p  DiK   Species’ i Knudsen diffusion coefficient in domain p s    m2 of phase f within domain r  Dir   Species’ i thermal diffusion coefficient of phase f within s   domain r

d p  m

Diameter of the catalytic pellet

p d pore m

Mean pore diameter in domain p

d r m

Diameter of the tubular reactor

 J of phase  within domain   H    Mass specific enthalpy of phase  within domain  kg of phase  within domain   


Page 38 of 46

Page 39 of 46

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


 J of species i in phase f  H i0   Species’ i standard molar enthalpy of formation  mol of species i in phase f   J of phase  within domain   h,i   Species’ i molar enthalpy of phase  within domain   mol of phase  within domain    J of phase  within domain   K   Mass specific kinetic energy of phase  within domain   kg of phase  within domain  

  J of phase  within domain  k   Thermal conductivity of phase  within domain     m  K  of phase  within domain    s     kg of species i in phase  within domain    ith species diffusion mass flux in phase  within j,i  2   m of phase  within domain    s   domain 

Lr  m 

 kg of i  Mi    mol of i 

Length of the tubular reactor

ith species molar mass

n  dimensionless  Unit vector direction of the differential area dA of the CS .  kg of species i in phase  within domain    ith species combined diffusion-convection mass n,i  2   m of phase  within domain    s   flux in phase  within domain 

 J of phase f within domain   p  3  Pressure of phase  within domain   m of phase f within domain  

 J of phase  within domain    Q  2   m of phase  within domain    s   

Heat flux into phase  within domain 

 J from phase f to s within domain    Heat transferred from phase f to s within domain  qf s  3   m of domain   s    



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

 J from domain r to domain p   Heat transferred from domain r to domain p due to convection. qcr  3   m of domain r  s    

 J from exterior to domain r   Heat transferred from the exterior to domain r . qer  3   m of domain r  s      J from domain r to domain p   Heat transferred from domain r to domain p due to mass flux qmr  3   m of domain r  s      mols of species i in phase  within domain    ith species volumetric generation rate in phase  R,i  3    m of domain    s   within domain 

 mols of species i in phase f within domain p  th R fp,i   i species phase s specific generation rate in kg of phase s within domain p  s     phase f within domain p

 J  R   mol.K 

Universal gas constant

r  m of domain p  spatial variable of domain p

r p m

Radius of the pellet

 kg of phase  within domain    Momentum source term of phase  within domain  S  2 2   m of domain   s      J of phase  to   within domain    Interphase energy transfer source term in domain  S   3 2   m of domain   s     T  K 

Temperature of phase  within domain 

Tc p  K 

Temperature of the composite phase in domain p


Page 40 of 46

Page 41 of 46

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


T w K 

Temperature of inner wall of the tubular reactor

 kg of phase  within domain     2     m of phase  within domain    s 

Viscous momentum flux tensor of phase  within

domain 

 J of phase  within domain   U    Specific internal energy of phase  within domain   kg of phase  within domain  

  J transferred from reactor's tubeto domain r  Uˆ r  Global heat transfer coefficient between the   m2  K  of interphase contact  s   





tubular reactor’s inner wall and domain r

V p  m3 of domain p  Total volume of domain p

m v   Mass average velocity of the phase  within domain  s  m v,i   ith species velocity in phase  within domain  s   kg of species i in phase  within domain   th w,i   i species mass fraction in phase  within domain kg of phase  within domain   

.  mol of species i in phase  within domain   x,i   mol of phase  within domain   

ith species molar fraction in phase  within

domain 

Greek symbols:

 m3 of phase  within domain    total m3 of domain   

 ,V 

Volumetric fraction of phase  within domain 



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

 m2 of phase  within domain    total m2 of domain   

 , A 

Surface fraction of phase  within domain 

 kg of phase  within domain    3  m of phase  within domain  

 

Mass density of phase  within domain 

 kg of species i in phase  within domain   th  i species mass concentration in phase  within m3 of phase  within domain   

,i 

domain 

 m of diffusion path length in phase f within domain p   Tortuosity of phase f within domain p . m of CV length in domain p  



 kg of species i in phase  within domain    Viscosity of phase f within domain  m of phase  within domain   s    

 f 

  dimensionless  Total number of species in the phase-domain  J of phase  within domain      Mass specific potential energy of phase  within domain   kg of phase  within domain  

  J from phase f within domain r   Interfacial heat transfer coefficient between phase f    m2  K  of phase s within domain p  s    rp





within domain r and phase s within domain p .

  J transferred from reactor's tubeto domain r    Interfacial heat transfer coefficient between the   m 2  K  of interphase contact  s    wr





tubular reactor’s inner wall and domain r


Page 42 of 46

Page 43 of 46

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


References

(1) (2) (3) (4) (5) (6) (7) (8) (9)

(10)

(11) (12)

(13) (14) (15) (16) (17) (18) (19)

(20) (21) (22) (23)

Cruz, F. E.; Júnior, S. de O. Petroleum Refinery Hydrogen Production Unit: Exergy and Production Cost Evaluation. Int. J. Thermodyn. 2008, 11, 187–193. Iordanidi, A. A. Mathematical Modeling of Catalytic Fixed Bed Reactors; Twente University Press, 2002. Coppens, M.-O. The Effect of Fractal Surface Roughness on Diffusion and Reaction in Porous Catalysts–from Fundamentals to Practical Applications. Catal. Today 1999, 53, 225–243. Keil, F. J. Diffusion and Reaction in Porous Networks. Catal. Today 1999, 53, 245–258. Cai, Y.; Zygourakis, K. A Multiscale Transient Model for Combustion of Highly Porous Chars. Ind. Eng. Chem. Res. 2003, 42, 2746–2755. Maxwell, J. C. On the Dynamical Theory of Gases. Philos. Trans. R. Soc. Lond. 1867, 157, 49–88. Jackson, R. Transport in Porous Catalysts; Elsevier Science & Technology, 1977; Vol. 4. Mason, E. A.; Malinauskas, A. P. Gas Transport in Porous Media: The Dusty-Gas Model; Elsevier: New York, 1983. Veldsink, J. W.; Van Damme, R. M. J.; Versteeg, G. F.; Van Swaaij, W. P. M. The Use of the Dusty-Gas Model for the Description of Mass Transport with Chemical Reaction in Porous Media. Chem. Eng. J. Biochem. Eng. J. 1995, 57, 115–125. Abashar, M. E.; Elnashaie, S. S. Mathematical Modelling of Diffusion-Reaction, and Solution Algorithm for Complex Reaction Networks in Porous Catalyst Pellets—steam Reforming of Natural Gas. Math. Comput. Model. 1993, 18, 85–100. Salmi, T.; Wärma, J. Modelling of Catalytic Packed-Bed Reactors—comparison of Different Diffusion Models. Comput. Chem. Eng. 1991, 15, 715–727. Elnashaie, S. S. E. H.; Abashar, M. E. E. Steam Reforming and Methanation Effectiveness Factors Using the Dusty Gas Model under Industrial Conditions. Chem. Eng. Process. Process Intensif. 1993, 32, 177–189. Xu, J.; Froment, G. F. Methane Steam Reforming: II. Diffusional Limitations and Reactor Simulation. AIChE J. 1989, 35, 97–103. Wesenberg, M. H.; Svendsen, H. F. Mass and Heat Transfer Limitations in a Heterogeneous Model of a Gas-Heated Steam Reformer. Ind. Eng. Chem. Res. 2007, 46, 667–676. Aris, R. On Shape Factors for Irregular Particles—I: The Steady State Problem. Diffusion and Reaction. Chem. Eng. Sci. 1957, 6, 262–268. Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design; John Wiley and Sons: New York, 1979. Wesenberg, M. H. Gas Heated Steam Reformer Modelling. Ph.D. Dissertation, Norwegian University of Science and Technology, Trondheim, Norway, 2006. Maroudas, D. Multiscale Modeling of Hard Materials: Challenges and Opportunities for Chemical Engineering. AIChE J. 2000, 46, 878–882. Quintard, M.; Whitaker, S. Transport in Ordered and Disordered Porous Media: Volume-Averaged Equations, Closure Problems, and Comparison with Experiment. Chem. Eng. Sci. 1993, 48, 2537– 2564. de Souza, S. M. A. G. U.; Whitaker, S. Mass Transfer in Porous Media with Heterogeneous Chemical Reaction. Braz. J. Chem. Eng. 2003, 20, 191–200. Balakotaiah, V.; Chakraborty, S. Averaging Theory and Low-Dimensional Models for Chemical Reactors and Reacting Flows. Chem. Eng. Sci. 2003, 58, 4769–4786. Chakraborty, S.; Balakotaiah, V. Spatially Averaged Multi-Scale Models for Chemical Reactors. In Advances in Chemical Engineering; Marin, G. B., Ed.; Academic Press, 2005, pp 205–297. Chakraborty, S.; Balakotaiah, V. Multi-Mode Low-Dimensional Models for Non-Isothermal Homogeneous and Catalytic Reactors. Chem. Eng. Sci. 2004, 59, 3695–3724.



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

(24) (25)

(26) (27)

(28) (29) (30) (31) (32)

(33) (34) (35) (36)

(37) (38) (39) (40) (41) (42) (43) (44)

(45) (46)

Vlachos, D. G. A Review of Multiscale Analysis: Examples from Systems Biology, Materials Engineering, and Other Fluid–surface Interacting Systems. Adv. Chem. Eng. 2005, 30, 1–61. Keil, F. J. Multiscale Modelling in Computational Heterogeneous Catalysis. In Multiscale Molecular Methods in Applied Chemistry; Topics in Current Chemistry; Springer, Berlin, Heidelberg, 2011; pp 69–107. Hansen, N.; Keil, F. J. Multiscale Modeling of Reaction and Diffusion in Zeolites: From the Molecular Level to the Reactor. Soft Mater. 2012, 10, 179–201. Hughes, T. J. R. Multiscale Phenomena: Green’s Functions, the Dirichlet-to-Neumann Formulation, Subgrid Scale Models, Bubbles and the Origins of Stabilized Methods. Comput. Methods Appl. Mech. Eng. 1995, 127, 387–401. Allaire, G.; Brizzi, R. A Multiscale Finite Element Method for Numerical Homogenization. Multiscale Model. Simul. 2005, 4, 790–812. Kevrekidis, I. G.; Gear, C. W.; Hummer, G. Equation-Free: The Computer-Aided Analysis of Complex Multiscale Systems. AIChE J. 2004, 50, 1346–1355. Reynolds, O.; Brightmore, A. W.; Moorby, W. H. Papers on Mechanical and Physical Subjects; The University Press, 1900. Leal, L. G. Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes; Cambridge University Press, 2007. Solsvik, J.; Jakobsen, H. A. Modeling of Multicomponent Mass Diffusion in Porous Spherical Pellets: Application to Steam Methane Reforming and Methanol Synthesis. Chem. Eng. Sci. 2011, 66, 1986–2000. Hartig, F.; Keil, F. J. Large-Scale Spherical Fixed Bed Reactors: Modeling and Optimization. Ind. Eng. Chem. Res. 1993, 32, 424–437. Kerkhof, P. J. A. M. A Modified Maxwell-Stefan Model for Transport through Inert Membranes: The Binary Friction Model. Chem. Eng. J. Biochem. Eng. J. 1996, 64, 319–343. Jiang, Y.; Khadilkar, M. R.; Al-Dahhan, M. H.; Dudukovic, M. P. CFD of Multiphase Flow in Packed-Bed Reactors: I. k-Fluid Modeling Issues. AIChE J. 2002, 48, 701–715. Su, Y.; Davidson, J. H. Introduction of Fluid Flow and Heat Transfer in Porous Media. In Modeling Approaches to Natural Convection in Porous Media; SpringerBriefs in Applied Sciences and Technology; Springer, Cham, 2015; pp 1–8. Whitaker, S. Mass Transport and Reaction in Catalyst Pellets. Transp. Porous Media 1987, 2, 269–299. Dixon, A. G.; DiCostanzo, M. A.; Soucy, B. A. Fluid-Phase Radial Transport in Packed Beds of Low Tube-to-Particle Diameter Ratio. Int. J. Heat Mass Transf. 1984, 27, 1701–1713. Wakao, N.; Kaguei, S.; Funazkri, T. Effect of Fluid Dispersion Coefficients on Particle-to-Fluid Heat Transfer Coefficients in Packed Beds. Chem. Eng. Sci. 1979, 34, 325–336. Xu, J.; Froment, G. F. Methane Steam Reforming, Methanation and Water-Gas Shift: I. Intrinsic Kinetics. AIChE J. 1989, 35, 88–96. Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems, 3 edition.; Cambridge University Press: Cambridge ; New York, 2009. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons, 2007. Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids; Mcgraw-hill: New York, 2001. Latham, D. A.; McAuley, K. B.; Peppley, B. A.; Raybold, T. M. Mathematical Modeling of an Industrial Steam-Methane Reformer for on-Line Deployment. Fuel Process. Technol. 2011, 92, 1574–1586. Pena Lopez, J. A.; Manousiouthakis, V. I. Natural Gas Based Hydrogen Production with Zero Carbon Dioxide Emissions. Int. J. Hydrog. Energy 2011, 36, 12853–12868. Erri, P.; Varma, A. Diffusional Effects in Nickel Oxide Reduction Kinetics. Ind. Eng. Chem. Res. 2009, 48, 4–6.


Page 44 of 46

Page 45 of 46

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


List of figure captions

Figure 1 - Packed-bed reactor multiscale model scheme ............................................................................ 12 p p Figure 2 - Variation of H2 gen. rate, at z Lr  0.05 , with  (a), d pore (b), and k sp (c) for constant  f ,V ... 20 p Figure 3 - Variation of H2 gen. rate, at z Lr  0.05 , with  fp,V (a), d pore (b), and k sp (c) for constant  .. 20 p p Figure 4 - Variation of H2 gen. rate, at z Lr  0.05 , with  f ,V (a),  (b), and k sp (c) for constant d pore ... 21

Figure 5 – Reactor species molar fractions at different z Lr , .................................................................... 22 Figure 6 – Species molar fraction pellet radial profiles at different z Lr , ................................................. 23 Figure 7 – Reactor-Pellet pressure difference  p p  r p , z   p p  r , z   radial profiles ................................... 24   Figure 8 - Reactor fluid temp. axial profile, for combinations 1 (straight lines) and 2 (dotted lines) ....... 26 Figure 9 - Reactor fluid-Pellet surface temperature difference axial profile, ............................................. 27 Figure 10 - Pellet surface – Pellet center temperature difference axial profile, .......................................... 28 Figure 11 – Reactor wall heat flux axial profile, ........................................................................................ 29 Figure 12 – Methane conversion axial profile, ........................................................................................... 30 Figure 13 – Species’ effectiveness factor axial profile, for combinations 1 (a) and 2 (b) .......................... 31 Figure 14 - Average species’ effectiveness factor axial profile, for combinations 1 (a) and 2 (b) ............. 33 Figure 15 – Cumulative transport-reaction rate axial profile, ..................................................................... 34 Figure 16 - Cumulative intrinsic reaction rate axial profile, ....................................................................... 35



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

For Table of Contents Only


Page 46 of 46

Parametric Studies of Steam Methane Reforming Using a Multiscale

Recommend Documents