Combined approach using mathematical modelling and artificial neural network for chemical industries: Steam methane reformer

Applied Energy 255 (2019) 113809

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Combined approach using mathematical modelling and artificial neural network for chemical industries: Steam methane reformer Nguyen Dat Voa, Dong Hoon Oha, Suk-Hoon Honga, Min Ohb, a b

⁎,1

T

, Chang-Ha Leea,

⁎,1

Department of Chemical and Biomolecular Engineering, Yonsei University, Seoul, Republic of Korea Department of Chemical and Biological Engineering, Hanbat National University, Daejeon, Republic of Korea

HIGHLIGHTS

SMR model with a multiscale reactor, wall and furnace was developed. • ATherigorous temperature, pressure, heat flux and mole fraction agree well with a reference. • A combined using an SMR dynamic model and ANN was suggested for industries. • The developedmethod predicts the output with 98.91% accuracy in a few seconds. • The method canapproach be used for the design and online optimization of H production. • 2

ARTICLE INFO

ABSTRACT

Keywords: Steam methane reforming Multiscale modelling Dynamic simulation Artificial neural network Stochastic simulation

The steam methane reformer (SMR) has become more attractive owing to the increasing importance of hydrogen production using natural gas. This study developed a rigorous dynamic model for an SMR including sub-models for a multiscale reactor, wall, and furnace. The developed SMR model was validated within a small error (lower than 4%) using the reference data such as temperature, pressure, mole fraction, and average heat flux. The results predicted by changing the catalyst parameters and operation conditions confirmed the reliability of the model. Therefore, the developed model was used to generate the SMR performance data using a deterministic and stochastic simulation with four main operating variables: the inlet flow rate, temperature, S/C ratio of the reactor side, and the inlet flow rate of the furnace side. To reduce the data dimensionality, the resultant dataset was analyzed using the principle components based on a singular value decomposition method. Artificial neural network (ANN) trained through 81 datasets was applied for the feed-forward back propagation of a neural network to map the relationship between the operating variables and predicted outputs. And the ANN relation predicted the outputs (temperature, velocity, pressure, and mole fraction of components) with higher than 98.91% accuracy. Furthermore, the computational time was significantly reduced from 1200 s (dynamic simulation) to 2 s (ANN). The developed methodology can be applied not only for the online operation and optimization of a reformer with high accuracy but also for the design of a hydrogen production system at low computational cost.

1. Introduction Hydrogen is widely used in many important industrial applications, such as petroleum refineries, methanol, and ammonia production [1,2]. In addition, hydrogen is also considered one of the most important clean energy source carriers [3–5]. Hydrogen is produced from various processes such as a reforming of hydrocarbons [6–8], gasification of coal [9–11], and biomass conversion [12–14]. Because the demand for

natural gas utilisation is growing, the reforming of natural gas is becoming the major process of industrial hydrogen production. Natural gas reforming can be conducted using steam reforming, thermal reforming, and partial oxidation [15]. Among the available processes, steam reforming is the most widely applied for industrial hydrogen production, demonstrating 50% of the required capacity [8]. A steam methane reformer (SMR) is classified based on its shape, firing direction, and direction of process gas [16]. The SMR includes a

Corresponding authors. E-mail addresses: [email protected] (M. Oh), [email protected] (C.-H. Lee). 1 Prof. Min Oh and Prof. Chang-Ha Lee contributed equally to the study as corresponding authors. ⁎

https://doi.org/10.1016/j.apenergy.2019.113809 Received 16 May 2019; Received in revised form 27 July 2019; Accepted 29 August 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.

Applied Energy 255 (2019) 113809

N.D. Vo, et al.

Nomenclature

f hf k

Upper case

A ANN BDF C CP CFD CO/ C CO2 / C D DAEs E F FDM FFBP H /C J K

Ka L M P PDAEs Pr Q R

Rr Re Sc S/C SMR SVD T U Ucv V

X

Y

area (m2), binary interaction parameter (–) artificial neural network backward differentiation formula concentration (mol m−3), cost function (–) heat capacity (J kg−1 K−1) computational fluid dynamic carbon monoxide/methane carbon dioxide/methane mass diffusion coefficient (m2 s−1) different algebraic equations activation energy (J mol−1) half-sum of forward and backward axial fluxes (W m−2), integration (–), mass flow rate (kg s−1) finite difference method feed forward back propagation hydrogen/methane Chilton-Colburn factor for mass transfer (–) thermodynamic equilibrium constant (Pa2, –), adsorption constant (Pa−1, –) gas absorption coefficient (m−1) length (m) molecular weight (kg mol−1), data matrix (–) pressure (Pa, bar, kPa) partial differential algebraic equations Prandtl number (–) heat release across flame (W m−3) gas constant (J mol−1 K−1), reaction rate (mol m−3 s−1, mol kgcat−1 s−1) the accuracy for reduced order of data matrix (–) Reynold number (–) Schmidt number (–) steam/methane steam methane reformer singular value decomposition temperature (K) matrix of left singular vector (–) heat transfer coefficient between gas and reactor tube wall (W m−2 K−1) variable (–), matrix of right singular vector (–), volume (m3) mass fraction (–), Conversion (%), data matrix after dimensional reduction (–) mole fraction (–)

n r s u x y (x ) w z

Greek letter

H P

µ

b d

heat of adsorption (J mol−1), heat of reaction (J mol−1) pressure drop (Pa, kPa, bar) parameter (–) parameter (–) parameter (–) Stefan-Boltzmann constant (W m−2 K−4) error (%, –) porosity (–), emissivity (–) thermal conductivity (W m−1 K−1) effectiveness factor for reaction (–) Thiele modulus (–) matrix of singular value (–) density (kg m−3) dynamic viscosity (Pa s) parameter (–) spherical coordinate (m)

Subscripts and superscripts

b cat data e f i in j k l L lower

mix out p r sim t upper w

Lower case

a av

probability density function, activation function heat transfer coefficient (W m−2 K−1) mass transfer coefficient (m s−1), rate constant (mol Pa1/ 2 kgcat−1 s−1, mol kgcat−1 s−1 Pa−1) number of cases (–) radial coordinate (m), radius (m) singular value (–) velocity (m s−1) input value of network (–) data for training and test of network (–) deterministic value for data generation (–), weight (–) axial coordinate (m), input of neuron in network (–)

output of neuron in network (–) external surface area per unit volume of catalyst bed (m2 m−3) bias (–) diameter (m)

catalytic reactor (tube) and furnace which interact through the wall of the tube. The mixture of fuel and air fed into the furnace side through the burner is combusted, and the heat generated is transferred to the reactor through the reactor wall by radiative heat transfer. The mixture of methane and steam is converted into hydrogen and carbon dioxide through catalytic reactions in the reactor [17,18]. Hydrogen with high purity can be obtained from hydrogen-rich gas using a separation unit [19–21].

adiabatic bed catalyst data effective furnace, flame species index inlet, inside neuron index, reaction index neuron index layer index output layer lower mixture outlet, outside particle reactor, reformer simulation tube upper wall

Most modelling studies have focused on the reforming reactor and can be classified into single- and multiscale models. A single-scale model, which links the distributed reactor domain with the non-catalyst domain or 0D (dimensional) catalyst domain, directly couples the reforming kinetics into a gas phase or lump (0D) model of catalyst particles for a computational fluid dynamic (CFD) simulation [17,22], online industrial control [16,23], and support for other experimental studies [8,24]. The multiscale model, which couples the distributed 2

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N.D. Vo, et al.

reactor domain with the distributed catalyst domain, links the reforming kinetics with a distributed model of catalyst particles for studying the impact of a feed disturbance [25], heat flux from the wall [26], and the catalyst parameters [27] on performance. Therefore, the multiscale reactor model can provide more information because it can reasonably couple the kinetics with catalyst particles, in which an endothermic reaction converts reactants into products and consumes the heat flux from the furnace side. Studies on a multiscale reactor model have mostly focused on the coupling of the 1D reactor domain and 1D catalyst domain. It has been pointed out that the reactor side requires at least the 2D domain to present the radial temperature gradients properly [28]. In contrast, a few studies on coupling the 2D or 3D reactor domain with the catalyst domain have been conducted, although the models were simplified under various assumptions. A model simulation, which coupled the 2D reactor domain with the 1D catalyst domain, was conducted under steady state conditions [29], but ignored the important factors, such as the axial dispersion and the heat and mass transfer resistances between the gas and solid phases [8,23]. In addition, it was reported that the furnace model also plays an important role in predicting the temperature disturbance of the furnace side [30]. Therefore, a reformer model that couples a 2D-1D (reactor domain-catalyst domain) multiscale reactor model with a furnace model through the reactor tube wall is necessary to efficiently and accurately predict the effects of the catalyst parameters and operating variables on the design, operation, and online optimization of a reformer used in industry. More recently, the potential of artificial neural network (ANN) has been increasingly highlighted regarding the prediction accuracy and saving of computational costs in many industrial fields. For the chemical processes, ANN has been used to estimate the outputs based on industrial data [31,32], reduce the high computational cost [33], and predict the behaviour for control and optimization systems [34–37]. For application in online systems, process design, and optimization, a simple model of steam methane reforming [16,26,30] is preferred because it saves computational costs compared to rigorous models. In addition, a simple 1D model of a reformer was used in combination with ANN for a hydrogen plant [32] to predict the steady-state outputs

without a data analysis. When ignoring radial heat and mass transfers in the reactor domain, and the heat and mass resistances between the solid and gas phases, simple models have limitations in terms of the accuracy of the reactor design and performance prediction. In addition, rigorous dynamic models can elucidate the processes in more detail for online systems, process design, and optimization. As a result, the combination of a rigorous reformer model and ANN can overcome the low accuracy and computational cost. However, few studies on the combination of a reformer model and ANN have been conducted by combining a modelling of the conservation law, data analysis, and ANN algorithm. This is the motivation for combining the rigorous model of a reformer and ANN to predict the dynamic behaviour of the reformer. This combination can help formulate the dynamic relation of the target outputs with high accuracy and low computational costs. Therefore, this methodology with high accuracy and a fast prediction is important for the online and control systems required to make a decision within a short time period and for the process design and optimization to upgrade the reformer efficiency [16]. In this study, a rigorous dynamic model for an SMR, including a 2D1D (reactor-catalyst) multiscale reactor, 2D reactor wall, and 1D furnace models, was developed using conservation equations. A typical steam methane reformer, a top-fired co-current furnace reactor, was chosen for the SMR modelling. The simulation was conducted by combining the sub-models validated using reference data. The inlet composition of SMR furnace was fixed because, in industrial operation, the off-gas (tail gas) of a hydrogen separation unit (typically, pressure swing adsorption) was supplied to the furnace. The roles of the main operating variables in terms of performance, such as the inlet flow rate, temperature, S/C ratio of the reactor side, and the inlet flow rate of the furnace side, were analyzed. In addition, the effects of the catalyst parameters, such as the catalyst porosity and particle diameter, on the performance were analysed. To achieve a methodology for prediction with high accuracy and low computational cost, ANN was applied using the data generated from deterministic and stochastic simulations, the singular value decomposition (SVD) analysis technique, and a feed forward back propagation (FFBP) neural network. The dynamic

Fig. 1. Model structure of top-fired co-current reformer. 3

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simulation of an SMR was conducted using gPROMS (PSE, UK) [38], and the SVD and FFBP were conducted using Tensorflow (Google, USA) [39].

Ucv (z ) = 0.4

ur (0, r ) = urin ,

r

•

z

b Ji

=

e b r (z ,

r)

Tr (z , r ) t

Sci =

(1)

2 T (z , r ) r z2

+

av =

Tr (z, 0) = 0, r

r

z

= Ucv (z )(Tw (z ,

ur (z, rtin ) = 0, r

z

9.49Di ur dp

r ) r (z, r ) Cp, r (z, r )

+

(5)

+ 0.365Sci

(6)

0.398

(7)

µr

µr

(8)

r Di

6 (1 dp

b)

(9)

0.41

Cp, r r ur

(10)

Pr 2/3

Cp, r µr

(11)

– Effective heat conduction:

Tr (z , r ) z

e r

1 Tr (z, r ) r r

[0, L]; Tr (z,

Tr (L , r ) = 0, z

z

b r

+ (1

(12)

b ) cat

[0, L]; r

[0, rtin];

[0, rp]

The catalyst porosity was considered a constant. When diffusion is the major phenomenon inside the catalyst particle [27], the mass and energy conservations are formulated, excluding the convection terms.

(0, rtin)

• Mass conservation:

Tr (z , rtin ) r

e in r (z , rt )

rtin)),

r

=

b. Catalyst domain (micro scale): z

(2)

Tr (z , r ))

(0, rtin );

[0, L]

(4)

r

2T (z , r ) r r2

rtin)

[0, L];

r ur dp

hf = 0.61Re

Boundary conditions:

Tr (0, r ) = Trin,

z

u r (L , r ) = 0, z

– Heat transfer coefficient [40,41]:

Ci, r (z , rtin) = 0, r

Pr =

b u r (z ,

+ a v hf (z , r )(Tp (z, r , rp)

(0, rtin);

– External surface area per unit volume of catalyst bed:

J m3 s

+

1+

0.82

[0, L]

r ) Cp, r (z, r )

r

Di dp

= 0.765Re

Re =

Ci, r (L , r ) = 0, z

[0, L];

r ) ur2 (z, r )

– Mass transfer coefficient [40]:

(0, rtin)

Energy conservation:

b r (z ,

(0,

r (z ,

0.5ur dp

Die, r = 0.73Di +

Boundary conditions:

rtin);

b)

– Effective mass diffusion coefficient [43]:

ki = Ji ReSci1/3

Ci, r (z, r ))

kg m2 s2

• Reactor coefficients:

mol m3 s

+ a v ki (z, r )(Ci, p (z, r , rp)

@r = rtin

,

r

(0, rtin)

r

[0, rtin]

[0, L]; r

0.4

(3)

Ci, r (z, r ) Ci, r (z, r ) + b u r (z , r ) t z 2C (z , r ) 2C (z , r ) 1 Ci, r (z , r ) i, r i, r e = b Di, r + + z2 r2 r r

Ci, r (z, 0) = 0, r

Cp, r µr

Boundary conditions:

The bed porosity in the reactor is considered a constant. The mass and heat transfers between the reactor and catalyst domains are linked through the heat and mass transfer coefficients [40,41]. For a porous zone filled by catalyst particles, the Ergun equation [42] has been employed to elucidate the pressure drop on the reactor side. The model of the reactor side is shown for a half-tube owing to the symmetry.

Ciin ,r,

r

0.8

r

ur (z, 0) = 0, r

a. Reactor domain (Macro scale): z

z

r

dp ur

1/3

Cp, r µr

2 Pr (z, r ) 150(1 1.75(1 b) = µr ( z , r ) u r (z , r ) + 3 2 3 z d b p b dp

2.1.1. Reactor side

• Species conservation:

1/3

r

• Momentum conservation:

A top-fired co-current SMR using eight reactor tube rows was selected in this study. The model structure of the reformer is shown in Fig. 1. A mathematical model for a reformer was formulated using heat, mass, and momentum conservations for a single reactor tube, assuming that all tubes have the same performance (temperature, pressure, velocity, mole fraction, etc.). In addition, the catalyst was assumed to have uniform particle size, porosity, and physical properties, and the ideal gas law was applied.

r

dp

dp ur

r

2.1. Steam methane reforming

Ci, r (0, r ) =

2.58

+ 0.094

2. Mathematical model

b

r

p

[0, L]

Ci, p (z , r , ) t

=

where the heat transfer coefficient between gas and reactor tube wall was evaluated based on a study by Ghouse [25]. 4

mol m3 s

e p Di, p (z , 2

r, )

2

Ci, p (z, r , )

+ (1

p ) cat Ri, p (z ,

r, )

(13)

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Boundary conditions:

Ci, p (z , r , 0)

e p Di, p (z ,

= 0;

r , rp)

z

f

z

Ci, p (z, r , rp)

=

r, )

2 F (z )

J m3 s

z2

e p (z ,

r , rp) z

= 0;

+ (1

p ) cat Rj (z ,

r , ) Hj

(14)

[0, rtin]

[0, L], r

= hf (z , r )(Tr (z, r )

[0,

=

r)

w (z ,

rtout )

t

r)

[0, Lf ] (16-2)

Tf (L)

;

z

Ci, f (z ) z

=0 mol m3 s

= Di, f (z )

2 C (z ) i, f z2

+ Ri, f (z )

(18)

(F (z )

in w (z , rt )

z

(z )

uf (z ) t

+

f

=0

( z ) u f (z )

z

z

uf (L)

uf (0) = ufin;

(15)

Tw (z , rtin) ; r

Tw4 (z , rtout ));

Ci, f (L)

u f (z ) z

=

Pf (z ) z

+ µ f (z )

2u

f

(z )

z2

(19)

Boundary condition:

1 Tw (z , r ) + r r

z

=0

• Coefficients for furnace:

– Adiabatic flame temperature [47]:

[0, L]

Tf =

[0, L]

in Cp, f Tmix

Yi, j Hj

(20)

Cp, f

– Furnace gas absorption coefficient [22]:

2.1.3. Furnace side: z [0, L] A modified Roesler flux-type model [44,45] was used to calculate the axial temperature profile in the furnace. However, the furnace gas absorption coefficient was evaluated by using Eq. (21) [22]. This relation was developed from an empirical model [46] by using the following assumptions: The pressure inside the furnace chamber is constant and close to the atmospheric pressure. The flame length is constant and the difference between the values of total emissivity at the furnace inlet and outlet is ignored. Then, the gas absorption coefficient becomes only the function of furnace temperature. In this study, the assumptions were reasonable for the system. The furnace side of the reformer was operated at a pressure close to the atmospheric condition. The flame length was generally considered as constant for computation [16,22] with the values from 4.5 to 6 m. Furthermore, since the main composition in the furnace side was the inert gases of nitrogen and argon with the mole fraction of up to 61%, ignoring the change of total emissivity between the inlet and outlet in the furnace was an acceptable assumption.

• Energy conservation:

z

z > Lf

• Momentum conservation: f

2T (z , w r2

Tr (z, rtin)) =

Tw (z, rtout ) = r

F (0)

Ci, f (0) = Ciin ,f ;

Boundary conditions:

Ucv (z )(Tw (z , rtin)

;

Boundary condition:

r ) Cp, w (z , r ) w (z ,

z2 L2f

z Lf

F (L) =0 z

+ u f (z )

t

J m3 s

Tw (z, r ) t 2T ( z , r ) w + z2

4

Ci, f (z )

Tp (z, r , rp));

2.1.2. Wall side: z [0, L]; r [rtin, rtout ] The heat transfer through the reactor tube wall between the furnace and reactor was considered through the following equations.

w (z ,

(16-1)

(17)

• Species conservation:

rtin]

• Energy conservation:

z

( F (z ) + )

F (0) = 0; z Tf (0) =

z

Tp (z, r , rp) [0, L], r

Tf (0))

Lf

=

Tf (z )

Boundary conditions:

Boundary conditions:

Tp (z , r , 0)

f uf Cp, f (T f

6

Q (z ) =

(z ) Cp, f (z ) uf (z )

0,

t Tp (z , r , )

2

2

Ci, p (z, r , rp));

f

T f4 (z )) + Q (z )

= 4K a ( F ( z )

Tp (z, r , )

r , ) Cp, p (z, r , ) e p (z ,

= ki (z, r )(Ci, r (z, r )

+

t

[0, rtin]

[0, L], r

• Energy conservation: p (z ,

[0, rtin]

[0, L], r

Tf (z )

(z ) Cp, f (z )

K a = 2.1

10 8T f2

2.06

10 4Tf + 0.456

(21)

– Alpha, beta, and gamma [23]:

=

(2K a + At + Ar ) 2

(22)

=

(4K a + 2 t At )

(23)

=

4K a T f4 (z )

+2

4 out t At Tw (z , rt )

(24)

2.1.4. Physical properties and evaluation The required physical properties of a fluid and solid were calculated using Eqs. (25)–(37) with the binary interaction parameters were evaluated by Mason-Saxena approach [48]. Meanwhile, the individual physical properties of the gas components (CH4, H2O, CO, H2 and CO2) were estimated based on a study by Ngo [49]. In addition, the conversion of CH4 and the relative error in the variables were evaluated using Eqs. (38) and (39).

J m3 s

• Physical properties: r

5

=

Ci, r Mi

(25)

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Cp, r =

Cp, i Xi, r n

Yi, r µi

µr =

n j =1

i=1 n r

i=1

p

Y j, r

( )

Mj 0.5

(27)

Mi

Yi, r i n Y A j = 1 j, r ij

= = (1

p ) cat

Cp, p = (1

+

(28)

Ci, p Mi

p

p ) Cp, cat

+

Cp, i Xi, p

p

(31) n

= (1

p ) cat

+

p i=1

f

=

(32) (33)

Cp, i Xi, f n

i=1

n j=1

n

Yj , f

Yi, f

= i=1

P=

V = wVlower + (1

(34)

w=

Yi, f µi

µf =

f

Yi, p i n Y A j = 1 j, p ij

Ci, f Mi

Cp, f =

2.2.1. Data generation A deterministic simulation was conducted under various operation conditions using Eqs. (40) and (41) below to create the training data for ANN. Meanwhile, the test data of the ANN was created through a stochastic simulation using random values for the operation conditions. The random value of the operation conditions was created based on the probability density function of a uniform distribution function [38] described in Eq. (42). The probability density function was integrated to form Eq. (43). This form was solved through a Monte Carlo integration to obtain a random value.

(30)

p

e p

2.2. Artificial neural network

(29)

p Di 1

Die, p =

chemical reaction and physical phenomena that occur in the catalyst domain. On the furnace side, the global kinetic model [22,51,52] was used to elucidate the combustion of fuel for heat generation during the reforming process. The main equations for the reforming kinetics on the reactor side and the combustion kinetics on the furnace side are listed in Table 1. Eqs. (A.1)–(A.11), which are related to the reaction rates and equilibrium constants, are presented in Appendix A.

(26)

n j=1

( )

Mj 0.5 Mi

CRT

(35)

f (V ) =

(36)

F=

in urin CCH 4, r

ur (z , r ) CCH4, r (z , r ) in urin CCH 4, r

Vlower 1

Vupper

Vlower

,

V

(Vupper , Vlower )

(42)

dV

(43)

[0, L], r

|Vsim Vdata | Vdata

VT

M=U

100, (38)

[0, rtin]

100

(44)

Here, U and V are left- and right-singular vectors of matrix M, and = diag (s1, s2, ...,sn) indicates singular values of the data matrix. The principle data can be analysed based on the ratio in Eq. (45). The value of Rr ( 1) is set as a standard to choose the rth rank of data matrix M. The reduced order of matrix M (Rmxn) becomes matrix X (Rmxr), which is used for training in the neural network.

– Relative error:

=

1 Vupper

2.2.2. Artificial neural network algorithm with dimensional reduction The data matrix, M, is decomposed using the singular value decomposition method to analyse the principle component in Eq. (44).

– Methane conversion:

z

(41)

(37)

• Evaluation: XCH 4 =

1 n

i

Yj, f Aij

(40)

w ) Vupper

(39)

Rr =

2.1.5. Kinetic model This study used an intrinsic kinetic model [50] to describe the

r j=1 n j=1

j j

=

r j=1 n j=1

s 2j s 2j

(45)

Table 1 Kinetics of steam methane reforming and fuel combustion [22,50–52]. Reaction Reactor side

Kinetic

CH4 + H2 O CO + H2 O

Furnace side

CO + 3H2 (1) CO2 + H2 (2)

CH4 + 2H2 O

CO2 + 4H2 (3)

CH4 + 1.5O2

CO + 2H2 O (4)

CO + 0.5O2 CO2

CO2 (5)

k1 1 P 2.5 H2

R2 =

k2 2P H2

R3 =

k3 3 P 3.5 H2

PCH4 P H2 O

(P

CO P H2 O

PCH4 PH2 2 O

3 P PH 2 CO K1 P H2 PCO2 K2

1 2

)

4 P PH 2 CO2 K3

R 4 = 1015.22 [CH4 ]1.46 [O2 ]0.5217 exp R5 = 1014.902 [CO]1.6904 [O2 ]1.57 exp

CO + 0.5O2 (6)

H2 + 0.5O2

R1 =

R 6 = 1014.349 [CO2 ] exp

H2 O (7)

R7 = 4.61

6

62281 Tf

1015 [H2 ][O2 ] exp

10080 Tf

1 2 1 2

20643 Tf 11613 Tf

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N.D. Vo, et al.

Fig. 2. General structure of multilayer neural network.

Fig. 3. Feed forward back propagation method.

ANN was used to map the input and output based on an FFBP neural network. The general structure of a multilayer neural network from the input to output layer at weight and bias is shown in Fig. 2. The bias for the jth neuron of the lth layer is defined as blj , and the weight for the jth neuron of the lth layer from the kth neuron of the (l − 1)th layer is defined as wljk . The input and output of the neurons are defined through Eqs. (46) and (47), respectively.

z lj

w ljk akl 1

=

+

blj

k

C=

w ljk akl k

1

+ blj = f (z lj )

y (x )

a (x )

2

x

(48)

The error definition of a neuron and layer, the error in the output layer, the error relation between layers, and the rate of change of a cost function with the bias and weight are presented in Eqs. (49)–(53). l j

C z jl

=

(46) L j

ajl = f

1 n

(47)

l j

The purpose of training of a neural network is to find the weight and bias that show the relation between the input and output. Initially, the feed forward procedure is used to calculate the output of a neural network (a(x)). The cost function was built from the average difference between the output of a neural network (a(x)) and the training data (y (x)), as presented in Eq. (48).

C ajL

=

=

[(w l )T l] f

C = bjl

l j

C = akl w ljk 7

(49)

ajL z jL

=

C f (z jL) ajL

(z l 1)

(50) (51) (52)

1 l j

(53)

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Fig. 4. Solution procedure for combination of SMR model and artificial neural network. Table 2 Dimensional and physical properties of catalyst and reformer [16,17,22,50]. Parameter

Table 3 Operation conditions for validation case [22] and artificial neural network.

Value −3

Catalyst

Density, (kg m ) Heat capacity, (J kg−1 K−1) Thermal conductivity, (W m−1 K−1) Catalyst porosity Particle diameter, (m)

3960 880 33 0.25 3.5

Reformer

Reformer dimension (m * m * m) Tube length, (m) Number of burners Number of tubes Bed porosity Inside tube diameter, (m) Outside tube diameter, (m) Density of tube, (kg m−3) Heat capacity of tube, (J kg−1 K−1) Thermal conductivity of tube, (W m−1 K−1) Flame length, (m) Tube emissivity

16 * 16 * 13 12.5 96 336 0.609 0.126 0.146 7720 502 29.58 5 0.85

Operation conditions

Validation case

Artificial neural network

Inlet reactor side

Pressure, (kPa) Temperature, (K) Mass flow rate, (kg s−1) Compositions: S/C H/C CO/C CO2/C

3038.5 887 39

3038.5 787–937 30.7–61.4

2.96 0.0072 0.0004 0.047

2–4 0.0072 0.0004 0.047

Pressure, (kPa) Temperature, (K) Mass flow rate, (kg s−1) Compositions: YCH4 YH2 YCO YCO2 YH2O YN2 YO2 YAr

131.3 532.9 36.3456

131.3 532.9 17.92–43.52

0.0501 0.0592 0.0208 0.0972 0.0039 0.6008 0.1610 0.0071

0.0501 0.0592 0.0208 0.0972 0.0039 0.6008 0.1610 0.0071

Inlet furnace side

A flow diagram of the FFBP method is shown in Fig. 3. The strategy used an iteration method was based on redesigning the bias and weight to achieve the gradients of cost function with the bias and weight which are lower than a setting tolerance. The optimal weight and bias would be used to formulate the relation between the input and output.

transform the PDAEs into differential algebraic equations (DAEs) [55]. The backward differentiation formula (BDF) method, Newton method, and LA solver were used to solve the DAEs [56]. Tolerances of 10−5 were used as the convergence criteria of the simulation. The SVD was used to analyse the principal component for choosing the necessary data. ANN was applied to map the relation between the operation conditions and desired outputs using the FFBP method with a convergence standard of 10−5 The solution procedure for the combina. tion of the SMR model and artificial neural network is shown in Fig. 4.

3. Solution strategy A coupling of the sub-model and finite difference method (FDM) [53,54] was used in this study. The rigorous dynamic model of the SMR includes the sub-models of a multiscale reactor, the tube wall, and the furnace models, which are linked based on the radial boundary conditions. Therefore, the rigorous model was converted into a system of partial differential algebraic equations (PDAEs). The FDM was used to 8

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Fig. 5. Dynamic profile of the reactor temperature and methane mole fraction.

4. Simulation basis

rigorous dynamic model was used to generate the data for ANN. A certain set of the data was used for the training data of the ANN in tensorflow. As a final step, the results from the ANN were compared with the test data generated from the dynamic simulation. The validated results and parametric study were discussed in later sections.

The simulation applied in this study was conducted using the dimensional and physical properties of the catalyst and reformer, as presented in Table 2. To validate the model simulation, the same operating conditions as the reference study [22] were applied to the SMR. For ANN, the simulation was carried out within the range of available operating conditions for the SMR, as listed in Table 3. On the reactor side, except for the S/C ratio, the composition of the inlet gas was fixed. This ratio was separately studied to prevent the formation of CO from the tube damage under high-temperature conditions [57]. The operating pressure of the reactor was recommended to have a high value (> 20 bar) because other units for hydrogen production are typically operated at high pressure to suppress the CO generation [26]. Therefore, to analyse the impacts of temperature, flow rate, and S/C ratio on the reactor side, the operating pressure was fixed at 30.385 bar [22]. On the furnace side, it was assumed that the fuel from the off-gas of a hydrogen separation unit (typically, pressure swing adsorption) was supplied to the furnace. Therefore, the composition, pressure, and temperature were fixed, and the flow rate was used to analyse the performance of the reformer [22].

5.1. Validation of model To evaluate the mathematical model accuracy, a simulation was conducted under the operating conditions of the validation case presented in Table 3 [22] including pressure, temperature, mass flow rate and inlet compositions at the reactor and furnace sides. Since the off-gas (tail gas) of a hydrogen separation unit was used for the SMR furnace in the validation case, pressure, temperature, flow rate and composition of the furnace side were fixed as shown in Table 3. On the other hand, a wider range of the operating conditions was applied for deep learning. Dynamic profiles of the reactor temperature and methane mole fraction are provided in Fig. 5. The axial temperature was initially reduced and thereafter increased owing to the endothermic reaction in the catalysts and the heat flux from the furnace. The minimum axial temperature of 780 K was observed at 0.25 m, which was close to that in [17] and [22] (799 K at 0.2 m). In addition, the temperature gradient between the reactor tube centre and inner tube wall was approximately 60–110 K. This result confirms the argument that the 2D reactor model is required to elucidate the mechanism properly owing to the radial temperature gradient [23,28,29]. Finally, the outlet temperature reached a steady-state value of 1150 K after 2100 s, and the methane mole fraction was reduced along the tube length. Because most studies have only focused on the steady-state results,

5. Results and discussion The procedure for the validation of the combined approach using the dynamic model and ANN for chemical industries is as follows: As a first step, the validation of a steam methane reformer was carried out by comparing the simulation results of the rigorous dynamic model with reference data [17,22] and the prediction accuracy was confirmed. Thereafter, the deterministic and stochastic simulated results of the 9

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Fig. 6. Steady-state profiles of reformer: (a)-(e) mole fraction of CH4, H2O, CO, H2, and CO2; (f)-(h) temperature, velocity, and pressure; (k) furnace temperature.

the steady-state results in Figs. 6 and 7 were analysed to validate the results with the reference studies. The temperature, pressure, velocity, and mole fractions in the reactor, and the temperature in the furnace, are shown in Fig. 6. The furnace temperature (Fig. 6(k)) increased along the flame length and reduced slowly thereafter owing to the heat transfer to the reactor. The temperature reached 1244 K at the outlet, which was close to that found in other studies [22,23] (1266–1323 K). Owing to the heat transfer from the furnace and catalytic endothermic reactions, the parabolic temperature profile and temperature gradient of a 2D reactor was as shown in Fig. 6(f), and almost the same velocity profiles were monitored, as indicated in Fig. 6(g). Owing to the packing and thermal resistances [42], the pressure decreased from 30.385 to 29.65 bar and the velocity increased from 1.17 to 2.31 m/s. The changes in concentration of methane and steam were significant between the inlet and middle of the reactor. The mole fraction of hydrogen increased along the reactor tube length and reached 0.4691 at the outlet (Fig. 6(d)). However, the range of CO production was relatively more extended (Fig. 6(c)), and the generation of CO2 was almost complete at the inlet part of the reactor, which is a similar region of the reactor at a relatively low temperature, as indicated in Fig. 6(k). The mole fraction variation of the components is compared with the reference in the following (Table 4). The mole fraction profile at the microscale of the catalyst particle, which is located at three different positions in the reactor tube, is shown in Fig. 7. To compare to the reference, the 1D reactor outlet results were derived from the average values of the 2D results. The gradient curve of the mole fraction inside the catalyst particles elucidated the mass transport tendency, which is controlled through diffusion. The reactants (CH4 and H2O) and products (H2, CO, and CO2) were transported to the inside and outside of the particles owing to the positive and negative gradients between the surface and centre of the particles, respectively. This gradient had a reducing tendency along the bed length, and agrees with [27]. Increasing the reactor temperature

(Fig. 6(f)) led to an improved reaction rate for the endothermic reaction in the particles. The outlet pressure and mole fraction were compared with industrial data [22] and the results from a CFD simulation for a single reactor tube [17]. As presented in Table 4, the mole fractions in the study can predict the industrial data within less than 3% error. The average heat flux across the reactor tube wall was 69.416 kW/m2, which is close to that of previous studies [16,17,22,58]. It was confirmed that the developed SMR model, including the 2D-1D multiscale reactor, 2D wall, and 1D furnace models, can accurately predict the reforming process, showing a slightly more accurate prediction with the industrial data than the 1D result in terms of the products. The other advantages of the developed model are the ability to observe the dynamic behaviour and 2D profiles as the catalyst parameters and operation variables are changed, which is discussed in the following section. 5.2. Effects of catalytic parameters and operating variables on SMR performance The effects of the catalyst parameters on the SMR performance were evaluated within the range of the operating variables, as presented in Table 3. The simulation results from the developed SMR model (multiscale reactor (2D-1D), wall (2D), and furnace (1D) models) were compared with those from the reformer model with the single-scale reactor (2D-0D), wall, and furnace models shown in Fig. 8. The multiscale and single-scale models showed a decreasing tendency of methane conversion with an increase in the catalyst particle diameter owing to the decreased contact area between the gas and solid phases [8]. The methane conversion in the multiscale SMR model was significantly changed with the porosity, which agrees with [27] because of the effect of the catalyst porosity on the diffusion coefficient in Eqs. (13)–(31). Meanwhile, the results in the single-scale model were almost 10

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Fig. 7. Steady-states profile for mole fraction of each component (CH4, H2O, H2, CO2, and CO) in catalyst particle: (a) z / L = 0.05; r / rtin = 0.5; (b) z / L = 0.25; r / rtin = 0.5; (c) z / L = 0.5; r / rtin = 0.5.

constant with a decrease in the porosity because the lump model for the catalyst particle, which did not consider the effects of the catalyst porosity, was used. The results indicate that the difference in methane conversion between the multiscale and single-scale models increases with a higher particle diameter. The results confirmed that the SMR model considering a multiscale reactor can predict the performance results more reasonably as the variables of the SMR are changed. The variations in methane conversion and pressure drop by the main operating variables are presented in Fig. 9, namely, the inlet flow rate, temperature, S/C ratio on the reactor, and the inlet flow rate on the furnace. The methane conversion and pressure drop were greater at a higher flow rate of the furnace, and the degree of variation among the flow rates of the furnace depends on the other operating variables. A higher inlet flow rate of the reactor led to a greater reduction of the residence time. As shown in Fig. 9(a), the methane conversion decreased significantly, and the difference in the pressure drop among the

Table 4 Comparison of the simulation, industrial data [22] and 1 tube CFD simulation [17] at the reactor outlet. Variable

1 tube CFD [17]

This simulation

Industrial data [22]

Error with industrial data: absolute (%)

P, (kPa) Heat flux, (kW m−2) Y_CH4 Y_H2O Y_CO Y_H2 Y_CO2

3044 68.972

2965.4 69.416

2879.8 67.125

85.59 (2.92%) 2.285 (3.41%)

0.0426 0.3467 0.0873 0.4645 0.0588

0.0440 0.3410 0.0904 0.4691 0.0555

0.0453 0.3377 0.0889 0.4713 0.0559

0.0013 0.0033 0.0015 0.0022 0.0004

(2.86%) (0.97%) (1.68%) (0.46%) (0.71%)

11

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Fig. 8. Effects of catalyst porosity and particle diameter on methane conversion in two reformer models: the multiscale (symbol) and single-scale (line) models for the reactor (Symbol: multiscale model, Line: single-scale model).

inlet flow rates in the furnace was minute. As shown in Fig. 9(b) and (c), the methane conversion increased almost linearly with an increase in the S/C ratio, the inlet gas temperature in the reactor, and the inlet flow rate in the furnace owing to the increase in the reactor temperature and reactant concentration. The pressure drop increased with the S/C ratio and decreased with an increase in the inlet temperature of the reactor. However, the variation was also small, as shown in Fig. 9(a). The variation in performance agreed with that of other studies [30,59] as the operating variables, that is, the key factors for the SMR performance, were changed. Therefore, it was confirmed that the model can predict the performance variation by controlling the key disturbance on the reactor and furnace sides. In addition, this implies that the model can be combined with ANN and achieve a highly accurate prediction. The four main operating variables were chosen as the input variables for the ANN to predict the target outputs of the SMR. It was reported that the operating temperature is usually higher than 800 °C [57], and the tube wall temperature should be lower than 1200 K for the lifetime of the tube [26]. Because the operational energy efficiency can be achieved through a low pressure drop and based on the amount of steam [57], the ranges of the inlet flow rate, temperature, S/C ratio of the reactor side, and the inlet flow rate of the furnace side for the ANN, as specified in Table 3, were selected as 30.7–61.4 kg/s, 787–937 K, 2–4, and 17.92–43.52 kg/s, respectively. Fig. 9. Effects of operating variables on methane conversion and pressure drop: (a) The inlet flow rate of the reactor and furnace at S/Creactor = 3 and Tin,reactor = 887 K, (b) S/C ratio of the reactor and the inlet flow rate of the furnace at Fin,reactor = 30.7 kg/s and Tin,reactor = 887 K, and (c) The inlet temperature of the reactor and the inlet flow rate of the furnace at Fin,reactor = 30.7 kg/s and S/Creactor = 3 (solid symbol: methane conversion, open symbol: pressure drop).

5.3. Artificial neural network in SMR The ANN presented in Figs. 2 and 3 was applied to map the relation between the operating variables and the outputs at the reactor outlet, including the temperature, velocity, pressure, and mole fraction of the components, followed by a combining of the SMR model with ANN, as shown in Fig. 4. Initially, for the ANN cases, the dynamic data at the reactor outlet were generated from the simulation under the different operating variables listed in Table 3. The principal components of the data were analysed using the SVD method in Eq. (44). Data on a total of 81 cases were chosen for the training under a standard accuracy of 98% accuracy, which demonstrates the relation in Eq. (45). This implies that only three different values of each operating variables are required. A comparison between the dynamic simulation and ANN results for the training and test cases are shown in Figs. 10 and 11. The operating conditions for the test were randomly estimated from Eqs. (42) and (43) in the input space. A total of 81 training cases and 32 test cases were applied in the study, but only three training cases, as shown in Fig. 10, and three test cases, as shown in Fig. 11, are presented as

representatives. The other cases and their accuracy are found in Appendix B (Table B.1). Excellent agreement (error < 2%) between the dynamic SMR model and ANN could be observed in the variation of temperature, velocity, pressure, and mole fractions of the components at the reactor outlet regardless of the operating variables. Therefore, using ANN to predict the results from dynamic to steady-state conditions highlights the potential of mapping between the operating variable and the target output. The mapping between the operating variables and outputs achieved an accuracy of 98.91% and reduced the computational cost 600-fold, namely, from 1200 to 2 s. It means that the second step validation 12

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Fig. 10. Comparison between dynamic model (line) and artificial neural network (symbol) results for the training cases (Train_28, Train_41, and Train_54).

achieved an high accuracy of 98.91%. As a representative case study, a comparison of temperature and mole fractions along the reactor among dynamic model, artificial neural network and reference data [22] are presented in Fig. 12. The accuracy of the dynamic model and ANN was confirmed even in the inside profiles of the reactor. Although the ANN results showed slightly higher deviation than the rigorous dynamic model, the difference was minute in the profiles and negligible at the exit of the reactor. These results suggest that combining the developed SMR model with artificial neural network, including a multiscale reactor, wall, and furnace models, can formulate the precise relation between the key operating variables and desired outputs within a few seconds. In addition, the developed methodology can be extended to the process design and

optimization for hydrogen production with high accuracy and a low computational cost. Furthermore, the potentials of the developed methodology can be applied to online control systems, which require reliable results for decision-making within an extremely short time period. 6. Conclusion In this study, an SMR model was developed, including a multiscale reactor, wall, and furnace models, based on the conservation equations of mass, momentum, and energy. The multiscale reactor model was composed of a 2D macroscale model for the reactor domain and a 1D microscale model for the catalytic particles. The dynamic behaviour could be successfully predicted from the combination of the sub-models and 13

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Fig. 11. Comparison between dynamic model (line) and artificial neural network (symbol) results for the test cases (Test_3, Test_10, Test_20).

finite difference method. The developed SMR model was validated using the reference data in terms of the temperature, mole fraction, pressure, and average heat flux from the furnace, and a good agreement was achieved within small error (lower than 4%). A sensitivity analysis was conducted using different catalyst parameters and operation conditions. The result showed the potentials of the model for a reasonable prediction and control of the performance variation. Therefore, the developed SMR model can be used to generate the performance data through a deterministic and stochastic simulation within the feasible range of four main operation variables, that is, the inlet flow rate, temperature, S/C ratio of the reactor side, and the inlet flow rate of the furnace side. The dimensionality of the simulated performance data was reduced

using the SVD method. A total of 81 cases with a standard accuracy of 98% were selected as the training data, which indicated only three different values for each operation condition for SMR training through artificial neural network. The artificial neural network was carried out on the TensorFlow platform based on the FFBP of a neural network. The mapping relation between the four main operating variables and the desired outputs could successfully predict the outputs of the temperature, velocity, pressure, and mole fraction of the components with 98.91% accuracy when the operating variables were changed. In addition, the computational time was reduced from 1200 s for a dynamic simulation to 2 s when applying artificial neural network. The results confirmed that the combined approach using the developed SMR model with artificial neural network can precisely 14

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Fig. 12. Comparison of temperature and mole fractions along the reactor among dynamic model, artificial neural network and reference data [22].

formulate the relation between the key operating conditions and the desired outputs within a few seconds. Therefore, the developed methodology can be applied for online control systems because industries need to make decisions within a short time period based on reliable data. Furthermore, the results can be extended to the process design and optimization of hydrogen production with high accuracy and a low computational cost.

Acknowledgement This work was supported by “the Industrial Strategic Technology Development Program-Engineering Core Technology Development Project” funded by the Korean Government Ministry of Trade, Industry & Energy (No. 10077467) and Korea Gas Corporation.

Appendix A. Kinetics for reforming The reaction rate for reforming (Eqs. (A.1)–(A4)) was estimated based on a study by Xu [50]. The effective factor (Eq. (A.5)) was evaluated using the general Thiele modulus [23]. The equilibrium, rate, and adsorption constants for the species and reactions are listed in Eqs. (A.7)–(A.11) [8].

R1 =

1

k1 PCH4 PH2 O PH2.52

PH3 2 PCO

R2 =

2

k2 PCO PH2 O PH2

PH2 PCO2 K2

R3 =

3

k3 PCH4 PH2 2 O PH3.52

K1

PH42 PCO2 K3

1 2

(A.1)

1 (A.2)

2

1 2

= 1 + K CO PCO + K H2 PH2 + K CH4 PCH4 + K H2 O

j

=

1 [3 j coth ( j ) 3 j2

j

=

Vcat Acat

RTp kj

PH2 O PH2

(A.4)

1]

(A.5)

1 + Kj Kj

(A.6)

26830 + 30.114 Tp

K1 = exp

K2 = exp

cat

Die, p

(A.3)

4400 Tp

(A.7)

4.036

(A.8) (A.9)

K3 = K1 K2

kj = ko, j exp

Ki = K o, i exp

Ej RTp

(A.10)

Hi RTp

(A.11)

15

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Appendix B. Accuracy of artificial neural network The accuracy of artificial neural network between Tensorflow and gPROMS is presented in Table B.1, which includes 81 cases for training, 32 cases for testing, and the last case for an average calculation of the 113 cases. Table B.1 Accuracy of artificial neural network between Tensorflow and gPROMS for training and test cases. Case

Flow rate of the reactor [kg/s]

Temperature of the reactor [K]

S/C ratio of the reactor

Flow rate of the furnace [kg/s]

Accuracy [%]

Train_1 train_2 Train_3 Train_4 Train_5 Train_6 Train_7 Train_8 Train_9 Train_10 Train_11 Train_12 Train_13 Train_14 Train_15 Train_16 Train_17 Train_18 Train_19 Train_20 Train_21 Train_22 Train_23 Train_24 Train_25 Train_26 Train_27 Train_28 Train_29 Train_30 Train_31 Train_32 Train_33 Train_34 Train_35 Train_36 Train_37 Train_38 Train_39 Train_40 Train_41 Train_42 Train_43 Train_44 Train_45 Train_46 Train_47 Train_48 Train_49 Train_50 Train_51 Train_52 Train_53 Train_54 Train_55 Train_56 Train_57 Train_58 Train_59 Train_60 Train_61 Train_62 Train_63 Train_64

30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7

787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 787 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 887 937 937 937 937 937 937 937 937 937 937

2 2 2 3 3 3 4 4 4 2 2 2 3 3 3 4 4 4 2 2 2 3 3 3 4 4 4 2 2 2 3 3 3 4 4 4 2 2 2 3 3 3 4 4 4 2 2 2 3 3 3 4 4 4 2 2 2 3 3 3 4 4 4 2

17.92 17.92 17.92 17.92 17.92 17.92 17.92 17.92 17.92 30.72 30.72 30.72 30.72 30.72 30.72 30.72 30.72 30.72 43.52 43.52 43.52 43.52 43.52 43.52 43.52 43.52 43.52 17.92 17.92 17.92 17.92 17.92 17.92 17.92 17.92 17.92 30.72 30.72 30.72 30.72 30.72 30.72 30.72 30.72 30.72 43.52 43.52 43.52 43.52 43.52 43.52 43.52 43.52 43.52 17.92 17.92 17.92 17.92 17.92 17.92 17.92 17.92 17.92 30.72

99.42 99.19 98.83 98.41 98.51 99.44 98.55 99.23 98.91 98.60 99.63 98.80 99.00 99.08 99.51 98.83 98.40 98.60 98.88 99.25 99.32 99.70 98.66 98.92 99.17 99.40 99.71 99.33 99.12 98.9 98.45 98.49 99.42 98.47 99.31 98.86 98.64 99.62 98.81 98.91 99.02 99.57 98.79 98.47 98.56 98.87 99.17 99.27 99.8 98.67 98.87 99.1 99.33 99.68 99.51 99.21 98.71 98.45 98.69 99.54 98.55 99.38 98.80 98.80

(continued on next page) 16

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Table B.1 (continued) Case

Flow rate of the reactor [kg/s]

Temperature of the reactor [K]

S/C ratio of the reactor

Flow rate of the furnace [kg/s]

Accuracy [%]

Train_65 Train_66 Train_67 Train_68 Train_69 Train_70 Train_71 Train_72 Train_73 Train_74 Train_75 Train_76 Train_77 Train_78 Train_79 Train_80 Train_81 Test_1 Test_2 Test _3 Test _4 Test _5 Test _6 Test _7 Test _8 Test _9 Test _10 Test _11 Test _12 Test _13 Test _14 Test _15 Test _16 Test _17 Test _18 Test _19 Test _20 Test _21 Test _22 Test _23 Test _24 Test _25 Test _26 Test _27 Test _28 Test _29 Test _30 Test _31 Test _32 Average

46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 30.7 46.05 61.4 56.71 53.56 38.38 34.49 37.8 33.19 57.95 52.7 41.71 53.74 55.79 36.67 58.84 52.75 40.66 55.29 60.09 54.66 59.17 56.81 40.57 37.81 46.49 55.02 57.84 47.9 46.06 38.02 58.37 44.38 52.23 44.82

937 937 937 937 937 937 937 937 937 937 937 937 937 937 937 937 937 887.47 824.41 886.15 890.72 863.06 788.66 836.76 827.9 921.9 889.2 822.35 818 794.25 867.47 906.24 913.57 872.56 863.51 931.14 885.82 857.5 928.12 821.17 846.4 840 871.35 922.07 788.83 907.9 934.77 924.41 849.15

2 2 3 3 3 4 4 4 2 2 2 3 3 3 4 4 4 3.34 3.16 2.52 2.75 2.49 3.84 3.08 3.41 3.56 3.48 2.95 3.3 2.35 2.17 3.97 2.88 2.28 2.99 2.63 2.98 2.44 2.83 2.37 3.96 3.54 3.88 3.86 3.9 2.89 2.87 3.4 2.43

30.72 30.72 30.72 30.72 30.72 30.72 30.72 30.72 43.52 43.52 43.52 43.52 43.52 43.52 43.52 43.52 43.52 31.68 23.04 23.68 42.24 20.16 28.48 42.88 22.72 28.48 32.64 25.6 36.48 22.4 33.6 43.2 30.08 29.44 21.12 28.48 41.92 42.88 33.6 29.12 35.84 19.52 38.08 31.68 31.36 33.6 39.36 37.12 27.2

99.73 98.94 98.88 99.12 99.67 98.97 98.63 98.52 98.99 99.31 99.26 99.98 98.78 98.70 99.19 99.39 99.53 98.21 98.61 99.01 97.9 98.76 98.08 98.43 98.59 98.8 99.27 98.69 98.56 99.31 98.48 98.06 98.32 99.3 98.82 98.93 98.92 98.16 98.11 98.32 98.35 98.96 98.49 98.2 98.62 98.1 98.48 98.27 98.88 98.91

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