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Bayesian Model Averaging for Estimating theTemperature Distribution in a Steam Methane
Reforming Furnace
Anh Tran1, Madeleine Pont1, Andres Aguirre1, Helen Durand1,Marquis Crose1 and Panagiotis D. Christofides1,2
1 Department of Chemical & Biomolecular Engineering, University of California, LosAngeles
2Department of Electrical Engineering, University of California, Los Angeles
Inaugural Data Science Workshop
August 7, 2018
Anh Tran Inaugural Data Science Workshop August 7, 2018 1 / 20
Uses and Applications of Hydrogen Gas
Hydrogen is one of the most important raw materials for thepetroleum refinery industry (Gupta, CRC Press, 2008)
Olefins +H2 (g) −→ ParaffinsR1−H2C−CH2−R2 (g)+H2 (g) −→ R1−H2CH (g)+HCH2−R2 (g)R − SH (g) + H2 (g) −→ R (g) + H2S (g)
Hydrogen is a precursor for many chemical industries, e.g.,ammonia production
3H2 (g) + N2 (g)∆H≪0−−−−→ NH3 (g)
Hydrogen is a carrier gas for the production of thin film solarcells (Crose et al., Chem. Eng. Science , 2015)
e− + H2 (g) −→ e− + 2H·
H· + SiH4 (g) −→ H2 (g) + (SiH3)·
Hydrogen is an efficient energy carrier for hydrogen-basedtechnologies (e.g. fuel cells)
Anh Tran Inaugural Data Science Workshop August 7, 2018 2 / 20
General Information of Hydrogen Production
In industry, hydrogen is produced by
Steam methane reforming (SMR) process, which accounts for 48%of world-wide hydrogen production (Ewan and Allen, Int. J. Hydrogen Energy, 2005)
CH4(g) + H2O(g) ↼−−−−−−−−⇁∆H≫0NiAl2O3
CO(g) + CO2(g) + H2(g) (1)
Top-fired reformer
Anh Tran Inaugural Data Science Workshop August 7, 2018 3 / 20
Industrial-scale Steam Methane Reformer
GeometryLength: 16 mWidth: 16 mHeight: 13 m
Components336 reforming tubes96 burners8 flue-gas tunnels
Daily hydrogen production of2.8×106 Nm3
Daily superheated steamproduction of 1.7×106 kg
Annual operating cost of $62×106
Anh Tran Inaugural Data Science Workshop August 7, 2018 4 / 20
Industrial-scale Steam Methane Reformer Mesh
The industrial-scale reformer mesh consists of 41 million grids
Mesh information
The reformer meshRecommendedrange
Min orthogonal factor 0.459 0.167 − 1.000
Max ortho skew 0.541 0.000 − 0.850
Anh Tran Inaugural Data Science Workshop August 7, 2018 5 / 20
Industrial-scale Reformer CFD Model
Modeling turbulent flowsStandard k − ǫ model with ANSYS enhanced wall treatment
Modeling the combustion phenomenaPremixed combustion assumptionGlobal kinetic model of CH4 combustion (D. G. Nicol, PhD Thesis, 1995)
Global kinetic model of H2 combustion (Bane et al., Technical Report, 2010)
FR/ED turbulence-chemistry interaction modelModeling thermal radiation
Empirical model for radiative properties (A. Maximov, PhD Thesis, 2012)
Beer’s law and Kirchoff’s lawDiscrete ordinate method
Anh Tran Inaugural Data Science Workshop August 7, 2018 6 / 20
Industrial-scale Reformer CFD Model
Modeling turbulent flowsStandard k − ǫ model with ANSYS enhanced wall treatment
Modeling the catalyst network of each reforming tubeA continuum approach using ANSYS porous zone functionEffectiveness factor and catalyst packing factor
Modeling the tube wall of each reforming tubeANSYS thin wall function
Modeling the SMR processGlobal kinetic model (J. Xu and G. F. Froment, AIChE Journal, 1989)
Anh Tran Inaugural Data Science Workshop August 7, 2018 7 / 20
CFD Model Validation with Plant Data
Simulation data generated by the reformer CFD model is in goodagreement with the data provided by industry
Reformer CFD model Industry Deviation
Fired duty (kW) 209474.8 211597.3 1.0 %
Total absorbedheat (kW)
113895.5 112246.2 1.5%
Fraction ofabsorbed heat (%)
54.4 53.1 2.4%
OD averageheat flux (kW/m2)
59.2 58 2.1%
ID averageheat flux (kW/m2)
69.5 75.7 8.2%
Average outletflue gas temp (K)
1243.1 1283 3.1%
x̄outletH2
46.5 46.8 0.6 %
Anh Tran Inaugural Data Science Workshop August 7, 2018 8 / 20
Motivations for Data-driven Modeling
The reformer service life is monitored by the system of infraredcameras that periodically record the outer wall temperatures(OTWTs) of the reforming tubes in real-time
Feedback from our third-party collaborator and publicly availableliterature suggest that OTWTs can be controlled by the total fuelflow rate and its spatial distribution inside the reformer
We developed an integrated model identification procedure todiscover the dependence of the OTWT distribution on the reformerinput using
Bayesian variable selection
Sparse nonlinear regressionBayesian model averaging
Theories of thermal radiation
Reformer geometry
Anh Tran Inaugural Data Science Workshop August 7, 2018 9 / 20
Data-driven Model for the ith OTWT
The data-driven model for therelationship between the i th
OTWT at a fixed height and thefurnace-side feed (FSF)distribution is formulated asfollows,
T̂ P,ni =
Ki∑
k=1
wPi,k T̃ P,n
i,k (2a)
where
Ki∑
k=1
wPi,k = 1 (2b)
T̃ P,ni,k =
G∑
g=1
(#»α
kgi
)T· fg
(#»
F n)+ αk
i (2c)
#»
F n =[F n
1 ,Fn2 , · · · ,F
n96
]T (2d)∥∥∥
#»
F n∥∥∥
1= F n
tot (2e)
T̂ P,ni is the BMA estimate of the ith
OTWT
T̃ P,ni,k is the estimate of the ith OTWT
based on the kth sub-model for theith OTWT
fg(
#»
F n)
is the gth basis function in
the library of transformation functions
Anh Tran Inaugural Data Science Workshop August 7, 2018 10 / 20
k th Sub-model for the ith OTWT (i.e., Mi,k )
T̃ P,ni ,k =
G∑
g=1
(#»α
kgi
)T· fg
(#»
F n)+ αk
i (3)
where
f1(
#»
F n)=
[F n
1 ,Fn2 , · · · ,F
n96
]T
f2(
#»
F n)=
[(F n
1
)2,(F n
2
)2, · · · ,
(F n
96
)2]T
f3(
#»
F n)=
[(F n
1
)3,(F n
2
)3, · · · ,
(F n
96
)3]T
f4(
#»
F n)=
[2√
F n1 ,
2√
F n2 , · · · ,
2√
F n96
]T
f5(
#»
F n)=
[3√
F n1 ,
3√
F n2 , · · · ,
3√
F n96
]T
f6(
#»
F n)=
[4√
F n1 ,
4√
F n2 , · · · ,
4√
F n96
]T
f7(
#»
F n)=
[5√
F n1 ,
5√
F n2 , · · · ,
5√
F n96
]T
f8(
#»
F n)=
[exp
(F n
1
), exp
(F n
2
), · · · , exp
(F n
96
)]T
G = 8 is the number of functions in the library of transformation functions
#»αkgi ∈ IR96×1 is the empirical vector of Mi,k corresponding to fg (·)
αki ∈ [298.15, 348.15] is the estimated ambient temperature of Mi,k
Anh Tran Inaugural Data Science Workshop August 7, 2018 11 / 20
Spare Nonlinear Regression with MLE
The formulation for the sparse nonlinear regression with MLE isproposed as follows
minαk
i ∈[298.15,348.15]α
kgij ∈[0,∞)
N∑
n=1
(T n
i − T̃ P,ni,k
)2
2(σn
i
)2 + λi
8∑
g=1
∥∥∥ #»αkgi
∥∥∥1
(6)
subject to8∑
g=1
αkgil fg
(F
0)=
8∑
g=1
αkgij fg
(F
0)
if dil = dij (7a)
8∑
g=1
αkgil fg
(F
0)≥
(dij
dil
)βl 8∑
g=1
αkgij fg
(F
0)
if dil < dij (7b)
8∑
g=1
αkgil fg
(F
0)≤
(dij
dil
)βu 8∑
g=1
αkgij fg
(F
0)
if dil < dij (7c)
F0=
F typtot
96(7d)
Anh Tran Inaugural Data Science Workshop August 7, 2018 12 / 20
Spare Nonlinear Regression with MLE
Regressors of the ith OTWT are defined asthe burners that directly control the ith OTWT
T̃ P,ni ,k =
G∑
g=1
(#̂»α
kgi
)T· fg
(#»
F n∣∣∣SiR
)+ α̂k
i (8)
SiR ∈ IRj×1 is the given set of regressors
#̂»αkgi is the MLE of #»α
kgi ∈ IRj×1
α̂ki is the MLE of αk
i ∈ IR
#»
F n∣∣∣SiR
∈ IRj×1 is the design matrix
Anh Tran Inaugural Data Science Workshop August 7, 2018 13 / 20
Model Identification Flow Diagram
Anh Tran Inaugural Data Science Workshop August 7, 2018 14 / 20
Simulation Inputs
21 reformer CFD data sets with varying FSF distributions and totalFSF flow rates are availableReformer CFD data sets are partitioned into two categories
A reformer CFD training data consists of 18 data setsA reformer CFD testing data consists of 3 data sets
Sλ = {0.1,0.2, · · · ,1.0,1.2, · · · ,2.0,5.0,10,20}, which controlsthe model complexity and goodness of fit
Small values of λi result in a low degree of shrinkage and favoroverfitting data-driven models with high goodness of fitLarge values of λi result in a high degree of shrinkage and favorunderfitting data-driven models with low levels of complexity
Leave-one-out cross validation is used to find the best λi
The 77th reforming tube is chosen as a representative examplebecause the number of sub-models with high goodness of fit (i.e.,4) and the number of predictors (i.e., 9) for the 77th OTWT
Anh Tran Inaugural Data Science Workshop August 7, 2018 15 / 20
Results
0.1 1 10LASSO parameter (λ77)
2
3
4
Mea
n sq
uare
d er
ror
(K2 ) Prediction error
Fitting error
The smallest fitting error correspondsto the smallest value of λ=0.1
The largest fitting error correspondsto the largest value of λ=10
The smallest prediction errorcorresponds to λ̂77=0.5
0 5 10 15 20Index of reformer CFD simulation
1100
1150
1200
1250
77th O
TW
T (
K)
CFD dataTraining dataForecasting data
The model for the 77th OTWT
T̂ P,n77 =0.01T̃ P,n
77,1 + 0.23T̃ P,n77,2
+ 0.29T̃ P,n77,3 + 0.47T̃ P,n
77,4
(9)
Anh Tran Inaugural Data Science Workshop August 7, 2018 16 / 20
Results
The data-driven model for the OTWT distribution correctly identifies thehot and cold regionsThe absolute maximum and average deviations from the reformer CFDdata are 20.1 K and 2.9 K, respectively
Anh Tran Inaugural Data Science Workshop August 7, 2018 17 / 20
Flowchart of The Furnace-balancing Scheme
The furnace-balancing scheme isdeveloped based on
The high-fidelity reformer CFD model(Tran et al., Chem. Eng. Sci., 2017)
The statistical-based modelidentification (Tran et al., Chem. Eng. Res. Des., in press)
The furnace-balancing optimizer (Tran et al.,
Comp. & Chem. Eng., 2017)
0 2 4 6 8 10 12Distance down the reforiming tube (m)
800
900
1000
1100
1200
1300
Tem
pera
ture
(K
)
Desgin outer wall temperatureMaximum outer wall temperatureAverage outer wall temperatureMinimum outer wall temperature
0 2 4 6 8 10 12Distance down the reforiming tube (m)
800
900
1000
1100
1200
1300
Tem
pera
ture
(K
)
Desgin outer wall temperatureMaximum outer wall temperatureAverage outer wall temperatureMinimum outer wall temperature
Anh Tran Inaugural Data Science Workshop August 7, 2018 18 / 20
Results
This result is obtained within a minute
The total FSF flow rate is increased by 40% from 98.113 to136.896 kg sec−1 without damaging the reforming tubes indicatedby the evidence that the maximum value in the OTWT distributionis 1288.35 K
0 1 2 3 4 5Iteration of the Heuristic Search
100
120
140
Tot
al F
uel F
low
Rat
e (k
g s-1
)
0 2 4 6 8 10 12Distance from the Ceiling (m)
900
1000
1100
1200
1300
OT
WT
(K
)T
maxwall
Tave
wall
Tmin
wall
Anh Tran Inaugural Data Science Workshop August 7, 2018 19 / 20
Conclusion
The integrated model identification procedure is structured to befully distributed, which allows the data-driven model for 336reforming tubes to be derived simultaneously from the trainingdata and independently from one another
Leave-out-one cross validation is successfully implemented to findthe optimal LASSO parameter for each reforming tube
The results from the goodness-of-fit and out-of-sample predictiontests of the data-driven model for the OTWT distributiondemonstrated the high effectiveness of the method proposed inthis work
AcknowledgmentsFinancial support from the Department of Energy is gratefullyacknowledged
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