2012 AIChE Meeting Pittsburgh PA2012 AIChE Meeting, Pittsburgh, PA
Enforcing Elemental Mass and Energy Balances for Reduced Order Models
Jinliang Ma 1, 4, Christopher Montgomery 1, 4, Khushbu Agarwal 2, g , p g y , g ,Poorva Sharma 2, Yidong Lang 5, David Huckaby1 ,
Stephen Zitney 1, Ian Gorton 2, Deb Agawal 3, David Miller 1
1U S DOE/National Energy Technology Laboratory Morgantown WV 26507U.S. DOE/National Energy Technology Laboratory, Morgantown, WV 265072U.S. DOE/Pacific Northwest National Laboratory, Richland, WA 993523U.S. DOE/Lawrence Berkeley National Laboratory, Berkeley, CA 947204URS Corporation, Morgantown, WV 265055Carnegie Mellon University Pittsburgh PA 152135Carnegie Mellon University, Pittsburgh, PA 15213
This technical effort was performed in support of DOE’s Carbon Capture Simulation Initiative(CCSI) project under the RES contract RES0004000.2.600.232.001
IntroductionMulti-Scale Models in Carbon Capture Simulation Initiative (CCSI)Multi-Scale Models in Carbon Capture Simulation Initiative (CCSI)
https:www.acceleratecarboncapture.orgPlant Scale
• Process (Aspen Plus, gPROMS)
Device Scale
( p g )
Device Scale• 1-D Reactor• Computational Fluid Dynamics
Scal
e
Single Particle Scale• Diffusion (Film, Pore)• Surface Adsorption/Reactions
S
Power Plant Process Model
Molecule Scale• Molecular Dynamics
C t ti l Ch i t
ROM
2
• Computational ChemistryCO2 AdsorberModel (CFD)
High-Fidelity Model Versus ROM Hi h Fid lit M d l High-Fidelity Model
• e.g. CFD, Ideal Reactors (Equilibrium, Plug Flow, CSTR)• Based on first principles• Usually conserves mass/energy• Usually conserves mass/energy
If converged tightly• Slow (CPU Intensive)
Reduced Order Model (ROM)( )• Based on mathematical regression/interpolation
Kriging Artificial Neural Network (ANN) Others
• Not necessarily conserves mass/energy• Possible unrealistic predictions (negative species mass flow rates)• Fast• Fast
ROM as a Bridge Between Multiple Scales• Needs tight mass/energy balances
Important for recycles
3
Important for recycles
REVEAL: CCSI’s ROM Generation SoftwareIn a form of unit operation model for PME (e g Aspen Plus ACM gPROMS)In a form of unit operation model for PME (e.g. Aspen Plus, ACM, gPROMS)
REVEAL
4
Enforcing Mass Balance
iN1
tN1
inm
in
in
N
NN
,2
,1
outn
out
out
N
NN
,2
,1
Species molar flow rate
FeedStreams
ProductStreams
inm, outn,
For each mole of species i, there are Ai,j moles of element j (j=1,2,…,l)e.g., for CH4, A1,1 =4, A1,2 =1, A1,3 =0, A1,4 =0
m
n
For element j:
Mass balance:
i
jiiniinj ANL1
,,,
i
jioutioutj ANL1
,,,
nm
ANAN (j 1 2 l)
5
Mass balance:
i
jioutii
jiini ANAN1
,,1
,, (j=1,2,…,l)
Correction Factors For Product Species
For product species i, Define:
Corrected Prediction
ROM Prediction
1,
,
,
,,
ROMouti
outiROMouti
ROMoutiouti
i NN
NNN
f
Corrected molar flow of species i: ROMoutiiouti NfN ,, 1
n
ROMm
Eqn. for solving fi:
Let
i
jiROMoutii
ijiini ANfAN
1,,
1,, 1
m
i
n
iji
ROMoutijiinij ANANL
1 1,,,,
(Mass imbalance)
Number of equations: l (one for each element)
(Based on element j)Then
i i1 1
01
,,
j
n
iji
ROMoutii LANf
Number of equations: l (one for each element)Number of unknowns: n (one for each product species)Notes: 1.Total mass will be balanced if individual elements are balanced.
2. if , set it to a small positive number. Use 0, ROMoutiN ROM
outiN ,01.0
6
Solving Correction FactorsScenario 1: n>lScenario 1: n>l
Approach: Find most reasonable correction factors by minimizingwhile enforcing mass balance for each element
n
iif
1
2
Algorithm: Lagrangian multiplier method (mass balance equations as constraints)
Lagrangian Function G:Lagrangian Function G:
Partial Derivatives of G:
l
j
n
ijji
ROMoutiij
n
iiln LANfffffG
1 1,,
1
22121 ,,,,,,,
Partial Derivatives of G:
l
jjji
ROMoutii
i
ANffG
1,, 02 ni ,,2,1
Total number of equations: n+l
n
ijiji
ROMouti
j
LfANG1
,, 0
lj ,,2,1
7
Total number of unknowns: n+l
Solving Correction FactorsScenario 2: n<lScenario 2: n<l
A h Fi d b t fit f ti f t b l t l ti
Example: CO2 and H2O as products (2 species, 3 elements)
Approach: Find best fit for correction factors by least square solution
ljLfAN j
n
iiji
ROMouti ,,2,1
1,,
bfM
(Matrix is )nl M
Algorithm: Minimize quadratic
i 1
Solve: (Product matrix is )
2bfM
bMfMM TT
nn MM T
(Linear Least Square Method)
Total number of equations: nTotal number of unknowns: n
Solve: ( ) bMfMM
Note: Applicable to non-reacting devices
8
Enforcing Energy BalanceW
inin
inin
ThmThm,,,,
2,2,2
1,1,1
outout
outout
ThmThm,,,,
222
1,1,1
StSt
Port 1Port 2
Port 1Port 2
outW
pinpinp Thm ,, ,,
,,
qoutqoutq
outout
Thm ,,
,,
,,
2,2,2
Stream enthalpy
Stream mass flow rate
Port p Port q
FeedStreams
ProductStreams
s To TdTChyThStream Enthalpy: (If ideal gas mixture)
lossQ
kT kpkfkmix TdTChyTh
1,,
0Stream Enthalpy: (If ideal gas mixture)
Enthalpy Rate In: p
iiniiniin hmH
1,,
qp
QWhh E B l
Enthalpy Rate Out:i 1
q
ioutioutiout hmH
1,,
9
lossouti
outioutii
iniini QWhmhm 1
,,1
,,Energy Balance:
Enforcing Energy BalanceO ti 1 Adj ti h t lOption 1: Adjusting heat loss
ROMq
i
ROMouti
p
iiniiniloss outouti
WhmhmQ 1
,1
,, ,
Option 2: Adjusting product stream enthalpy/temperature
ROMloss
ROMoutinout QWHH Total Enthalpy Rate
of Products:
Total Enthalpy Rate Correction:
ROMout
ROMloss
ROMoutin
ROMoutoutout HQWHHHH
tim
E th l C ti P
outq
joutj
outiouti H
m
mH
1
,
,,Enthalpy Rate
Correction for Port i:
HEnthalpy Correction Per Unit Mass for Port i :
q
joutj
outouti
m
Hh
1,
,
Solve Temperature Ti iT
10
Solve Temperature Tifor Port i : TdTCh i
ROMi
T
T ipouti ,, (for product port i)
Implementations
CAPE-OPEN Unit Operation Model• Aspen Plus, gPROMS, COFE
Generation of Vendor Specific Source Code (Custom Model) Generation of Vendor Specific Source Code (Custom Model)• Aspen Custom Modeler (Equation-Oriented)• gPROMS (Equation-Based)
A Pl th h CAPE OPEN C C
11
Aspen Plus through CAPE-OPEN ACM through Custom Model
Example: Equilibrium Flow Reactor
CH4+Air→Products• Const p Adiabatic
kg/s1
K300
4
4
CH
CH
m
T
)8,1(for??
imT
i
exhaust
atm1p Model Predictions
Const p, Adiabatic• Reactants: CH4, O2, N2 (m=3)• Products: CH4, O2, N2, H2, H2O,
CO, CO2, NO (n=8) kg/s8.20,9.13
K420,280
i
air
mT
19.3
20.0
20.8
g/s)
, 2, ( )• Elements: C, H, O, N (l=4)• High-Fidelity Model: Aspen Plus
Latin Hypercube Sampling (LHS)
kg/s8.20,9.13airm
109
8
16.2
17.0
17.7
18.5
Flow
Rate (kgyp p g ( )
• 10 samples• Two input variables
Air Temperature airT
airm
76
54
13.9
14.6
15.4
280 296 311 327 342 358 373 389 404 420
Air p
Air Mass Flow Regression Method
• Kriging
32
1
12
Air Temperature (K)• ANN
Example: Equilibrium Flow ReactorResponse Surface (Kriging)Response Surface (Kriging)
K280airT
0 6
0.7
0.8
0.9
st (kg/s)
0.3
0.4
0.5
0.6
e of O
2in Exhau ROM (Uncorrected)
ROM (Corrected)
Aspen Plus
‐0.1
0
0.1
0.2
Flow
Rate
Flow rate of O2 in exhaust versus air temperature and air flow rate
Flow rate of O2 in exhaust versus air flow rate (280 K air temperature)
13 15 17 19 21
Air Flow Rate (kg/s)
13
air temperature and air flow rate air flow rate (280 K air temperature)
Example: Equilibrium Flow ReactorResponse Surface (Kriging)Response Surface (Kriging)
K280airT
0 6
0.7
0.8
0.9
ust (kg/s)
0.3
0.4
0.5
0.6
e of CO in
Exhau
ROM (Uncorrected)
ROM (Corrected)
Aspen Plus
‐0.1
0
0.1
0.2
Flow
Rat
Flow rate of CO in exhaust versus air temperature and air flow rate
Flow rate of CO in exhaust versus air flow rate (280 K air temperature)
13 15 17 19 21
Air Flow Rate (kg/s)
14
air temperature and air flow rate air flow rate (280 K air temperature)
Example: Equilibrium Flow ReactorResponse Surface (Kriging)Response Surface (Kriging)
K280airT
2200
2250
e (K)
2100
2150
ust Tem
peratur
ROM (Uncorrected)
ROM (Corrected)
2000
2050Exha
u
Aspen Plus
13 15 17 19 21
Air Flow Rate (kg/s)
Exhaust temperature versus air temperature and air flow rate
Exhaust temperature versus air flow rate (280 K air temperature)
15
temperature and air flow rate flow rate (280 K air temperature)
Conclusions
Enforcing elemental mass balance for ROM• Enforcing positive species flow rate
L i M lti li M th d (# f d t i # f l t )• Lagrangian Multiplier Method (# of product species > # of elements)• Least Square Method (otherwise)
Enforcing energy balance• Adjust heat loss• Adjust product enthalpy/temperature
Implementations Implementations• CAPE-OPEN unit operation model• Custom model in ACM and gPROMS languages
Corrected ROM predictions are usually closer to Corrected ROM predictions are usually closer to high-fidelity model predictions• Especially in regions with negative product flows predicted by ROM
16
Disclaimer
This presentation was prepared as an account of worksponsored by an agency of the United States Government.N ith th U it d St t G t th fNeither the United States Government nor any agency thereof,nor any of their employees, makes any warranty, express orimplied, or assumes any legal liability or responsibility for theaccuracy completeness or usefulness of any informationaccuracy, completeness, or usefulness of any information,apparatus, product, or process disclosed, or represents that itsuse would not infringe privately owned rights. Reference hereinto any specific commercial product process or service byto any specific commercial product, process, or service bytrade name, trademark, manufacturer, or otherwise does notnecessarily constitute or imply its endorsement,recommendation, or favoring by the United States Government, g yor any agency thereof. The views and opinions of authorsexpressed herein do not necessarily state or reflect those ofthe United States Government or any agency thereof.
17