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Transfer Learning for Efficient Meta-Modeling of Process
Simulations
Yao-Chen Chuang a, Tao Chen b, Yuan Yao a* , David Shan Hill Wong a*
a Department of Chemical Engineering, National Tsing Hua University, Hsinchu
30013, Taiwan, ROC
b Department of Chemical and Process Engineering, University of Surrey, Guildford,
GU2 7XH, UK
ABSTRACT
In chemical engineering applications, computational efficient meta-models have been
successfully implemented in many instants to surrogate the high-fidelity
computational fluid dynamics (CFD) simulators. Nevertheless, substantial simulation
efforts are still required to generate representative training data for building meta-
models. To solve this problem, in this research work an efficient meta-modeling
method is developed based on the concept of transfer learning. First, a base model is
built which roughly mimics the CFD simulator. With the help of this model, the
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feasible operating region of the simulated process is estimated, within which
computer experiments are designed. After that, CFD simulations are run at the
designed points for data collection. A transfer learning step, which is based on the
Bayesian migration technique, is then conducted to build the final meta-model by
integrating the information of the base model with the simulation data. Because of the
incorporation of the base model, only a small number of simulation points are needed
in meta-model training.
KEYWORDS
Meta-model; transfer learning; model migration; computational fluid dynamics
(CFD); chemical processes; Bayesian inference.
* Correspondence information:
Y. Yao: [email protected]
D. S. H. Wong: [email protected]
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INTRODUCTION
Computational fluid dynamics (CFD) is a powerful tool for analyzing fluid flow
and transport phenomena. Its implementation to the chemical processes has been
widely researched and many successful applications 1-3 were reported in the past
decade. A notable advantage of CFD is its capability of high-fidelity modeling of
complex chemical processes involving multi-phase flows, mixing of fluids,
heterogeneous reactions, intricate reactor geometry, etc. Such high-fidelity CFD
models generally require a large number of ordinary differential equations (ODEs)
and partial differential equations (PDEs) to characterize various physical factors and
spatial-temporal variations of the system. Substantial computational resources and
time (hours to days) are needed for even one simulation. Hence the simulation study
becomes a long and arduous task if it has to be performed many times for some
specific applications, e.g., sensitivity analysis 4, 5, model calibration 6, consequence
analysis 7 and optimization 8-11.
Recently, meta-modeling has been introduced as a useful methodology to reduce
the computation demand of CFD simulations 12-16. The main purpose of meta-
modeling is to use a number of computer simulation data to develop surrogate models
(models of model, or meta-models) that predict the system input-output relationship
with very little computational cost. There have been already many developments and
investigations on meta-modeling, and a comprehensive review of meta-model
representation, construction and evolution can be found in a recent survey 17. In
general, the predictive performance of meta-models depends on the choice of how the
computer simulations are designed (i.e. design of experiments, DoE) and what types
of meta-models are adopted. If the training data are sufficiently representative of the
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input space and the meta-model is flexible enough to capture input-output
relationships, the meta-model developed can be an accurate surrogate of CFD
simulations.
Nevertheless, to build meta-models, substantial simulation efforts are inevitably
required to generate representative training data. To address this problem, the concept
of transfer learning 18-20 is adopted here to develop meta-models with a reduced
amount of simulation data. Specifically, the technique of Bayesian model migration 21,
which belongs to the family of parameter-based transfer learning, is employed. Model
migration is an evolutionary approach that allows us to leverage knowledge learned
from a previous process (cast in the form of a base model) in the model development
of a new but “similar” process being investigated. Gao and coworkers 22, 23 recognized
this problem and revealed that model migration is efficient to reduce data requirement
for new process model construction. It should be pointed that process similarity does
exist in many problems in chemical process engineering such as scale-up, product
grade change for various customers with slightly different specifications, etc.
Although the results are quite good, the aforementioned migration studies do not
reveal how such similarity helps in model building or whether a migration from a base
model dissimilar to the investigated process is detrimental. In addition, in previous
research the model migration technique has seldom been utilized for meta-modeling
of CFD simulations.
Furthermore, many real-world engineering problems consist of explicit
constraints on the system inputs and implicit constraints on system outputs. The
explicit constraints define the search space, which can be addressed with existing DoE
methods. However, it is unclear how implicit constraints should be handled. For
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example, in an exothermic reaction system, the pressure, inlet/outlet flow rate, inlet
reactant concentration and cooling system are manipulated to maximize the reactant
conversion as well as preventing the thermal runaway. If an improper operation
condition was implemented to CFD simulations, the runaway situation may cause
numerical error, and even if the simulation converges the results are not very useful in
terms of learning response surface in the feasible region. The challenge is thus how to
generate initial experiments that are as feasible as possible over the complete domain
of interest while minimizing the simulation cost. So far, to the best of our knowledge,
there has been no work that addresses meta-model construction for high-fidelity CFD
model with implicit constraints.
In this paper, a base model and a Bayesian migration scheme is integrated for
efficient meta-modeling of complex chemical process simulations, where the base
model is a computationally efficient model describing a specific problem that is
similar to the high-fidelity CFD one being studied. Generally, such a base model can
be obtained from an existing well-studied model or be developed by fundamental
theories. The aim of using the base model is to give fast and rough prediction of the
high-fidelity CFD simulation, and at the same time use its feasibility information to
assist the computer DoE. In addition, the proposed Bayesian migration scheme
implements a functional scale-bias correction to merge the base model into a flexible
Gaussian process regression (GPR) meta-model, and applies Bayesian inference with
the computer DoE data for meta-model training. Expectedly, the quality of the base
model, as measured by its similarity to the high-fidelity CFD model, will have
significant impact on the number of computer experiments (i.e. expensive CFD
simulations) required for reliable meta-model development. This has been explored
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by using base models that encode the “right” or “wrong” physical mechanisms, and a
base model that does not rely on any physical mechanism. The results show that, as
expected, a base model with good physical basis is beneficial to meta-modelling. It is
also assuring that even if a wrong physical base model is used, the resulting meta-
model is no worse than when no physics is used (i.e. the model purely based on the
computer experimental data). In other words, no negative transfer is observed in the
case studies. Finally, it appears that the correct identification of the search space and
the process constraints is at least as important as the absolute accuracy of the base
model. These observations will be elaborated and discussed subsequently.
The remainder of this paper is organized as follows. In the Methodology section,
the transfer learning based meta-modeling scheme is introduced in detail.
Subsequently, a complex CFD model of a non-isothermal continuous stirred tank
reactor (CSTR) is described in Case Study section, together with several base models
with various forms. In the Results and Discussions section, the effects of choosing
different base models in model migration are studied. In addition, a comparative study
is conducted by comparing the results of migration and a conventional meta-modeling
approach. Finally, the Conclusions section concludes the paper with remarks.
METHODOLOGY
Figure 1 depicts the overall flowchart of the transfer learning based meta-
modeling strategy which integrates two specific parts, i.e., base model setup and
model migration. The base model is used to roughly mimic the input-output
relationship of the high-fidelity CFD simulations as well as help to design the
computer experiments. The model migration step incorporates the base model in a
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flexible GPR structure and applies Bayesian inference to obtain the final meta-model
from the CFD experimental results.
The key advantage of the proposed strategy is that, as long as there is sufficient
similarity between the base model and the CFD model, only a small number of CFD
simulations are required to build a meta-model with high accuracy. In other words, the
computational time to run the CFD experiments is largely reduced. The entire
procedure and the details of different steps are presented in the following subsections,
whereas the effects of the quality of the base model are discussed in the Case Study
section.
Generation of Base Model
Generation of the base model is an important start to the proposed meta-
modeling strategy especially for the CFD problems with implicit constraints. As
mentioned above, the target of the base model is to give fast and rough predictions of
the CFD simulator and at the same time use its feasibility information to guide the
computer DoE. For this purpose, the base model must be computationally efficient
enough to quickly complete the exploration of the entire design space. Meanwhile, the
more similarity there is between the base model and the CFD one, the more valuable
the data obtained from the DoE is (i.e. the higher chance the designed CFD
simulations are feasible). However, a ready-made base model may not be available
especially when the problem described by the CFD simulator is new and complicated.
For this situation, a base model will be developed by simplifying the CFD one via the
following approaches.
Idealize the physical, chemical and material properties. In a high-fidelity CFD
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model, most factors describing the physical, chemical and material properties are
characterized by ordinary differential equations and/or expressed as functions of
temperature, pressure, velocity, etc. To decrease the amount of calculation, these
factors can be set to constant values which approximate the described properties
under certain operating conditions.
Idealize the transport phenomena. A high-fidelity CFD model concerns the
exchange of momentum, mass and energy between the observed systems. To
obtain a simplified base model, ideal assumptions can be adopted to reduce the
transport complexity and eliminate the interactions between different engineering
systems. For example, in a CSTR system the heat transfer between the inner
reactor wall and the cooling jacket can be simplified with a uniform and constant
heat flux. In this way, the fluid dynamics related to the cooling jacket are
ignored.
Simplify model geometry. The model geometry in high-fidelity CFD may be
extremely detailed, and its intricate structure will lead to very complex meshing.
Deleting the tiny edges and faces can improve the mesh quality. Besides, the
mesh size can be reduced by merging the small pieces close to each other into
one chunk. To substantially reduce the computational resource, one can even
simplify a three-dimensional model to two-, one- or non-dimensional model.
Following the above discussions, the base model is developed through the
simplification of the high-fidelity CFD model. There are certainly similarities
between these two models, because they fundamentally describe the same process.
Also, there is a large chance that the feasible design spaces of both models overlap
with each other in some degree. Therefore, in the situation that the feasible region of
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the CFD simulator is unknown, it is reasonable to conduct the CFD experiments in the
feasible region of the base model.
Computer DoE
Space-filling design has been proved as an effective and reliable approach to
allocate sampling points within the design space. The most widely used space-filling
design techniques include Latin hypercube sampling (LHS) 24, Hammersley sequence
sampling (HSS) 25, uniform design (UD) 26, etc. LHS, HSS and UD perform well
when the design space is rectangular. However, these strategies are not designed to
deal with the problem of uniform sampling within a constrained and non-rectangular
design region. To solve this problem, in this paper, a two-step space-filling approach
is adopted. First, the feasible region is estimated based on the base model. As
discussed in the previous subsection, a good base model is computationally
inexpensive. Therefore, it is easy to explore the feasible region by exhaustive search.
In detail, a number of different input values (i.e. operating conditions), which are
determined by applying HHS, are submitted to the base model; and then the
corresponding outputs are compared to the process constraints to determine the
boundary of the feasible region. In the second step, the Fast Flexible Filling (FFF)
design 27 is applied to quickly generate space-filling designs that have the flexibility to
accommodate the non-rectangular design region. For more details about the FFF
design, please refer to the cited reference.
As discussed in the previous subsection, in this research work the computer DoE
is carried out in the feasible design space determined with the base model. Because
the base model is computationally inexpensive, the feasible region can be explored by
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exhaustive search. In detail, a number of different input values (i.e. operating
conditions) are applied to the base model, and the corresponding outputs are
compared to the process constraints to determine the boundary of the feasible region.
After the DoE is complete, CFD experiments are conducted at the designed
operating conditions to generate the training data for the following model migration
step.
Model migration
In model migration, the base model gives a priori predictions of the CFD outputs
at given operating conditions, and the responses of the final meta-model are obtained
by doing a scale-and-bias transformation on the prediction results of the base model.
Given a set of training data {( y1, x1)⋯ ( y N , x N )} that are collected through the
computer experiments as introduced in the previous subsection, the meta-model is
structured as a scale-and-bias correcting function 28 of the base model:
y ( x i)=α ( xi ) zi+ β ( x i ), (1)
where z i are the based model predictions at x i, and i=1⋯N .
The bias adjustment β ( xi ) G (0 ,C ) is chosen as a zero-mean Gaussian process
(GP), where C is an N × N covariance matrix in which the ij-th element is defined by
a covariance function: C ij=C ( x i , x j ). In this work, the following second order
covariance function 29 is used:
C ( x i , x j )=ao+a1∑k=1
d
( x ik−x jk )+νo exp(∑k=1
d
w k ( x ik−x jk )2)+σ2 δij,
(2)
where x ik is the k-th variable of the input vector x i of dimension d , δ ij is the Kronecker
10
delta function, and Θ=[ ao , a1 , νo ,w1⋯wd , σ 2 ]T are known as the “hyper-parameters”
defining the covariance function. The four terms in Eq. (2) account for the effects of
constant bias, linear correlation, non-linear correlation and random noise,
respectively.
The scale correction is chosen to be a linear function:
α ( xi )=α o+∑k=1
d
α k x ik. (3)
Because β (∙ ) is a GP, the resulting meta-model is also a GP with a discrete form:
y= [ y1 ,⋯ yN ]T G ( Γα ,C ), (4)
where
Γ=[ z1
⋮zN
x11 z1
⋮xN 1 zN
⋯⋱⋯
x1 d z1
⋮x Nd zN
], (5)
and α=[ αo , α1⋯α d ].
In order to estimate the parameters, a Bayesian approach 21 is used to integrate
out the regression coefficients α. This is an effective method to fully incorporate the
parameter uncertainty. In particular, an independent prior distribution is assigned to
each element of α : α j G (0 , λ2 ), then
p ( y|λ , Θ )=∫ p ( y|α , Θ ) p ( α|λ )d α (6)
and
y= [ y1 ,⋯ yN ]T G (0 , λ2 ΓΓT+C ). (7)
λ2 and the hyper-parameters Θ can be obtained by maximizing the log-likelihood
function log p ( y|λ , Θ ). This is a non-linear optimization problem that can be solved
by using the gradient based methods, e.g. the conjugate gradient method 29. Finally,
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for a new data point ~x , the predictive distribution of the response conditional on the
training data is also Gaussian, of which the mean y (~x ) and the variance σ 2 (~x ) are
calculated as follows:
y (~x )=k (~x )T (λ2~Γ ΓT +C)−1 y (8)
σ 2 (~x )=C (~x ,~x )−k (~x )T (λ2~Γ ΓT+C)−1 k (~x ) (9)
where k (~x )=[C (~x , x1 )⋯C (~x , x N ) ].
CASE STUDY
CFD Model of a Non-Isothermal Continuous-Stirred Tank Reactor
In this section, a non-isothermal CSTR system is used to illustrate the proposed
method. A full-scale three-dimensional (3-D) model of the system and the associated
system parameters are illustrated in Figure 2 and Table 1, respectively. Here, V is the
volume of the reactor, F is the inlet and outlet volumetric flow rate, C Ai, CBi and CCi
are the inlet concentrations of species A, B and C, respectively, T i is the inlet
temperature, −2000 (kJ/min) ≤ Q ≤0 (kJ/min) is the heat removal rate by the
cooling jacket, S is the stirring speed of impeller, and τ=V / F is the time constant of
the CSTR. Physical properties such as the material density ρ, specific heats CP,
thermal conductivity κ and dynamic viscosity η are assumed to be constants with
respect to temperature and compositions. This system consists of a first-order
sequential reaction, A→B→C, taking place in the CSTR. k A , k B are the pre-
exponential Arrhenius constants, whereas EA, EB are the activation energies. HR A,
HRB are the molar heats of the reactions.
For more details about the CFD model building and the governing equations (e.g.
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continuity, momentum, energy, mass, k-ε turbulence, and rotating machinery) used for
the simulation, please refer to the ANSYS Fluent manual 30, 31 and the related tutorials
32, 33. To ensure the accuracy of the simulation results, the mesh of the CFD model was
validated under a high density condition (622,774 domain elements, 28,715 boundary
elements and 1,792 edge elements). The mesh density under such condition gives the
variation of the predictive concentration and temperature less than 0.001%. The
simulation terminates when all the concentrations and temperature in the domain drop
to less than 10−6, and each simulation run costs around 10 hours to obtain the steady-
state solutions on a desktop computer running Windows 8.1 with an Intel i7-4930K
CPU @3.4GHz and 32GB RAM.
In this study, the objective is to predict the yield of species B (ξ AB=CB /C Ai¿
under a given operation condition x=[C Ai , F ,T ]. Please note that the heat removal
capacity of cooling jacket is constrained in the range of
−2000 (kJ/min) ≤ Q ≤ 0 (kJ/min) . Because of the existence of the constraint, the
feasible operating region is only a subset of the design space, the shape of whose
boundary is unknown and could be irregular.
As mentioned above, the time consumption of each CFD simulation run is quite
long (around 10 hours). As a result, this model cannot be directly used in many
specific engineering applications, such as optimization, sensitivity analysis, etc. To
overcome this problem, model migration should be implemented to build a meta-
model with a small number of computer experiments.
Base Model Setup
In the following, the effect of the quality of the base model on migration is
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investigated. Four different base models are considered here.
Base model 1 captures the essential physics of the process by assuming that the
CSTR is perfectly mixed and isothermal. The following steady-state equations can be
obtained by applying the conservation law to the total energy and mass.
Q=Fρ CP ( T−T i )+V (r1 HR1+r 2 HR2 ), (10)
C A=CAi
1+τ k1 exp(−E1
RT ) , (11)
CB=τ k 1exp(−E1
RT )C Ai
1+τ k 2exp(−E2
RT ), (12)
where
r1=−k1 exp(−E1
RT )CA, (13)
and
r2=k1 exp(−E1
RT )CA−k2 exp(−E2
RT )CB. (14)
Base model 2 is with wrong physics, in which the reactions forming B and C are
assumed to be parallel rather than sequential. Hence we have
C A=C Ai
1+τ (k1 exp(−E1
RT )+k2exp(−E2
RT )) , (15)
CB=τ k1 exp(−E1
RT )CA, (16)
r1=−k A exp(−E2
RT )C A, (17)
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and
r2=−k2 exp(−E2
RT )CA. (18)
Conducting a computer experiment using Base model 1 or 2 costs about 2.49*10-6
seconds, which is significantly faster than a CFD simulation run.
Base model 3 is a model with no physics, i.e. the predictions from this model are
just random numbers. It takes only 7.22*10-8 seconds to generate a random number in
the range of [0 3500], where 0 represents the situation that all species A is converted
to species C; while 3500 corresponds to largest possible value of the yield of B.
Please be noted that in practice one would never migrate form a random number
generator to a serious meta-model. Here, this model is adopted to test the performance
of the proposed method in the extreme case.
Base model 4 is a two-dimensional (2-D) and axisymmetric CFD model
describing the same process. Comparing to the 3-D CFD model, some details of the
process operation are lost in Base model 4. This 2-D CFD model is a more accurate
approximation of the CSTR system than Base model 1. The price is the computational
burden. A simulation experiment using Base model 4 costs 3 hours, which is faster
than that based on the 3-D CFD simulator but much slower than that using Base
model 1 or 2.
Figure 3 illustrates the prediction performance of each base model, which plots
the CFD outputs of CBversus the predicted values. The test data were randomly
generated from the CFD simulations in the feasible operating region, whose sample
size is 100. Obviously, Base model 4 provides most accurate predictions, while Base
model 3 performs worst because it only generates random numbers without any
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physical characteristics. Such conclusions are confirmed by the root mean square
error (RMSE) values annotated in the figure. The problem of finding the boundary of
the feasible region will be discussed in the next section.
RESULTS AND DISCUSSIONS
Feasible Region Estimation
In order to migrate the base model to an accurate meta-model of the CFD
process, it is important to explore the feasible operating region with a small number of
computer experiments. However, the actual boundary of the feasible region is
unknown in prior, in order to determine which a large number of CFD experiments
should be conducted in the design space. Such a solution is impractical in terms of
computing time for realistic instances. As introduced in the subsection of Computer
DoE, an alternative way is to estimate the feasible region by running computer
experiments using the base model. Here, the feasible region can be estimated by using
either Base model 1 or 2. Base model 3 is not useful in such estimation, because its
outputs are purely random and with no physical background. Base model 4 is not able
to be used either because of its computational burden, despite the fact that it is the
most accurate model among all the alternatives.
After estimating the feasible region, the FFF design was implemented to generate
computer experimental data from the CFD model for model migration. Three cases
are considered here for comparison. In the first case, the entire design space was
treated as the feasible region without the help of any base model. The computer DoE
generated 15 points, only 6 of which were feasible as determined by the expensive
CFD simulations. In the second and third cases, Base models 1 and 2 were
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respectively used for feasible region estimation, as shown in Figure 4 and Figure 5.
In these two figures, the estimated feasible operating regions are indicated in colors.
In the estimated feasible region using Base model 1, 14 out of 15 experiments
provided outputs within the actual process constraint. In contrast, 7 out of 15
experimental points designed based on Base model 2 were feasible, only slightly
better than the result in the first case. Obviously, a good base model provides a better
estimation of the boundary of the feasible operating region. As a result, more valid
information can be collected with the same number of computer experiments. In other
words, fewer CFD simulation experiments are needed to be conducted to collect
enough information for model migration. Considering that each CFD simulation run
costs 10 hours, a good base model is critically important for achieving an efficient
migration.
Performance of Migration
In the following, the performance of the meta-models migrated from different
base models are compared by using RMSE as the criterion. For a fair comparison, all
the migrations were put into effect based on the same 3-D CFD experiments which
were designed and conducted in the feasible region estimated using Base model 1.
The results of additional 100 experiments were used as the test dataset to evaluate the
prediction capabilities of the obtained meta-models.
The results are summarized in Table 2 and Figure 5, where the meta-model
without migration is a zero-mean GP 29 with covariance function C parameterized as
in Eq. (2) and trained based on the CFD experimental data only. The results clearly
show that a base model with a sound physical background (i.e. Base model 1 or 4) can
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significantly reduce the number of computer experiments for constructing a meta-
model with reasonable accuracy. Specifically, the meta-models migrated from Base
models 1 and 4 only require 15 DoE data points to achieve better prediction accuracy
than the other three models constructed with a training dataset of size 30. Between
these two meta-models, the former is recommended although the model migrated
from Base model 4 has a better performance in terms of prediction accuracy. The
computational burden of Base model 4 limits its applicability.
Another interesting finding is that the base models with wrong physics or no
physics do not have negative impacts on the obtained meta-models, although they do
not provide any useful information. In Figure 5, the meta-models migrated from Base
models 2 and 3 have almost identical performance with that of the model purely based
on the DoE data. By checking the model parameters, it is found that, when Base
model 2 or 3 is used as a start point of migration, λ2 in Eq. (7) shrinks to 0
automatically, leading to α ( xi ) ≈ 0. As a result, the impact of the base model is
excluded from the final meta-model.
CONCLUSIONS
In this research work, a fast and accurate meta-modeling approach is developed
to surrogate high-fidelity CFD simulations based on the concept of transfer learning.
In the proposed method, the final meta-model is migrated from a base model with the
help of a small number of training data collected during CFD computer experiments.
The issues of base model building, feasible region estimation, and Bayesian migration
are discussed in details. The applicability and effectiveness of the proposed method
are demonstrated through a full-scale CFD model of a CSTR system. The case studies
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show that the influence of a base model with little similarity to the high-fidelity CFD
model can be automatically eliminated during the migration process. In other words, a
low-quality base model does not have negative effects on the performance of the final
meta-model. In the worst case, the model migration procedure results in a meta-model
with similar prediction accuracy to that purely based on the computer experimental
data. However, a high-quality base model can be migrated to an accurate meta-model
with fewer training data points. More importantly, with a high-quality base model, the
feasible operating region defined by implicit process constraints can be identified
more accurately. As a result, more valid CFD experiments are designed in the
following computer DoE step, accelerating the entire procedure. It is noted that, even
using a fairly good base model, there is no guarantee that all DoE data points will be
valid. In such situation, an incremental DoE algorithm is needed to explore the
boundary of the actual feasible region as well as generate the designed points step by
step. This issue will be discussed in the future research work, to keep this paper
within a reasonable length without losing focus.
ACKNOWLEDGMENTS
This work was partially supported by an International Exchange grant co-funded
by the UK Royal Society (Grant number: IE140859) and the Ministry of Science and
Technology, ROC (Grant number: MOST 105-2911-I-007-504).
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Figure 1 The flowchart of the proposed meta-modeling approach
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Figure 2 The schematic diagram and 3-D model of the CSTR process
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Figure 3 Prediction accuracy of each base model
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Figure 4 Estimated feasible operating region by Base model 1
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Figure 5 Estimated feasible operating region by Base model 2
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10 15 20 25 30 35 40 45 50 556
8
10
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14
16
18
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Number of CFD ecperimental data points used in meta-modeling
RMSE
Meta-model migrated from Base model 1Meta-model migrated from Base model 2Meta-model migrated from Base model 3Meta-model migrated from Base model 4Meta-model without migration
Figure 6 Performance comparison between meta-models
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Table 1 System variables and their values used in this study
Parameters Values Units Descriptions
kA 8.4105 1/min Physical constant
kB 7.6104 1/min Physical constant
HRA -2.12104 J/mol Physical constant
HRB -6.36104 J/mol Physical constant
EA 3.64104 J/mol Physical constant
EB 3.46104 J/mol Physical constant
ρ 1180 kg/m3 Physical constant
CP 3.2103 J/kg/K Physical constant
κ 0.61 W/m/K Physical constant
η 0.0008 Pas Physical constant
R 8.314 J/mol/K Physical constant
Ti 300 K Physical constant
V 0.004 m3 Physical constant
S 120 rpm Physical constant
CAi 2000~3500 mol/m3 Design variable
F 0.001~0.1 m3/min Design variable
Q -2000~0 kJ/min Implicit constraint
T 300~370 K Design variable
CA mol/m3 System output
CB mol/m3 System output
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Table 2 Comparison of model prediction performance
ModelsNumber of CFD experimental data points used
15 20 30 50
Meta-Models
Migrated From
Base Model 1 19.18 14.06 9.28 7.80
Base Model 2 24.39 21.75 19.92 13.07
Base Model 3 24.39 21.75 19.92 13.07
Base Model 4 18.44 10.33 8.21 7.47
Meta-Model without Migration 24.39 21.75 19.92 13.07
31