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

1

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: yyao@mx.nthu.edu.tw

D. S. H. Wong: dshwong@che.nthu.edu.tw

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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

13

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)

14

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).

REFERENCE

1. Chen C-T, Tan W-L. Mathematical modeling, optimal design and control of an SCR

reactor for NOx removal. Journal of the Taiwan Institute of Chemical

19

Engineers. 2012;43:409-419.

2. Chuang Y-C, Chen C-T. Mathematical modeling and optimal design of an MOCVD

reactor for GaAs film growth. Journal of the Taiwan Institute of Chemical

Engineers. 2014;45:254-267.

3. Pan H, Chen X-Z, Liang X-F, Zhu L-T, Luo Z-H. CFD simulations of gas–liquid–

solid flow in fluidized bed reactors — A review. Powder Technology.

2016;299:235-258.

4. Kajero OT, Thorpe RB, Yao Y, Hill Wong DS, Chen T. Meta-Model-Based

Calibration and Sensitivity Studies of Computational Fluid Dynamics

Simulation of Jet Pumps. Chemical Engineering & Technology. 2017;40:1674-

1684.

5. Razavi S, Tolson BA, Burn DH. Numerical assessment of metamodelling strategies

in computationally intensive optimization. Environmental Modelling &

Software. 2012;34:67-86.

6. Kajero OT, Thorpe RB, Chen T, Wang B, Yao Y. Kriging meta-model assisted

calibration of computational fluid dynamics models. AIChE Journal.

2016;62:4308-4320.

7. Loy YY, Rangaiah GP, Lakshminarayanan S. Surrogate modelling for enhancing

consequence analysis based on computational fluid dynamics. Journal of Loss

Prevention in the Process Industries. 2017;48:173-185.

8. Chen C-T, Chen M-H, Horng W-T. Reliability-based design optimization of pin-fin

heat sinks using a cell evolution method. International Journal of Heat and

Mass Transfer. 2014;79:450-467.

9. Chuang Y-C, Chen C-T, Hwang C. A simple and efficient real-coded genetic

20

algorithm for constrained optimization. Applied Soft Computing. 2016;38:87-

105.

10. Manfren M, Aste N, Moshksar R. Calibration and uncertainty analysis for

computer models – A meta-model based approach for integrated building

energy simulation. Applied Energy. 2013;103:627-641.

11. Boukouvala F, Ierapetritou MG. Derivative-free optimization for expensive

constrained problems using a novel expected improvement objective function.

AIChE Journal. 2014;60:2462-2474.

12. Wang GG, Shan S. Review of Metamodeling Techniques in Support of

Engineering Design Optimization. Journal of Mechanical Design.

2006;129:370-380.

13. Wang K, Chen T, Kwa ST, Ma Y, Lau R. Meta-modelling for fast analysis of

CFD-simulated vapour cloud dispersion processes. Computers & Chemical

Engineering. 2014;69:89-97.

14. Jia G, Taflanidis AA, Nadal-Caraballo NC, Melby JA, Kennedy AB, Smith JM.

Surrogate modeling for peak or time-dependent storm surge prediction over an

extended coastal region using an existing database of synthetic storms.

Natural Hazards. 2016;81:909-938.

15. Jiang C, Soh YC, Li H, Masood MK, Wei Z, Zhou X, Zhai D. CFD results

calibration from sparse sensor observations with a case study for indoor

thermal map. Building and Environment. 2017;117:166-177.

16. Klingenberger M, Hirsch O, Votsmeier M. Efficient interpolation of precomputed

kinetic data employing reduced multivariate Hermite Splines. Computers &

Chemical Engineering. 2017;98:21-30.

21

17. Kajero OT, Chen T, Yao Y, Chuang Y-C, Wong DSH. Meta-modelling in chemical

process system engineering. Journal of the Taiwan Institute of Chemical

Engineers. 2017;73:135-145.

18. Pan SJ, Yang Q. A survey on transfer learning. IEEE Transactions on knowledge

and data engineering. 2010;22:1345-1359.

19. Weiss K, Khoshgoftaar TM, Wang D. A survey of transfer learning. Journal of Big

Data. 2016;3:9.

20. Tsung F, Zhang K, Cheng L, Song Z. Statistical transfer learning: A review and

some extensions to statistical process control. Quality Engineering. 2017:DOI:

10.1080/08982112.2017.1373810.

21. Yan W, Hu S, Yang Y, Gao F, Chen T. Bayesian migration of Gaussian process

regression for rapid process modeling and optimization. Chemical

Engineering Journal. 2011;166:1095-1103.

22. Lu J, Gao F. Process modeling based on process similarity. Industrial &

Engineering Chemistry Research. 2008;47:1967-1974.

23. Lu J, Yao Y, Gao F. Model migration for development of a new process model.

Industrial & Engineering Chemistry Research. 2009;48:9603-9610.

24. McKay MD, Beckman RJ, Conover WJ. A Comparison of Three Methods for

Selecting Values of Input Variables in the Analysis of Output From a

Computer Code. Technometrics. 2000;42:55-61.

25. Kalagnanam JR, Diwekar UM. An Efficient Sampling Technique for Off-line

Quality Control. Technometrics. 1997;39:308-319.

26. Fang K-T, Dennis KJL, Winker P, Zhang Y. Uniform Design: Theory and

Application. Technometrics. 2000;42:237-248.

22

27. Lekivetz R, Jones B. Fast Flexible Space-Filling Designs for Nonrectangular

Regions. Quality and Reliability Engineering International. 2015;31:829-837.

28. Qian Z, Seepersad CC, Joseph VR, Allen JK, Jeff Wu CF. Building Surrogate

Models Based on Detailed and Approximate Simulations. Journal of

Mechanical Design. 2005;128:668-677.

29. Rasmussen CE, Williams C. Gaussian processes for machine learning.

Cambridge, MA, USA: The MIT Press, 2006.

30. ANSYS FLUENT 17.0 - User´s guide: ANSYS Inc., 2016.

31. ANSYS FLUENT 17.0 - Theory guide: ANSYS Inc., 2016.

32. ANSYS FLUENT 17.0 - Tutorial guide: ANSYS Inc., 2016.

33. Bakker A. Reacting Flow Lectures

(http://www.bakker.org/dartmouth06/engs199/), 2006.

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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

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10

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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