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Applied Energy

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Dual mixed refrigerant LNG process: Uncertainty quantification anddimensional reduction sensitivity analysisMuhammad Abdul Qyyuma,1, Pham Luu Trung Duongb,1, Le Quang Minhc,1, Sanggyu Leed,Moonyong Leea,⁎

a School of Chemical Engineering, Yeungnam University, Gyeongsan 712-749, Republic of Koreab Engineering Product Development, Singapore University of Technology and Design, Singapore 487372, Singaporec Department of Chemical and Biomedical Engineering, West Virginia University, Morgantown, WV 26506, USAdGas Plant R&D Center, Korea Gas Corporation, Incheon 406-130, Republic of Korea

H I G H L I G H T S

• Uncertainty quantification and sensi-tivity analysis for LNG process.

• Standard Monte Carlo (MC) method isutilized.

• Relative percentage of the Sobol totaleffect indices for DMR LNG process.

• Probability distribution of the ap-proach temperature for DMR lique-faction process.

• Global sensitivity analysis with lesscomputational effort.

G R A P H I C A L A B S T R A C T

LNG-DMR process

MITA 1 (oC)

-25 -20 -15 -10 -5 0 5

Den

sity

func

tion

f

0.0

0.2

0.4

0.6

0.8

1.0

Specific energy (kW)0 5 10 15 20 25 30

Den

sity

func

tion

f

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

MITA 2 (oC)

0 1 2 3 4 5

Den

sity

func

tion

f

0.0

0.2

0.4

0.6

0.8

Uncertainty quantification and global sensitivity analysis

Uncertainty quantification

Global sensitivity analysis

A R T I C L E I N F O

Keywords:DMR natural liquefaction processUncertainty quantificationSensitivity analysisMonte CarloMultiplicative dimensional reduction method

A B S T R A C T

The dual mixed refrigerant (DMR) liquefaction process is complicated and sensitive compared to the competitivepropane pre-cooled mixed refrigerant liquefied natural gas (LNG) process. When any uncertainty is introduced tothe process operation conditions, it is necessary for the DMR process to be re-optimized to maintain efficientoperation at a minimal cost. However, in actual operation, re-optimization is a challenging task when the processoperational input variables are varied, typically owing to the lack of information regarding the nature, impact, andlevels of uncertainty. Within this context, this study investigates the uncertainty levels in the overall energy con-sumption and minimum internal temperature approach (MITA) inside LNG heat exchangers with variations in theoperational variables of the DMR processes. Moreover, a global sensitivity analysis is conducted to identify theinfluence of random inputs on the process performance parameters. The required energy is significantly influencedby the variations in the variables in the cold mixed refrigerant (approximately 63%), while changes in the warmmixed refrigerant (WMR) section only slightly affect the uncertainty of the required specific energy. Furthermore,the probability distribution of the approach temperature (MITA1) inside the WMR exchanger is mainly affected bychanges in the compositions of methane, ethane, and propane, as well as the high pressure of the cold mixedrefrigerant (approximately 97%). Conversely, the flow rate of ethane and low pressure of the WMR significantlyaffect the uncertainty of the approach temperature (MITA2) inside the cold mixed refrigerant exchanger.

https://doi.org/10.1016/j.apenergy.2019.05.004Received 4 February 2019; Received in revised form 5 April 2019; Accepted 1 May 2019

⁎ Corresponding author.E-mail address: mynlee@yu.ac.kr (M. Lee).

1 These authors contributed equally.

Applied Energy 250 (2019) 1446–1456

Available online 16 May 20190306-2619/ © 2019 Elsevier Ltd. All rights reserved.

T

1. Introduction

As a relatively clean energy source, natural gas (NG) has attractedsubstantial attention with respect to satisfying the increasing globalenergy demand. Electrical power can be produced from NG with 50%less greenhouse gas emissions than when using coal [1]. Therefore,rapid growth has occurred in the NG trade compared to those of oil andcoal [2,3], and NG is often referred to as the bridge fuel to a renewablefuture, mainly owing to its lower air-pollutant emissions [4]. To date,NG transportation in the form of liquefied NG (LNG), which has a vo-lume of 600 times less than the gaseous state, is considered as one of themost promising and economic transportation approaches [5,6]. Amongthe well-established LNG processes, the dual mixed refrigerant (DMR)process has been the most favored candidate for onshore and large-scaleLNG production, mainly owing to its relatively high energy efficiencycompared to other available liquefaction processes [7–9]. Indeed, theenergy efficiency of the DMR process should be high owing the to tworefrigeration loops (warm and cold) that use two different mixed re-frigerants: one for pre-cooling, namely the warm mixed refrigerant(WMR), and another for subcooling, namely the cold mixed refrigerant(CMR), followed by the liquefaction step. Nevertheless, the DMR pro-cess is considered to be more complicated and sensitive than its com-peting propane pre-cooled mixed refrigerant (C3MR) liquefaction pro-cess. This is mainly owing to the significantly different compositions ofthe two mixed refrigerants, with varying evaporation and condensationpressure levels in the WMR and CMR loops. For example, when theambient temperature changes considerably, the concentrations of themixed refrigerants (WMR and CMR) in the liquefaction process need tobe adjusted (re-optimization) accordingly. However, in a practicalscenario, the resumption of the operation of a DMR process at or closeto an optimal state is considered as a challenging task.

In practice, operational parameter optimization is a challengingissue, mainly owing to the lack of information regarding the variousdecision variables, such as those in complex models including the DMRliquefaction process. Numerous optimization algorithms [10–13] havebeen developed to achieve efficient operation. However, it is often in-feasible and compulsory to include all of these variables in a givencalibration model. In general, unimportant variables should be main-tained at their mean values when estimating the model parameters. Toidentify the influential parameters, sensitivity analysis (SA) is used toreduce the dimensions of a high-dimensional uncertainty quantificationproblem, in order to quantify sensitive performance precisely [14]. Theapplication of global SA (and the closely related uncertainty quantifi-cation (UQ)) has been researched extensively, particularly with orwithout a meta-model for process system engineering-related problems,including chemical models [15], process design [16], enhanced oil

recovery [17], energy distributed systems [18], high-density fluid ra-dial-inflow turbines for renewable low-grade temperature cycles [19],and other renewable energy technologies [20]. The design of zero/low-energy buildings has been optimized using sensitivity analysis (SA) ofthe design parameters [21]. Moreover, recently, Ali et al. [22], pre-sented the UQ with simultaneous determination of sensitivity indices(SI) to evaluate the operational reliability for the optimal single mixedrefrigerant (SMR) liquefaction process.

However, to the best of our knowledge, there exist no detailed in-vestigations in the open literature regarding the impact and uncertaintylevels of operational parameters (mixed refrigerant compositions, eva-poration pressure, and condensation pressure) of the DMR LNG processfor the overall power and approach temperatures, namely the minimuminternal temperature approach (MITA) values. Moreover, the impact ofthe design and operational variables on the overall energy consumptionand feasibility (the approach temperature values inside the LNG ex-changer) of the process has not been studied. Therefore, a detailed in-vestigation regarding global SA of the operational parameters of theDMR liquefaction process, which considers UQ, is required to achieveprocess optimization for greater stability and viable operation withminimal energy consumption. Within this context, a surrogate modelbased on the multiplicative dimension reduction method (MDRM) wasadopted in this study for the SA. The Monte Carlo (MC) [23,24] andquasi-Monte Carlo (QMC) [25–28] methods were employed to examinethe effects of the uncertainties. In the analysis, uniform distributionswere assumed for the uncertainty inputs, while a standard MC methodwas employed using 10,000 simulations. The relative percentage of theSobol total effect indices was calculated to compare the influences ofthe uncertain inputs on the process output. This work investigated atime-efficient methodology for identifying the uncertainty levels of theoperational variables in the DMR liquefaction process, corresponding tothe overall energy consumption and approach temperature inside thecryogenic heat exchangers. This approach avoids unwanted processoutputs (for example, overall energy consumption and approach tem-peratures) under the uncertainty of the input operational variables,which will ultimately facilitate the development of rigorous stochasticmodels. Such a method will simplify the subsequent stochastic modelby neglecting non-influential parameters during the operational opti-mization. The results of this study are expected to be useful in terms ofaddressing global SA with less computational effort, using a multi-plicative dimension-reduction approach.

Nomenclature

DMR dual mixed refrigerantLNG liquefied natural gasMITA1 minimum internal temperature approach inside pre-

cooling cryogenic exchangerMITA2 minimum internal temperature approach inside sub-

cooling cryogenic exchangerCMR cold mixed refrigerantWMR warm mixed refrigerantOP operating pressureUQ uncertainty quantificationSA sensitivity analysisSI sensitivity indicesMDRM multiplicative dimension reduction methodMC monte carloQMC quasi monte carlo

C1 methaneC2 ethaneC3 propanei-C4 iso-butanen-C4 normal butaneN2 nitrogen

= x xX [ , , ]nT

1 vector of uncertainties= = p xp(X) ( )i

ni i1 pdf of independent uncertainties

Y = f(X) output of processg(Y) function of output modelE(g(Y)) expectation of output functionSi first order Sobol’s index (Noninteraction)Ti total effect Sobol’s index (Total contribution of xi)wil lth weight of quadrature for ith variable xixil lth node of quadrature for ith variable xiq number of quadrature nodesQ number of MC/QMC samples

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2. Methodology

2.1. Uncertainty quantification using Monte Carlo/Quasi Monte Carlo

Uncertainty is unavoidable in process design, owing of naturalvariation, measurement errors, or simply a lack of suitable technology.The purpose of UQ is to quantify the degree of confidence in theavailable result data for the overall DMR process. Numerous influentialuncertainties may be listed, such as the flow rate, feedstock composi-tion, operating conditions, feed, and product prices and/or supply ofutilities. To achieve appropriate results, the feasibility area and prob-ability distribution of each uncertain variable need to be quantifiedcorrectly, using existing knowledge and experience. Based on thegoodness-of-fit analysis, the distribution of a random variable is ap-proximated using a probability density function (PDF) [29] to estimatethe response distributions. In this study, the MC simulation was used toapproximate a quantitative problem, in which the number of outputswas evaluated based on random performance for sampling from theprocess input probability distributions. These evaluation results couldbe used to determine the uncertainty and perform SA of the model. Anuncertainty propagation scheme is illustrated in Fig. 1.

Consider a process system with a mathematical model, as follows:

= XY f ( ), (1)

where =X x x x[ , , ..., ]nT n

1 2 is the vector of uncertain inputs that canbe assumed as stochastic parameters and are used to characterize thestudied system. Based on the recognized uncertainties, f X( ) is calcu-lated to predict a corresponding quantity of interest (QoI), Y. However,under numerous circumstances, the values of X cannot be determinedprecisely because of missing information. Thus, they are only defined ina statistical manner using a PDF function, Xp ( ). Nevertheless, as anexpression of the uncertain inputs X , the QoI is also uncertain. Invarious UQ problems, it is necessary to quantify the influence of theinput uncertainties on the QoI. Meanwhile, the expression of the QoI isillustrated by a mathematical expectation (so-calledg Y( )):

= = XE g Y g f X p dX( ( )) ( )( ) ( ) , (2)

For example, Eq. (2) allows the kth raw statistical moment of Y to becalculated when =g Y Y( ) k, and yields the probability of Y exceeding acertain threshold value Y* if = >g Y Y( ) 1 ( )Y Y (that is, an indicatorapproximation). In general, model (1) lacks an analytical expression,and can be evaluated only using computer simulation. Therefore, ex-pectation (2) must also be computed using a numerical approach.

For UQ problems, the MC/QMC method operates by generating a setof QoI estimations, Y Y Y, , ..., M1 2 . The prediction f X( ) is repeated withindependent and identically distributed probabilities of the uncertaininputs x x x, , ..., n1 2 , which are sampled from the prerequisite PDFp X( ).The expected value from Eq. (2) is then approximated as

=Mg Y1 ( ),

i

M

i1 (3)

The MC method has a convergence rate of O((ln(M))d/M), which isapproximated by O(1/M). To obtain an accurate approximation of thetrue solution to the problem, the MC method generally requires a largenumber of samples M, and hence, numerous simulations are required toobtain satisfactory precision.

2.2. Global SA of DMR process

Global SA focuses on identifying the important variables affectingthe process performance, considering all potential variations in theinput parameters. In particular, it aims to characterize the behavior ofthe output response to alterations of the inputs, based on identifying themost influential inputs [30]. An evaluation of the global SA is con-ducted according to variance-based indices, including the MC method[31], fast amplitude sensitivity test (FAST) [32,33], high-dimensionalmodel representation (HDMR) [34], and polynomial chaos [35,36]. TheFAST method is based on design points selected from a space-fillingcurve, designed to explore each uncertainty using different integerfrequencies. Although the number of design points (number of simu-lations) in the FAST is low, its accuracy is also low and the algorithm fordetermining non-interference frequencies for all uncertainties is com-plex. The random balance method applies a simplified algorithm for

Uncertainty factorXn

Uncertainty factorX2

Uncertainty factorX1

Process simulationYi=f (fX1, X2,….., Xn)

Uncertain inputs

Processing unit

Output response

Uncertain outputs

???

Fig. 1. Concept of UQ propagation scheme.

M.A. Qyyum, et al. Applied Energy 250 (2019) 1446–1456

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identifying the non-interference frequencies, but it can only be used tocalculate the relative importance of the uncertainties alone (non-in-teraction). Polynomial chaos has demonstrated its advantages in termsof both accuracy and computational cost in cases with a small numberof uncertainties (less than 5) [35]. The sparse polynomial chaos [36]method has achieved high accuracy with a moderate computationalcost.

A common popular framework for global SA is provided by the MCor QMC methods. This approach enables statistical estimators of partialvariances to quantify the sensitivities of all uncertain inputs and theirgroups according to multi-dimensional integrals. However, this requiresinordinate simulation times for complex problems. Sepulveda et al. [37]applied global SA by comparing various methods, which facilitated theidentification of the stage recoveries as critical factors in flotation cir-cuits. To address the problem of excessive computational times, anaccurate and surrogate model that offers evaluation efficiency can beused in place of time-consuming simulations [38]. The main conceptunderlying this approach is that an HDMR can be decomposed into ahierarchy of low-dimensional functions in an additive expansion. Forexample, Zhang and Pandey [39] developed a simple approximationusing an MDRM to simplify the variance-based global SA. In general,the MDRM can provide moderate accuracy with a low computationalcost (linearly). This method was successfully applied in a chemicalprocess design by our group [40]. In recent years, an efficient two-stagepolynomial chaos method was proposed to study the SA of several in-dustrial processes in order to address complex chemical processes [40].A significant reduction in the computational time (approximately 95%)was demonstrated. Furthermore, the results indicated strong agreementbetween the proposed method and conventional approach (MC/QMC)in terms of the simplified model. Moreover, after GSA is performed, it isimportant to assess the correctness of the result provided by GSAthrough a carefully UQ. Normally the MC /QMC is used as referencesolution for this purpose [41,42].

2.2.1. Global SA using MDRMThis section briefly describes the MDRM for the global SA from Ref.

[39]. In the MDRM, the product of the one-dimensional functions isused to approximate a high-dimensional function of the random inputs.Therefore, derivation of simple algebraic expressions is used for thefirst-order effect and total effect sensitivity coefficients, in addition to asignificant decrease in time-consuming simulations. This approach fa-cilitates moderate accuracy and has a low computational cost. TheMDRM is then used to detect influential inputs that can also be used asoptimization variables. In this paper, we briefly discuss MDRM-orientedapplications. For further details, the reader is encouraged to review theMDRM description provided in Ref. [39].

From Eq. (1), a logarithmic transformation is obtained as follows:

=X y X( ) log(| ( )|), (4)

With the anchor point =a a a( , ..., )n1 , an additive dimensional reductionmethod can be applied to the logarithm transform X( ) to obtain:

=X t x a n( ) ( , , ) ( 1) ,

i

n

i i1

0(5)

where the following functions are related to those in the original do-main:

== +

aa

yx y a a x a a

log(| ( )|)( , ) log(| ( , ..., , , , ..., )|),i i i i i i n

0

1 1 1 (6)

The original model output can be rewritten as

= +e ee X n a a x a a( ( )) (1 ) ( ,..., , , ,..., )in

i i i i n0 1 1 1 1 (7)

Using Eqs. (5) and (6), the approximation of the model output is ex-plained in the form of:

== +

=

aa

y X y y a a x a ah y x

( ) ( ) ( , ..., , , , ..., )( , ),

nin

i i i nn

in

i i

11 1 1 1

01

1 (8)

where = +a a a a a a( , , ..., , , ..., )i i i n1 2 1 1 .This approximation model of the input–output correlation is re-

cognized as the univariate MDRM.The mean and mean square (μi and σi, respectively) of the ith di-

mensional function are defined as:

= == =

aa

µ E y x E h xE y x E h x

[ ( , )] [ ( )][ ( , ) ] [( ( )) ],

i i i i i

i i i i i2 2 (9)

Using Eqs. (5) and (6), the mean and mean square of the model outputcan be approximated as follows:

=

=

=

=

µ h µ

E y X hD µ µ

^

( ( ) )(^ ) [ /( ) 1]

.y

nin

in

in

i

y y in

k k

01

12

02 2

12

12

(10)

Using the MDRM approximation in Eq. (8), the conditional momentscan be obtained as follows:

= ==

=

E y X x h y x µ

E y X x h µ µ µ

[ ( | )] ( )

[( ( | )) ] ( ) (^ ) /( ) ,i

ni i k k i

nk

in

i k k in

k y i i

01

1,2

02 2

1,2 2 2

(11)

The primary variance is estimated as follows:

D µ µ(^ ) ( / 1),i y i i2 2

(12)

Thereafter, the first-order effect Sobol sensitivity index is calculated as

=+

+ =S D D

µµ

/1 /( )

1 /( ),i i y

i i

i kn

k k

2

2 (13)

Using the MDRM standard, and following several algebraic manipula-tions, the total effect sensitivity index is approximated for parameter xias follows:

=T

µµ

1 ( ) /1 ( ( ) / )

,ii i

kn

k k

2

12 (14)

The moments of the dimensional function in Eq. (9) with a Gaussianquadrature with q nodes =w x{ , }il il l

q1 can be expressed as

==

= +

= +

µ w y a a x a aw y a a x a a

( , ..., , , , ..., )( ( , ..., , , , ..., )) ,

i lq

il i il i n

i lq

il i il i n

1 1 1 1

1 1 1 12 (15)

From Eqs. (13)–(15), it can be observed that only nq evaluations arerequired to estimate the sensitivity indices.

Meanwhile, (n+2)Q simulations are required if QMC is used tocalculate these Sobol indices. It should be noted that Q=5000–10,000for the QMC method [43], while the MDRM only requires q= 5–10(number of quadrature nodes). The MDRM can easily be extended toestimate the Sobol indices of multiple QoIs simultaneously, as in [40].

2.3. DMR process description

The DMR process was introduced in 1978. According to the name ofthe process (“dual mixed refrigerant”), two different refrigeration loopsexist: one consists of a WMR, while the other includes a CMR to providecold energy for NG liquefaction. The WMR refrigeration loop is used topre-cool the feed NG as well as the mixed refrigerant of the CMR loop,while the CMR refrigeration loop is used to liquefy as well as sub-coolthe NG. The WMR contains methane, ethane, propane, iso-butane, andn-butane, while the CMR comprises methane, nitrogen, ethane, andpropane [7]. The major units associated with the DMR process are themulti-stage compression units, including inter-stage cooling, LNG heatexchangers (CHX-01 and CHX-02), and Joule–Thomson valves (JTVs)for isenthalpic expansion, as illustrated in Fig. 2. According to this

M.A. Qyyum, et al. Applied Energy 250 (2019) 1446–1456

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figure, the compression of the stream-12 (WMR) initiates the WMRrefrigeration loop, and the CMR loop begins from stream-1. The WMRand CMR are both compressed to condensation pressure (which is alsoone of the decision variables) using inter-stage cooling-assisted multi-stage compression units. The stages/number of compressors can bedetermined based on the optimal pressure ratio, which varies between 2and 5. Stream-16 at its optimal condensation pressure passes throughthe pre-cooling heat exchanger CHX-01, and after lowering the pressurethrough JTV-1, it exchanges cold energy with the NG (stream-18 and19) and CMR (stream-7 and 8). Stream-18 (feed NG) at 55.0 bar and25.0 °C, stream-7 (CMR), and stream-15 (high-pressure WMR) enter theCHX-01 and exchange heat with stream-17 (expanded WMR). Stream-12 exits from the CHX-01 as a superheated vapor and is recycled forcompletion of the WMR loop. The precooled NG stream-19 emergesfrom the pre-cooling heat exchanger (CHX-01) with a liquid fraction of5–10%, and the CMR stream-8 is obtained with a liquid fraction of25–45%, depending on the optimal pre-cooling temperature and com-position. Subsequently, stream-19 and 8 are introduced into the CHX-02, where NG is liquefied and sub-cooled by exchanging latent heatfrom the vaporization with latent heat from the liquefaction of ex-panded stream-10, which consists of a vapor fraction of 5.0–15.0%.

Stream-20 exits from the CHX-02 as sub-cooled LNG, while stream-11becomes superheated vapor and is recycled to complete the CMR loop.Stream-20 (sub-cooled LNG) is introduced into the end flash valve(JTV-3) to lower its pressure to slightly above atmospheric pressure;that is, 1.1–2.0 bar, and the resulting stream-21 is secured as finalproduct LNG with a liquid mole fraction of ≥0.9.

2.4. Process simulation

The prominent commercial simulator Aspen HYSYS® v10 was usedto develop a steady-state model for the DMR process. ThePeng–Robinson [44] fluid package with the option of the Lee–Kesler[45] equation, which rigorously estimates the enthalpy and entropy ofthe process streams, particularly at high pressures, was used. Table 1lists the simulation basis and feed conditions, which were adopted froma recent investigation [7] regarding the DMR liquefaction process.Furthermore, the following assumptions were considered in the simu-lations:

• Negligible heat losses occurred.• The compressor isentropic efficiency was selected as 75%.

Fig. 2. Basic configuration of DMR process [7].

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• The inter-cooler outlet temperature was fixed at 30 °C.• The proposed analysis was performed considering the feasibleminimum internal temperature approach (MITA) value of 3 °C.• The end flash gas vapor fraction was assumed as 3.7%.

3. Process analysis

The main objective of this study was to determine and evaluate theuncertainty levels of the operational decision variables, and their im-pacts on the performance parameters (overall energy consumption andMITA values inside the LNG heat exchangers) in the DMR liquefactionprocess. A rigorous HYSYS (v10) simulation was integrated with theglobal SA coded in the MATLAB R2017b environment, while the in-dependent input variables were assumed to be uniformly distributed.An ActiveX/COM functionality was executed between HYSYS andMATLAB, and all of the variables were linked in a spreadsheet–vari-ables window in HYSYS. The MATLAB-based pseudo-code for the SAusing the MDRM method is presented in Supporting Information underAppendix S-A. Furthermore, the proposed analysis was performed on a64-bit operating system (x64-based processor – Intel® Core™ i7-4790CPU) with 16.0 GB RAM. To understand the uncertainty levels of theoperational decision variables, and their sensitive nature with respect tothe overall energy consumption and feasible approach temperature(1–3.0 °C), the analysis was categorized into the following three groups:

I. CMR flow rates of each ingredient, namely C1, C2, C3, and N2;II. WMR flow rates of C1, C2, C3, i-C4, and n-C4; andIII. operating pressures (OPs), namely evaporation and condensation

pressures of the CMR and WMR loops.

Table 2 lists the upper and lower bounds for the process inputs. Thespecific energy requirement and MITA of two LNG cryogenic heat ex-changers (MITA1 and MITA2) were calculated as the output in HYSYSspreadsheets, and used as the output performance for the UQ and globalSA. It should be noted that, in UQ, the specific energy requirement issacrificed for the constraint condition (MITA greater than 3). Further-more, the Sobol indices were estimated according to Eqs. (13) and (14).The implication of the small values of the Sobol sensitivity indices isthat the uncertainty of the corresponding input is trivial in terms of theoutput response. Therefore, the input xi can be maintained at its

nominal value and further studies can be directed towards the con-secutive optimization steps [16].

The process constraints were the MITAs of the LNG heat exchangers1 and 2 (CHX-01 and CHX-02, respectively).

All independent variables were randomly varied along their upperand lower boundaries. To proceed with the UQ analysis, random samplepoints of 10,000 simulations were generated from the Halton sequence[46], which uses a low-discrepancy sequence and is exploited for outputperformance [40,47]. For the MC/QMC methods, the number of eva-luations was selected according to the Chernoff bound [48] for an ac-curate probability approximation. As discussed previously, three sub-groups (two mixed refrigerant cycles and their operating pressures)influenced the overall performance of the DMR process. The first cycleinvolved pre-cooling the NG and condensing the refrigerant of thesecond cycle. The second cycle (CMR) liquefied and sub-cooled the NGto LNG. In this study, the refrigerant compositions in each cycle andtheir operational conditions were uncertain. Therefore, the aim of thisstudy was to quantify the uncertainty of each subgroup for the processoutputs. It was also extended to investigating the interaction effects ofthese uncertain subgroups. This study considered three cases: UQ forthe required specific energy, minimum approach temperature of theLNG (MITA), and minimum approach temperature of the LNG1(MITA2). For each case, the MC method was used to analyze the resultsof the single effect of each subgroup, interaction effect of two sub-groups, and total effect of all three subgroups, respectively.

In this study, the Sobol indices computed using the MDRM wereused to identify the influence of the process inputs, which mainly af-fected the output response (required energy and minimum approachtemperature) [31,49]. If the total Sobol index is equal to zero, this in-dicates that the corresponding input uncertainty is negligible and canbe maintained at any point in its distribution, without affecting theoutput variance. The Sobol indices can also be used to identify the mostimportant process variables for the design and optimization of com-plicated problems.

4. Results and discussions

4.1. UQ for required specific energy

Figs. 3–5 illustrate the UQ results regarding the single effect of eachsubgroup, interaction effect, and total effect of the uncertainty sub-group on the required specific energy. As observed in Fig. 3, two sub-groups, namely the CMR and OP, had a significant influence on therequired specific energy. Meanwhile, the flow rates of the four re-frigerants in the CMR subgroup had the greatest effect on the outputperformance (overall energy). In particular, the PDF was widely

Table 1Simulation basis and feed condition for DMR process [7].

Property Condition

NG feed conditionFlow rate 124,654.33 kg/hTemperature 25.0 °CPressure 55 bar

NG feed composition Mole %Methane 87.36Ethane 6.69Propane 3.5n-Butane 0.89i-Butane 0.59n-Pentane 0.19i-Pentane 0.29Nitrogen 0.49

Pressure drops across cryogenic exchanger (CHX-01)“Stream-18” to “Stream-19” 1.0 bar“Stream-7” to “Stream-8” 0.5 bar“Stream-15” to “Stream-16” 0.5 bar“Stream-17” to “Stream-12” 0.1 bar

Pressure drops across cryogenic exchanger (CHX-02)“Stream-19” to “Stream-20” 1.0 bar“Stream-8” to “Stream-9” 1.0 bar“Stream-10” to “Stream-11” 0.1 bar

Table 2Upper and lower bounds for all random input variables.

Base case Lower bound Upper bound

Cold mixed refrigerant (CMR)Flowrate of N2 (kg/h) 0.377 0.100 0.450Flowrate of C1 (kg/h) 0.962 0.250 1.000Flowrate of C2 (kg/h) 0.953 0.300 1.200Flowrate of C3 (kg/h) 2.213 0.800 2.700

Warm mixed refrigerant (WMR)Flowrate of C1 (kg/h) 0.016 0.005 0.350Flowrate of C2 (kg/h) 0.163 0.050 0.600Flowrate of C3 (kg/h) 0.319 0.100 2.300Flowrate of i-C4 (kg/h) 0.401 0.100 1.500Flowrate of n-C4 (kg/h) 0.401 0.100 1.000

Operating pressures (OP)CMR Inlet, P1 3.250 1.500 6.000CMR Outlet, P6 55.000 35.000 65.000WMR Suction P12 1.300 1.200 4.000WMR Discharge P13 24.000 15.000 35.000

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distributed from 0 to 12.8 kW. The single effect of the OP uncertainsubgroup was considered as the second most important subgroup whenthe PDF range was 0 to 10.7 kW. The single effect of the WMR sub-groups on the required specific energy was the smallest. As such, Fig. 4presents the interaction effect of each pair of subgroups on the requiredspecific energy. Moreover, the UQ results indicate that the processoutput was significantly influenced by the variation in the CMR sub-group interacting with the others. As indicated in the previous analysis,the UQ was expected to be influenced by the variations in the CMR andOP subgroups. The simulation results demonstrate that the CMR–OPsubgroup pair had the greatest influence on the output. As observed inFig. 4(b), the required specific energy distribution varied from 0 to23.3 kW. This is because a sum of the distributions of the single effectexisted as a result of the CMR and OP subgroups. Fig. 5 illustrates therequired specific energy distribution of the total effect for the threesubgroups. The required specific energy varied from 0 to 27.3 kW,which was the sum of the effects owing to the three subgroups, whilethe effect of the CMR subgroup was the most significant in the requiredspecific energy disturbance.

Specific energy (kW)0 2 4 6 8 10 12

0.00

0.05

0.10

0.15

0.20

0.25

Specific energy (kW)0 1 2 3 4 5

0

1

2

3

4

5

Specific energy (kW)0 2 4 6 8 10 12 14

Den

sity

func

tion

f

0.00

0.01

0.02

0.03

0.04

0.05

0.06

(a) CMR (b) WMR (c) OP

Fig. 3. Probability distribution profile of single effect of uncertain subgroup on required overall energy.

Specific energy (kW)0 2 4 6 8 10 12 14 16 18 20

Den

sity

func

tion

f

0.00

0.01

0.02

0.03

0.04

0.05

Specific energy (kW)0 5 10 15 20 25

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Specific energy (kW)0 2 4 6 8 10 12 14 16

0.00

0.05

0.10

0.15

0.20

0.25

(a) CMR-WMR (b) CMR-OP (c) WMR-OP

Fig. 4. Probability distribution profile of interaction effect of uncertain subgroups on required overall energy.

Fig. 5. Probability distribution profile of total effect of uncertain subgroups onrequired overall energy.

MITA 1 (oC)

-12 -10 -8 -6 -4 -2 0 2 4 6

Den

sity

func

tion

f

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

MITA 1 (oC)

-2 -1 0 1 2 3 40

1

2

3

4

5

MITA 1 (oC)

-8 -6 -4 -2 0 2 4 60.0

0.1

0.2

0.3

0.4

0.5

0.6

(a) CMR (b) WMR (c) OP

Fig. 6. Probability distribution profile of single effect of uncertain subgroups on MITA1.

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4.2. UQ for MITA1

Figs. 6–8 depict the disturbances of the MITA1 when the threesubgroups were varied. Fig. 6 illustrates the single effect of each sub-group on the MITA1. Similarly, the effect of the CMR was the mostinfluential on the PDF of the MITA1. For the MITA1, the changes in fourrefrigerant compositions in the CMR resulted in a probability widelydistributed from −9.5 to 3.4 °C. The second most important subgroupwas the variation in the OP, which resulted in an MITA1 range of −7.0to 3.6 °C. Finally, with the variation in the WMR subgroup, the MITA1ranged from −1.0 to 3.1 °C. Clearly, the MITA1 was highly sensitive tochanges in each subgroup, and the CMR had the greatest impact. Fig. 7

illustrates the interaction effect of each pair of subgroups on the MITA1,including the pairs of CMR–WMR, CMR–OP, and WMR-OP, respec-tively. It can be observed that the MITA1 tended towards negativevalues when both subgroups were changed together. This implies thatthe constraint condition was easily violated. As a result, the MITA1 wasdistributed from −18.9 to 3.5 °C when the pair CMR–OP was changed(see Fig. 7(b)). This pair also introduced the greatest effects on theMITA1. Consequently, Fig. 8 illustrates the distribution of the MITA1for the three (total) subgroups, which varied from −23.2 to 3.5 °C.When the three subgroups varied simultaneously, the results indicatethat the disturbance of the CMR and OP subgroups still had the greatesteffect on the MITA1 performance. The total effect was not a summationeffect of each individual subgroup, as opposed to the case of the re-quired specific energy. The correlation between the total effect andsingle effect by the corresponding input could be in the exponentialorder, but the investigation of the numerical results is beyond the scopeof this study.

4.3. UQ for MITA2

A different trend was exhibited in the uncertainty of the MITA2compared to the previous two cases. Figs. 9–11 only present the dis-turbance in the MITA1 when the three subgroups of the WMR and OPwere varied. The single effect of the CMR subgroup on the MITA2disturbance was trivial and is not presented in the main text; instead,the simulation data are listed in the Supporting Information underTable S1. When all other decision variables remained constant, theprecooling temperatures significantly affected the MITA2 value insidethe CHX-02. These precooling temperatures were a function of theWMR subgroup rather than the CMR subgroup, which is the reasonunderlying the trivial effect of the individual CMR subgroup on theMITA2. Moreover, the MITA2 ranged from 2.3 to 3.6 °C when the singleeffect of the WMR was varied, as indicated in Fig. 9. The MITA2 was

MITA 1 (oC)

-16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6

Den

sity

func

tion

f

0

1

2

3

4

MITA 1 (oC)

-20 -15 -10 -5 0 50.0

0.2

0.4

0.6

0.8

1.0(a) CMR-WMR (b) CMR-OP

MITA 1 (oC)

-12 -10 -8 -6 -4 -2 0 2 4 60.0

0.1

0.2

0.3

0.4

0.5(c) WMR-OP

Fig. 7. Probability distribution profile of interaction effect of uncertain subgroups on MITA1.

MITA 1 (oC)

-25 -20 -15 -10 -5 0 5

Den

sity

func

tion

f

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 8. Probability distribution profile of total effect of uncertain subgroups onMITA1.

Fig. 9. Probability distribution profile of single effect of uncertain subgroup on MITA2.

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more widely distributed, from 0.6 to 4.5 °C, when the single effect of theOP was changed. Therefore, the single effect of the OP subgroup washigher than that of the WMR subgroup, which became the dominantinfluence on the MITA2. As such, Fig. 10 presents the interaction effectof each pair of subgroups on the MITA2, including the pairsCMR–WMR, CMR–OP, and WMR–OP, respectively. Given that the CMRsubgroup did not have an effect on the MITA2, the interaction effect ofeach pair containing CMR, namely CMR–WMR and CMR–OP, was theresult of the single effect caused by the variation in the WMR and OP,respectively. That is, an interaction effect occurred between the WMRand OP subgroup. As a result, the distribution of the MITA2 was variedfrom 1.1 to 4.5 °C, while the effect of the OP subgroup was moredominant on the MITA2 disturbance. Similarly, Fig. 11 illustrates thedistribution of the MITA2 (from 1.1 to 4.5 °C), which was mainly in-fluenced by the effects of the WMR and OP subgroups.

4.4. Global SA of required specific energy, MITA1, and MITA2

In this study, a rigorous simulation in HYSYS v10 was integratedinto MATLAB to perform a global SA. The uncertain subgroups in-cluding 13 variables (as listed in Table 2) were uniformly distributed.The connection between HYSYS and MATLAB was facilitated by meansof the ActiveX/COM function. According to the MDRM criteria, asample of 130 simulations was executed in MATLAB [50], and the re-sults were passed to HYSYS. Table 3 lists the Sobol sensitivity indices,namely the Si and Ti indices, respectively. The first-order sensitivityindex only examines the influence of the key effect of each uncertainvariable on the output response. As such, the total effect sensitivityindex is defined as the summation of all the order effects, including thefirst, and its interaction order effects. The arithmetic difference between

Si and Ti identifies the extent of the interactions between the input xiand other input variables. Furthermore, Si is necessary for determiningthe most important input uncertainties. The total sensitivity index (Ti) isessential for reducing the uncertainty in the output when identifyingthe importance of the input and its interactions. If Si is small, the cor-responding input is unimportant. However, this approach does not takeinto account any interactions. Then, Ti is estimated. If Ti is also small,the interaction of the input xi with the other inputs is negligible. In thiscase, apart from being unimportant, the specific input does not affectthe uncertainty of the output, and can be maintained at the nominalvalue in future studies [16].

For the SA, Si and Ti are meaningful when the arithmetic value ishigher than 0.1. As indicated in Table 3, the Si values were very small,which implies that there were no important uncertainties input into the

MITA 2 (oC)

2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8

Den

sity

func

tion

f

0.0

0.2

0.4

0.6

0.8

1.0

1.2

MITA 2 (oC)

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

MITA 2 (oC)

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8(a) CMR-WMR (b) CMR-OP (c) WMR-OP

Fig. 10. Probability distribution profile of interaction effect of uncertain subgroups on MITA2.

MITA 2 (oC)

0 1 2 3 4 5

Den

sity

func

tion

f

0.0

0.2

0.4

0.6

0.8

Fig. 11. Probability distribution profile of total effect of uncertain subgroups onMITA2.

Table 3Global sensitivity analysis of all factors.

Variables Abbreviation Sobolindices

Specificenergy

MITA1 MITA2

Flow rate of N2 inCMR

CMR-N2 S1 0.0041 0 0

Flow rate of C1 inCMR

CMR-C1 S2 0.0031 0.002 0

Flow rate of C2 inCMR

CMR-C2 S3 0.0092 0.0001 0

Flow rate of C3 inCMR

CMR-C3 S4 0.0146 0.0185 0

Flow rate of C1 inWMR

WMR-C1 S5 0.0004 0 0.0144

Flow rate of C2 inWMR

WMR-C2 S6 0.0033 0 0.1130

Flow rate of C3 inWMR

WMR-C3 S7 0.0054 0 0.0378

Flow rate of n-C4 inWMR

WMR-nC4 S8 0.0025 0 0.0544

Flow rate of i-C4 inWMR

WMR-iC4 S9 0.0028 0 0.0320

Low P of CMR CMR-LP S10 0.0034 0 0High P of CMR CMR-HP S11 0.0144 0.0154 0Low P of WMR WMR-LP S12 0.0070 0 0.7187High P of WMR WMR-HP S13 0.1076E-03 0 0.0022

T1 0.2573 0.0358 0T2 0.2059 0.2381 0T3 0.4380 0.1335 0T4 0.5551 0.9735 0T5 0.0353 0 0.0164T6 0.2197 0.0005 0.1271T7 0.3120 0.0144 0.0430T8 0.1746 0.0243 0.0617T9 0.1932 0.0238 0.0364T10 0.2212 0.0139 0T11 0.5508 0.9681 0T12 0.3720 0.0001 0.7449T13 0.0090 0.0001 0.0003

Computational cost(second)

73.6

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required specific energy and MITA1. Exclusively, the low pressure ofthe WMR cycle was the most important uncertain variable for theMITA2 variance (Si=0.7187). In contrast, the Ti values of the totaleffect of each uncertain variable on the required specific energy,MITA1, and MITA2 were significant. For the required specific energy,the interaction effects could be determined using these variables, in-cluding the flow rates of the four components in the CMR cycle, and theflow rates of C2, C3, nC4, and iC4 in the WMR cycle. Furthermore, theinteractions of the low and high pressures of the CMR cycle and lowpressure of the WMR were influential in terms of the required specificenergy, while the MITA1 variance was affected by the interactions ofthe flow rates of C1, C2, and C3 in the CMR cycle. Moreover, the MITA1was highly influenced by the interaction effect of the high pressure ofthe CMR. For the uncertainty of the MITA2, the interaction effects ofthe C2 flow rate and low pressure of the WMR were significant. Forimproved clarity, Fig. 12 illustrates the relative percentages of the totaleffect indices for the 13 uncertain variables. It can be observed that theuncertainty of the input onto the output was strongly associated with ahigher percentage value of the corresponding total effect index. More-over, it is indicated that the uncertainty of the input onto the outputwas strongly associated with a higher percentage value of the corre-sponding total effect index. In particular, Fig. 12(a) illustrates that theC2 and C3 compositions of the mixed refrigerant, as well as the highpressure of the CMR cycle and low pressure of the WMR cycle, wereinfluential in terms of the uncertainty of the required specific energy.Similarly, the MITA1 distribution was sensitive to the high pressure, inaddition to the C1 and C3 compositions of the mixed refrigerant in theCMR cycle, respectively (see Fig. 12(b)). Nevertheless, Fig. 12(c) in-dicates the importance of the total effect of the low pressure and C2composition in the WMR cycle on the MITA2 variance.

5. Conclusions

The aim of this study was to determine the model variables with thegreatest influence on the output uncertainty. This information thenallowed for non-influential parameters to be maintained in the model,in addition to providing a direction for further research to reduceparameter uncertainties and increase the model accuracy. Prior to theSA, the UQ was exploited to determine the output distribution when theuncertain inputs were varied. This established a new trend in the studyof LNG with respect to the identification of the most important para-meters in a model, and is critical in supporting effective para-meterization and model development. The MDRM, a variance-based SAmethod, was used to obtain information regarding the SA indices,which quantify the influence of each input on the output variance. Forthis specific DMR process, important variables were identified based onthe Sobol indices. As a result, based on the UQ and SA, the specificenergy was completely influenced by the variable changes in the CMRsection (approximately 63%), which included the flow rates of N2, C1,C2, and C3, as well as the low and high pressures of the CMR. Thespecific energy was also slightly influenced by the variation in the flowrates of C2, C3, nC4, and iC4, as well as the low pressure of the WMR(37%). Furthermore, the uncertainty of the MITA1 was mostly affectedby the changes in C1, C2, and C3, and the high pressure of the CMR(approximately 97%). Conversely, the flow rate of C2 and the lowpressure of the WMR were completely influential in terms of the MITA2uncertainty. Based on these observations, robust optimization of theDMR process can be performed in future work. Finally, our perspectiveson research issues relating to SA were presented for complex LNGmodels.

7%6%

12%

16%

1%6%9%

5%

5%

6%

16%

11%CMR-N2CMR-C1CMR-C2CMR-C3WMR-C1WMR-C2WMR-C3WMR-nC4WMR-iC4CMR-LPCMR-HPWMR-LPWMR-HP

1%

10%

5%

40%

1%1%

1%1%

40%

(a)(b)

(c)

2%

12%

4%

6%

4%

72%

Fig. 12. Percentage sensitivity effect of each variable on DMR process: (a) required specific energy, (b) MITA1, and (c) MITA2.

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Declaration of Competing Interest

The authors declare no competing financial interests.

Acknowledgment

This research was supported by the Basic Science Research Programthrough the National Research Foundation of Korea (NRF) funded bythe Ministry of Education (2018R1A2B6001566) and the PriorityResearch Centers Program through the National Research Foundationof Korea (NRF) funded by the Ministry of Education(2014R1A6A1031189).

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.apenergy.2019.05.004.

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