Characterization and prediction of flow-conditions in the hot-leg of PWR during loss of coolant accident

Nuclear Engineering and Design 359 (2020) 110446

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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes

Characterization and prediction of flow-conditions in the hot-leg of PWR during loss of coolant accident

T

Kumar Samal, Suman Ghosh

⁎

Department of Mechanical Engineering, National Institute of Technology, Rourkela 769008, India

GRAPHICAL ABSTRACT

ARTICLE INFO

ABSTRACT

Keywords: LOCA PWR-hot-leg Plugging/blocking Finite volume-based VOF PD & VF GA tuned MLFFNN

Numerical and computational attempts are made to investigate and characterize the flow-condition in the hotleg of a Pressurized Water Reactor (PWR) during the loss of coolant accident (LOCA). Finite Volume-based Volume of Fluid (VOF) model is employed for transient numerical simulations of the two-phase hydrodynamics in hot-leg during LOCA. The turbulence effect is captured using ‘k-w’ model. Effect of individual fluid-flowrates and stored water (initial water-level), on the developed counter-current two-phase flow-structures in hot-leg are extensively studied. Variations of the spatial distribution of phases with time are evaluated. Flow-structures in the hot-leg are characterized and parameterized in terms of statistical parameters extracted from the time variation of pressure drop (PD) and volume fraction (VF) across the flow domain. Computational intelligencebased methodologies are developed to capture the complex nonlinear relationship between obtained flowconditions (occurrence or absence of plugging/blocking) in the hot-leg and the extracted statistical parameters. A computational approach is also developed to find the dependency of occurrence or absence of plugging on the physical working conditions. Plugging is found to be most responsive to the gas-flowrate. Plugging/blocking is easily occurred at high gas-flowrate. It may occur even at moderate gas-flowrate when initial stored water-level in hot-leg crosses the threshold limit. It is found that the developed methodologies are able to perfectly capture the relationship between the flow-condition (occurrence or absence of plugging) in hot-leg and the said statistical parameters. The relation between the flow-condition in the hot-leg and the working conditions is also perfectly correlated by the developed computational approach.

1. Introduction The nuclear power station is one of the conventional power plants, which generates electricity from nuclear energy. One of the principal processes in a nuclear power plant, is the evaporation of water into steam using nuclear energy. Heat is generated in the reactor core by the

⁎

fission process. The generated steam is used to rotate turbine-blades and produce the desired output (electricity). The above process consists of two circuits (primary and secondary) as shown in Fig. 1a. The primary circuit is filled with water (green color in Fig. 1a) at high pressure (~15 MPa). This pressurized water takes heat from the reactor in the reactor pressure vessel (RPV) and goes to the steam generator (SG). In

Corresponding author. E-mail address: [email protected] (S. Ghosh).

https://doi.org/10.1016/j.nucengdes.2019.110446 Received 19 February 2019; Received in revised form 15 November 2019; Accepted 18 November 2019 Available online 31 December 2019 0029-5493/ © 2019 Elsevier B.V. All rights reserved.

Nuclear Engineering and Design 359 (2020) 110446

K. Samal and S. Ghosh

Nomenclature

Yk Yw

Symbols Description nb

p

b BHj BO1 F Gk Gw HIj HOj Ht hw I¯

IIi IOi Is k Ku Lr ṁa Me Mf Mp ṁw N Ng OI1 OO1 Ot P Ps R Sc Sd Sk Sk Sw Ts W1ij W2j1

Dissipation of ‘k’ due to turbulence Dissipation of ‘ω’ due to turbulence

Greek Symbols Description

Influence coefficients of the neighboring cells Centre coefficient Contribution of the constant part of the source term Bias value added to the jth neuron in the hidden layer Bias value added to the neuron in the output layer External body forces (N) Generation of ‘k’ due to mean velocity gradients (kg/ms3) Generation of ‘ω’ due to mean velocity gradients (kg/ms3) Input of the jth neuron in the hidden layer Output of the jth neuron in the hidden layer Transfer function co-efficient in the hidden layer Initial water-level in the horizontal part of hot-leg (m) Unit tensor Input of the ith neuron in the input layer Output of the ith neuron in the input layer Constant slope in the input layer Turbulence kinetic energy (J/kg) Kurtosis Learning rate Mass flowrate of air at the inlet (kg/s) Mean Momentum factor GA mutation probability Mass flowrate of water at the inlet (kg/s) Number of neuron in the hidden layer Number of generations of GA Input of the neuron in the output layer Output of the neuron in the output layer Transfer function co-efficient in the output layer Numbers of training or test scenarios GA population size Unscaled residual Source term Standard Deviation Skewness User-defined source terms for ‘k’ User-defined source terms for ‘ω’ GA tournament size Connecting weight between ith neuron in the input layer and jth neuron in the hidden layer Connecting weight between jth neuron in the hidden layer and the neuron in output layer

p

s

µp µs µt p

s

k w

p

ω

Under-relaxation factor Primary phase volume fraction Secondary phase volume fraction Viscosity of the primary phase (m2/s) Viscosity of the secondary phase (m2/s) Turbulent viscosity (m2/s) Density of the primary phase (kg/m3) Density of the secondary phase (kg/m3) Turbulent Prandtl numbers for ‘k’ Turbulent Prandtl numbers for ‘ω’ Computed change in Variable at a cell p Specific rate of dissipation (1/s)

Abbreviations Description AIAD ANN BPA CCFL CCTPF CI DV ECCS EES ESWS FVM GA HPHL HSC IPHL LOCA MLFFNN PD PISO PRESTO PWR RPV SG VF VOF

the SG, pressurized hot water of the primary circuit exchanges heat with water of the secondary circuit without mixing. The secondary circuit contains water (blue color) approximately at 70 bar pressure. The water in the primary circuit at high temperature remains as liquid water due to the high pressure but the water in the secondary circuit is evaporated in SG and generates steam as it is in the lower (~70 bar) pressure. Generated steam (orange color) is used to rotate the turbineblade and the turbine drives a generator for generating electricity. The RPV is connected with the SG through the hot-leg conduit. Hot-leg conduit consists of a horizontal conduit, one or more elbow, and an inclined or vertical conduit. Enormous amount of heat is generated in the core of a nuclear reactor and the removal of this heat from the core is essential in accidental scenarios. One of the important safety issues for a nuclear reactor is related to the loss of coolant accident (LOCA). Because of leakage in the primary circuit during LOCA, there is a possibility of getting counter-current two-phase flow (CCTPF) through the hot-leg of PWR.

Algebraic Interfacial Area Density Artificial Neural Network Back Propagation Algorithm Counter-Current Flow Limitation Counter-Current Two-Phase Flow Computational Intelligence Decoded Value Emergency core cooling system Emergency electrical systems Essential service water system Finite Volume Method Genetic Algorithm Horizontal Part of the Hot-Leg High-Speed Camera Inclined Part of the Hot-Leg Loss of Coolant Accident Multi-Layered Feed Forward Neural Network Pressure Drop across the flow domain Pressure-Implicit with Splitting of Operators PREssure STaggering Option Pressurized Water Reactor Reactor Pressure Vessel Steam Generator The Volume Fraction of air across the flow domain Volume of Fluid

Due to the leakage, depressurization and vaporization of water take place in the primary circuit. The generated steam will flow towards SG through the hot-leg. Some part of the steam will condensate (reflux condensation) due to heat-exchange between the steam and water in SG. This condensate water will flow back towards RPV through the same hot-leg due to gravity. This will create a CCTPF in the hot-leg during LOCA. The flow path of steam and water in this reflux condensation mode during LOCA is shown in Fig. 1b. From the safety point of view of the plant, hot-leg plays a crucial role during LOCA. If the counter-current two-phase flow-structure in the hot-leg is stratified in nature, then some part of the condensate water will be able to flow towards RPV and it will help to remove the heat generated in RPV. However, the stratified two-phase flow exists (is stable) only for a certain range of the individual fluid (gas and water) flowrates. Plugging/blocking may start in the hot-leg when flowrate of any one of the phases increases and reaches a sufficient value. Gasflowrate generally increases with time during LOCA, and consequently 2

Nuclear Engineering and Design 359 (2020) 110446

K. Samal and S. Ghosh

Fig. 1. Schematic representation of the (a) working principle of a nuclear power plant (Observatory, 2019) and (b) flow path of steam and water in reflux condensation mode during LOCA (Deendarlianto et al., 2008).

slugging criteria to solve the momentum balance equations (third viewpoint) were implemented by Lopez-De-Bertodano (1994). Wang and Mayinger (1995) made 2D CFD simulation and analysis using a two-fluid model for thermal–hydraulic phenomena in a PWRhot-leg during small break LOCA. CFD simulations of the counter-current two-phase flow through the hot-leg (during LOCA) have been investigated by Utanohara et al. (2009) using the Euler-Euler model of FLUENT6.3.26. The effects of the gas–liquid interfacial friction correlations were evaluated by them. Experiments and CFD simulation on counter-current gas–liquid two-phase flow in a PWR-hot-leg (during LOCA) were conducted by Minami et al. (2010). Three-dimensional two-fluid model using FLUENT6.3.26 was implemented by them to predict the two-phase flow patterns and CCFL characteristics in the hotleg. 3D two-fluid model using FLUENT6.3.26 was employed by Kinoshita et al. (2010) to numerically simulate the CCFL in a PWR hotleg under reflux cooling. Numerical simulation of 1/15th scaled model of the PWR-hot-leg under reflux condensation mode using CFD software (with two-fluid model) was carried out by Utanohara et al. (2011). The water-levels and the wave heights during LOCA were measured by them to understand the characteristics of the interfacial drag. They observed that wave heights are strongly related to the water-level and interfacial drag. For a better understanding of the CCTPF characteristics during LOCA, CFD simulation of 1/3rd scale model of the hot-leg of a GermanKonvoi PWR with rectangular cross-section was performed by Deendarlianto et al. (2011) and Höhne et al. (2011). In both of the investigations, numerical simulations were performed using ANSYS CFX 12.1 with the multi-fluid Euler-Euler model. Surface drag was approached by them using a new correlation inside the Algebraic Interfacial Area Density (AIAD) model. Using the VOF model of FLUENT6.3.26, CCFL in a PWR hot-leg was numerically investigated and a correlation for CCFL was developed by Murase et al. (2012). Numerical simulations of CCFL in PWR during LOCA were conducted and the steam-water two-phase flow characteristics were analyzed by Utanohara et al. (2012) using VOF and two-fluid model. RPV, SG, and hot-leg were considered as the computational domain in their work. It was found that VOF simulations are more capable than the two-fluid simulations to reproduce the fluid properties. According to the best of the authors’ knowledge, though adequate experimental studies and a few numerical investigations have been conducted on CCTPF characteristics and CCFL mechanisms in the hotleg of a PWR during LOCA, the dynamics has not yet been revealed completely. In most of the numerical works, the two-fluid model was used. Few work using the VOF model has been found (Utanohara et al., 2012), where CCTPF has been modeled using a single momentum equation. However, Utanohara et al. (2012) found that the VOF model (using single momentum equation) is more capable to reproduce the fluid properties on CCPTF and CCFL during LOCA in PWR, compare to the two-fluid model. Again, as per the best of the authors’ knowledge,

the water flow is restricted by the increased gas flow. Water flow will be completely stopped when gas-flowrate increases and reaches a limiting value. In this situation, water flow will be carried over by the gas flow and may flow in the reverse direction. This is called counter-current flow limitation (CCFL) or onset of plugging/blocking in CCTPF and is the limiting point of stability of the CCTPF (Deendarlianto et al., 2008). During plugging, the water-level in the reactor core continuously decreases which is very undesirable for the reactor regarding safety issues. Because in this situation, heat-removal is very difficult which may lead to failure of the whole system. Therefore, CCFL or plugging/blocking in the hot-leg of PWR during LOCA is an undesirable situation. Some experimental investigations have been carried out on the CCFL and plugging phenomenon during LOCA in the PWR of a nuclear reactor as it is directly related to the safety issue of the nuclear reactor plant. CCFL and flooding in the nuclear reactor plant (considering air–water fluid pair) was investigated by Richter et al. (1978), Krolewski (1980), and Wan and Krishnan (1986). Wongwises (1994, 1996) made experimental studies on CCFL during LOCA in the PWR of nuclear reactor plants. At low water-flowrate, the hydraulic jump was observed by them at the end of the horizontal part of the hot-leg (HPHL). Geffraye et al. (1995) performed experimental investigation for the better understanding of geometrical and scale effect on the CCFL behavior in PWR. To examine the effects of geometrical parameters on CCFL, Kang and No (1999) performed experimentations and proposed a correlation, which was later modified by Kim and No (2002) using different CCFL experimental data from the literature. The CCFL at hotleg in reflux condensation mode during LOCA of a nuclear reactor was predicted by Jeong (2002). Navarro (2005) experimentally investigated the effect of the inclination angle of the riser, water inlet condition, and length of the horizontal part of the flow channel on CCFL. Experimental investigation on CCFL was made by Deendarlianto et al. (2008), Vallée et al. (2009), and Lucas et al. (2017) using a 1/3rd scaled model of German Konvoi PWR. High-speed camera (HSC) pictures of the hot-leg during CCFL were presented by them to understand the hydrodynamics of the flow pattern. A detailed description of flow patterns, CCFL mechanism, and high-quality HSC imaging of the interfacial structure in a large diameter hot-leg geometry was presented and analyzed by Issa and Juan (2014) using 1/3.9 scale of a PWR geometry. To solve the mass and momentum conservation equations for the CCFL phenomenon in a PWR-hot-leg, the one-dimensional stratified two-fluid model was used by several researchers but in different points of view based on CCFL or flooding mechanisms. Ardron and Banerjee (1986) provided the first viewpoint where they considered that flooding coincides to the slugging inception in the lower leg of the elbow near the bend. The envelope theory to solve the momentum balance equation (second view point) between the two-phase in the HPHL of the PWR model was suggested by Ohnuki et al., 1988; Geweke et al., 1992; Minami et al., 2008, and Nariai et al. (2010). The available 3

Nuclear Engineering and Design 359 (2020) 110446

K. Samal and S. Ghosh

till date, no such attempt is found regarding characterization or categorization of flow-condition (flow-structure) in the hot-leg (based on the individual fluid flowrates and stored water in hot-leg) during LOCA. Moreover, no attempt is found to predict the occurrence or absence of CCFL and plugging/blocking in the hot-leg by knowing the nature of fluctuation in the time variation of gas volume fraction (VF) and pressure drop (PD) across the PWR or by knowing the working-conditions (individual fluid-flowrates and stored water-level in hot-leg) in the PWR. Therefore, using single momentum equation with finite volume based VOF model, attempt is made here to numerically simulate the CCTPF structure and CCFL (plugging/blocking) in 1/3rd scaled model of German Konvoi hot-leg of PWR (Deendarlianto et al., 2008) by rigorously varying the gas and water flowrates and stored water-level in HPHL to provide more generalized solutions and features for the said complex phenomena. Effect of the flowrate of gas and water as well as the stored water-level in HPHL on the flow-structure (occurrence or absence of plugging) in hot-leg during reflux condensation mode in LOCA are thoroughly analyzed. Plugging/blocking in the hot-leg is characterized in terms of the nature of fluctuation in the time variations of VF and PD across the flow domain. An effort is also made to parameterize the flow-condition (occurrence or absence of plugging) in the hot-leg in terms of the statistical parameters extracted from the time variation of the said PD and VF. At last, hybrid methodology employing neuro-biological scheme with the evolutionary-based algorithm is developed using computational intelligence (CI) to predict the flow-condition (occurrence or absence of plugging) in the hot-leg either by knowing those statistical parameters or by knowing the working conditions of the PWR (individual fluid-flowrates and stored water-level in the hot-leg).

high-pressure emergency core cooling system (ECCS) and of the main feed pumps (Vallée et al., 2009; Höhne et al., 2011). Emergency core cooling system (ECCS), Essential service water system (ESWS), Emergency electrical systems (EES), etc. are generally used as emergency procedures during LOCA. As the interfacial behavior of air–water CCTPF is exactly similar to that of steam-water CCTPF and air–water CCTPF closely resembles the steam-water CCTPF, hydrodynamics of CCTPF in PWR during LOCA have been studied by considering air–water as the working fluid-pair in various literature (Deendarlianto et al., 2008; Wan and Krishnan, 1986; Wongwises, 1994; Jeong, 2002; Navarro, 2005; Vallée et al., 2009; Issa and Juan, 2014; Ohnuki et al., 1988; Geweke et al., 1992; Minami et al., 2010; Utanohara et al., 2011; Deendarlianto et al., 2011; Höhne et al., 2011; Utanohara et al., 2012) to avoid unwanted complexities. Interfacial hydrodynamics is the main concern in the current analysis and as per that, here also air–water fluid-pair is chosen as the working fluid-pair. Air enters through the upper part of the RPV (air-inlet section as shown in Fig. 2 using red color) and flows towards SG through the hot-leg. The water enters through the lower part of the SG (water-inlet section as shown in Fig. 2 using green color) and moves towards RPV through the same hot-leg. As a result, a CCTPF (air and water flow in the opposite direction) is established in the hot-leg. Air leaves through the upper part of the SG (airoutlet section as shown in Fig. 2 using red color). During plugging/ blocking, water flow may be completely stopped due to a very high gasflowrate. In this circumstance, water flow may be carried over by the high gas flow and may flow in the reverse direction. In this situation, undulation in the PD and VF across the flow domain are observed and consequently, PD and VF across the flow domain will start to fluctuate with time. These fluctuations in the time variation of PD & VF depend on the flow-condition in hot-leg. Here, the first problem is formulated to numerically simulate the above-mentioned hydrodynamics of CCTPF and plugging during LOCA in 1/3rd scaled model of a German Konvoi hot-leg (Deendarlianto et al., 2008) by rigorously varying the gas and water flowrates and stored water-level in HPHL. The 2nd problem is defined as to study the effect of working conditions of the PWR (like gas mass flowrate (ṁa), water mass flowrate (ṁw), and stored (initial water-level) in the HPHL (hw)) on the flow-structure (flow-condition in the hot-leg) in terms of the time variation of PD and VF across the flow domain. At first, the effect of ṁa on the flow-condition in hot-leg is studied by varying the ṁa at a fixed combination of ṁw and hw. Next, this study (effect of ṁa) is repeated for some other fixed combinations of ṁw and hw. Similarly, the effect of ṁw is studied for some fixed combinations of ṁa and hw. The effect of the hw is also studied at some fixed combinations of ṁa and ṁw. The 3rd problem is formulated as to parameterize the flow-conditions (flow-structures) in the hot-leg in terms of the statistical parameters extracted from the time variation (fluctuation) of PD and VF

2. Problem formulation 1/3rd model of German Konvoi PWR (Deendarlianto et al., 2008) with rectangular cross-section is considered for the present numerical investigation. This PWR consists of different components like hot-leg, RPV, and SG as shown in Fig. 2. RPV is connected with the HPHL and SG is connected with the inclined part of the hot-leg (IPHL). Dimensions of these components are shown in Fig. 2. RPV and SG are identical in shape and size. The length and width of the HPHL are 2120 and 250 mm, respectively. The length and expansion angle of the IPHL is 230 mm and 7.5°, respectively. The inclined part makes an angle of 500 with the horizontal plane. The radius of curvature of the inner and outer bends of the elbow (connecting part between HPHL and IPHL) of hot-leg are 250 and 500 mm, respectively. The present investigation is made for large break LOCA. The LOCA is either caused by the leakage at any location in the primary circuit (Wongwises, 1994; Deendarlianto et al., 2012) or due to the failure of

Fig. 2. Schematic representation of the flow domain with dimensions.

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Nuclear Engineering and Design 359 (2020) 110446

K. Samal and S. Ghosh

across the flow domain. Here, the nature and intensity of the PD and VF fluctuations are expressed in terms of four statistical parameters (mean (Me), standard deviation (Sd), skewness (Sk), and kurtosis (Ku)) extracted from the time-series variation of PD and VF across the flow domain. As a 4th problem, it is decided to develop methodologies to predict the flow-condition in the hot-leg (occurrence or absence of plugging) by knowing the above-mentioned statistical parameters (extracted from the time variation of PD and VF across the flow domain). Lastly, the 5th problem is defined as to develop methodologies to find the relation between flow-condition developed (plugging or non-plugging) in the hot-leg and the working conditions maintained in the PWR (ṁa, ṁw, and hw).

domain, a single momentum equation is solved as shown below (Eq. (2)) and the resulting velocity field is shared among the phases.

t

t

t

s s Vs )

=0

s s

=

s s

+ (1

s) p

s ) µp

and µ (2)

( k) +

(

)+

xi

xi

( kui) =

(

ui ) =

xj

xj

µ+

µt k

µ+

k xj

µt xj

+ Gk

Yk + Sk

+G

Y +S

(3)

(4)

Here, Gk represents the generation of turbulence kinetic energy due to mean velocity gradients, Gω represents the generation of ω. k and are the turbulent Prandtl numbers for k and ω, respectively. µt is the turbulent viscosity. Yk and Yω represent the dissipation of k and ω, respectively due to turbulence. Sk and Sω are user-defined source terms. The energy equation is not considered in the present study as the focus of the investigation is concentered only on the hydrodynamics of CCTPF and plugging. 3.1.2. Meshing The geometry (shape) of the chosen computational domain (hot-leg together with the RPV and SG) is irregular in nature. To track the complicated interface between gas and water in the chosen intricate domain, very fine triangular mesh is chosen in the whole computational domain after a rigorous grid-independent test (as described in subsection 4.1). A proper study to choose the size of the grids towards the wall was also made to handle the near-wall issues. The ‘k-ω’ turbulence model is coupled with ‘Enhanced wall treatment function’ to capture the nearwall issues (wall boundary layers effect). y+ (=ρuτy/μ) is a non-dimensional distance between the cell centroid for wall-adjacent cells and the wall. For the wall adjacent cells, y+ ≈ 1 is the most desirable condition for near-wall modeling using the ‘k-ω’ turbulence model. For the successful application of the ‘k-ω’ turbulence model, the grid size along the walls is chosen in such a way that the numerically obtained y+ value along the wall remains nearly equal to 1 (y+ ≈ 1). Though it is very small, this size of the grids adjacent to the wall (required to obtain y+ ≈ 1 along the wall), is chosen to create uniform meshing (grid size is not increased toward the outer part of the near-wall region) throughout the flow domain. Uniform meshing with a very smaller size of grids is used (though it may take larger computational time) to achieve a smooth and accurate interfacial distribution throughout the flow domain using the said finite volume based VOF model. The resolution of obtained interfacial distributions using finite volume based VOF, increases as the grid size decreases. Fig. 3 shows this mesh pattern employed for the present numerical study.

3.1.1. Governing equations In the present numerical simulation, the mixture of two immiscible fluids (gas and water) are modeled by solving single continuity and single momentum equation and then by tracking (transient tracking) the volume fraction throughout the flow domain. The tracking of the interface between the phases is accomplished by solving the continuity equation for the volume fraction of the secondary phase (water) only as shown below.

·(

+

2 ( · V ) I¯ 3

T

V )

This momentum equation depends on the volume fraction of the phases through the properties of ρ and μ. Here, I¯ is the unit tensor, F is the external body forces. Turbulent modeling is done using ‘k-ω’ model. ‘k-ω’ turbulence model with shear flow correction is considered for the turbulence calculations. The standard ‘k-ω’ model is based on the Wilcox ‘k-ω’ model (Wilcox, 1998), which incorporates modifications for low-Reynolds number effects, compressibility, and shear flow spreading. The transport equations for this turbulence model as used in the present study are given below.

To capture the complex, nonlinear, unsteady hydrodynamics and characteristics of CCTPF and plugging/blocking (in hot-leg of PWR during LOCA), the interfacial distributions are tracked through transient 2D Finite Volume-based Volume-of-Fluid (VOF) method. The transient pressure-based solver is used, where, the pressure–velocity coupling is made using the Pressure-Implicit with Splitting of Operators (PISO) scheme. The PISO scheme is based on the higher degree of approximate relation between the corrections for pressure and velocity. The least-squares cell-based method is used for spatial discretization of the gradient. Spatial discretization of volume fraction is made using the Geo-Reconstruct scheme whereas PRESTO (PREssure STaggering Option) scheme is used for spatial discretization of pressure. 2nd order upwind scheme is employed for spatial discretization of momentum, whereas 1st order upwind scheme is hired for spatial discretization of turbulent kinetic energy (k) and specific dissipation rate (ω). The firstorder implicit scheme is employed for transient formulation. In the present study, gas is considered as the primary phase and water is chosen as the secondary phase. As the computational domain, hot-leg together with RPV and SG (as shown in Fig. 2) are included to avoid conflicts and confusion regarding boundary conditions. Since the flow phenomena are complex and nonlinear in nature, time-varying (transient) analysis is made for a better understanding of the flow behavior.

+

p p

= µs µs + (1

3.1. Numerical Modelling:

s s)

. (¯¯) + g + F

P+

=

This section deals with the detailed numerical scheme and model employed to simulate the said hydrodynamics of CCTPF and plugging/ blocking in the HPHL under different physical situations and statistical analysis to parameterize the said time variation of PD and VF. The details of the developed hybrid computational intelligence-based methodologies to predict the flow-condition (occurrence or absence of plugging) are also discussed here.

(

. ( VV) =

where, ¯ = µ ( V +

3. Methodology

t

( V)+

(1)

The volume fraction equation is not solved for the primary phase (gas). The primary phase (gas) volume fraction is computed based on the constraint: p + s = 1. Here, ‘p’ denotes primary phase, ‘s’ denotes secondary phase, and ‘ ’ denotes volume fraction. Throughout the flow

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Nuclear Engineering and Design 359 (2020) 110446

K. Samal and S. Ghosh

Fig. 3. Grid pattern employed for the present numerical investigation.

3.1.3. Boundary conditions The mass flowrate of air and water are specified at the air-inlet and water-inlet section (as mentioned in section 2), respectively. At the airoutlet section (as mentioned in section 2), atmospheric pressure (P = 1 atm) is specified. At the above-mentioned inlet and outlet sections, turbulent intensity and hydraulic diameter are specified. Remaining all the boundaries of the computational domain are considered as stationary walls with no-slip (Ualong the wall = 0) and no penetration (Unormal to the wall = 0) boundary conditions. As the initial condition, the stored (initial) level of water at HPHL, SG, and RPV are specified.

unscaled residual value. The scaled residual value may be the appropriate choice as converging criteria. Here, the scaled residual value is chosen as converging measures. A scaling factor representative of the flowrate of ϕ throughout the domain is used to scale the residual. The scaled residual is calculated as

Rscalled =

p

=

(anb

nb)

+b

(5)

nb

where ap =

anb

SP

nb

In this equation, ap is the center coefficient of the cell p, anb is the influence coefficients of the neighboring cells, and b is the contribution of the constant part of source term Sc in S = Sc + SP . Unscaled residual is defined here as

R =

anb cells P

nb

nb

+b

ap

p

nb

nb

+b

ap

p

÷

|ap p| cells P

(7)

The convergence of the solution is achieved when all the scaled residual values (i.e. continuity, u-phase1, u-phase2, v-phase1, v-phase2, volume fraction, k, and ω) reach below a sufficiently small magnitude: 10-6. Because of the nonlinearity of the equation set being solved, it is required to control the change in ϕ. This is achieved here by using the under-relaxation of variables (also referred to as explicit relaxation), which reduces the change in ϕ produced during each iteration. The new value of the variable ϕ within a cell depends upon the old value: ϕold, the computed change in ϕ: Δϕ, and the under-relaxation factor: α. In a simple form, it can be expressed as:

3.1.4. Residual and convergence criteria Conservation equation for a variable ϕ at a cell p as used here can be written as

ap

anb cells P

=

old

+

(8)

3.2. Parameterization of the time variation of PD and VF For various working conditions of the PWR (by considering different combinations of ṁa, ṁw, and hw covering a wide range), time variations (fluctuations) of VF and PD across the flow domain are evaluated and corresponding flow-condition in the hot-leg are noted using transient

(6)

It is tough to decide the convergence of a solution based on the

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K. Samal and S. Ghosh

numerical simulations (as described earlier). For each of the cases (combinations), statistical parameters (such as Me, Sd, Sk, and Ku) of the said time-series fluctuations of VF and PD are extracted using MATLAB coding. Those statistical parameters bear the signature of the fluctuating nature of the said VF and PD fluctuations. The nature of fluctuations in the time variation of VF and PD strictly depends on the flow-condition (occurrence or absence of plugging/blocking) in the hotleg. Ultimately, the extracted statistical parameters and the corresponding flow-condition (plugging or non-plugging) in the hot-leg under different working conditions (for different combinations of ṁa, ṁw, and hw) are tabulated.

when any one of the inputs is negative. The output of the MLFFNN (i.e. the output of the neuron in the output layer) denotes the flow- condition/situation (i.e. plugging or non-plugging of the hot-leg). Plugging or non-plugging of the hot-leg is expressed by an indicator, which can take two numerical values. A value of 0.5 is assigned to indicate the occurrence of plugging and a value of 1.0 is assigned to indicate the nonplugging situation of the hot-leg in PWR. This indicator is considered as the output of the neuron in the output layer. Inputs of MLFFNN (inputs of the neurons in the input layer) are assigned in two distinct ways as explained below. 3.3.1. Using statistical parameters extracted from time-series VF and PD fluctuation as inputs Here, the network is designed to predict the flow-condition/situation (plugging or non-plugging) in hot-leg by knowing the statistical parameters extracted from the time-series fluctuation of VF and PD and consequently, these parameters are used as the inputs of MLFFNN. To find the most favorable combination of statistical features, four different sets of inputs using four different combinations of statistical features are considered in the input layer as described below. Case-A: At first, Me and Sd extracted only from the time-series fluctuation of VF are used as the inputs in the input layer. Case-B: Secondly, Me and Sd extracted only from the time-series fluctuation of PD are used as the inputs in the input layer. Case-C: Then, Me and Sd extracted both from VF and PD fluctuation are considered as the inputs in the input layer. Case-D: At last, all the four statistical parameters namely Me, Sd, Sk, and Ku extracted both from the VF and PD fluctuation are considered as the inputs in the input layer. PD may have negative value and sometimes the value of skewness of any fluctuation may be negative. Considering these facts, the log-sigmoid transfer function is used for the neurons in the hidden layer only when Case-A (as both of the inputs: Me and Sd of VF fluctuation are always positive) is considered and the tan-sigmoid transfer function is used in the hidden layer for all the remaining cases (Case-B–D).

3.3. Neuro biological scheme with evolutionary-based algorithms (using CI) A relation may exist between the flow-condition/situation (occurrence or absence of plugging) in the hot-leg and those VF and PD fluctuations, which are parameterized in terms of the statistical parameters as described in the previous sub-section. Therefore, it may be possible to predict the flow-situation (plugging or non-plugging of the hot-leg) by knowing the set of those statistical parameters under different working conditions. Here, methodologies are developed by adopting Multi Layered Feed Forward Neural Network (MLFFNN) with Back Propagation Algorithm (BPA) and tuned by the Genetic Algorithm (GA) to capture that complex nonlinear relationship and consequently to predict the flow-condition in the hot-leg by knowing those statistical parameters. Multiple attempts are made to develop them by considering different combinations of those statistical parameters. During LOCA, the occurrence or absence of plugging/blocking in the hot-leg depends on different working conditions (physical situations/working atmosphere) in different parts of the PWR. In the present investigation, these working conditions (physical situations/working atmosphere) are considered using some physical parameters (ṁa, ṁw, and hw). Therefore, an attempt is also made to find a direct correlation between the flowconditions/situation in hot-leg (plugging or non-plugging of the hotleg) and the working conditions (physical parameters of the PWR) using the above-mentioned GA tuned MLFFNN with BPA. The complete methodologies (MLFFNN with BPA tuned by GA) are indigenously developed using ‘C’ programming language in the Linux atmosphere. MLFFNN is a computing system inspired from biological neural networks. MLFFNN is formed by a connection of nodes or artificial neurons and each connection can transmit signals from one node to another. Between the input and output nodes, some hidden layers of neurons are used. MLFFNN is capable of learning, remembering, thinking, and making decisions with the help of proper training with appropriate training data. The performance of the MLFFNN strongly depends upon the connecting weights between the neurons. During training (learning), the mathematical structure of MLFFNN may change iteratively depending upon the input parameters (training data) into the system and ultimately an optimized set of connecting weights is obtained through BPA. Both incremental and batch modes of training are utilized to train the MLFFNN through BPA. In batch training, changes in weights are gathered together for all the training data and weights are updated once based on the average effect of the whole training set. In the case of incremental training mode, weights are updated for each of the training data. Only one hidden layer is considered for the present MLFFNN structure. Only one output and consequently only one neuron is considered in the output layer of MLFFNN. Biases are assigned to all the neurons except the neurons in the input layer. The linear transfer function (constant slope) is assigned to each of the neurons in the input layers whereas log-sigmoid transfer function is used for the neuron in the output layer. For the hidden layer, either log-sigmoid or tan-sigmoid transfer function is used based on the requirement. For a case when all the inputs are positive, a log-sigmoid transfer function is used for each of the neurons in the hidden layer. On the other hand, the tansigmoid transfer function is used for all the neurons in the hidden layer

3.3.2. Using the working conditions of the PWR as inputs Here, the network is designed to capture the dependency of flowcondition (plugging or non-plugging) developed in hot-leg on the working conditions (physical situations) maintained at different parts of the PWR. Here, physical parameters like ṁa, ṁw, and hw are considered as the inputs in the input layer. As only one hidden layer is considered, the proposed MLFFNN contains a total of three layers. The number of nodes or neurons in the input layer depends on the situations as described in sub-sections 3.3.1 and 3.3.2. Case-A and B have two nodes in the input layer, Case-C has four nodes in the input layer, and Case-D has eight nodes in the input layer. MLFFNN by considering the working conditions, has three nodes in the input layer. For each of the cases, the number of nodes or neurons in the hidden layer is optimized by the GA individually. Each of the above cases (as described in sub-sections 3.3.1 and 3.3.2) contains only one node or neuron in the output layer. The main architecture of the GA-MLFFNN by considering ‘n’ number of inputs and ‘N’ number of neurons in the hidden layer is shown below in Fig. 4a. In this architecture, ‘IIi’ and ‘IOi’ are the input and output of the ith neuron in the input layer, respectively. In the same way ‘HIj’ and ‘HOj’ denote the input and output of the jth neuron in the hidden layer, respectively. Lastly, ‘OI1′ and ‘OO1′ represent the input and output of the neuron in the output layer, respectively. W1ij is the connecting weight between the ith neuron in the input layer and jth neuron in the hidden layer. In the same way, W2j1 is the connecting weight between the jth neuron in the hidden layer and the neuron in the output layer. BHj is the bias value, added to the jth neuron in the hidden layer. Similarly, BO1 is the bias value, added to the neuron in the output layer. The constant slope in the input layer, transfer function co-efficient in the hidden and output layer are denoted by ‘Is’, ‘Ht’, and ‘Ot’, respectively. The forward 7

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K. Samal and S. Ghosh

In order to minimize the prediction error corresponding to pth training scenario in the tth iteration of BPA, the change in connecting weight between jth neuron of hidden layer and the output neuron is calculated as follows:

[ W j21]p (t ) =

E Wj21

Lr

(t ) + Mf [ W j21]p (t

1)

(16)

p

where

E W j21

E OO1 OI 1 × × OO1 OI 1 W j21

(t ) = p

= [ (TO1

(t ) p

OO1) × {Ot × OO1 × (1

OO1)} × (HOj )]p (t ) (17)

Now, the average change of the same connecting weight for the whole scenarios in a batch mode of training is determined like the following.

1 P

[ W j21 ]avg (t ) =

P

[[ W j21 ]p (t )] (for the batch mode of training ) p=1

(18) Finally, the updated connecting weight between the jth hidden neuron and the output neuron in the tth iteration of BPA using incremental training is calculated as follows:

[Wj21 ]1 (t ) = [Wj21 ]P (t [W 2j1 ]p (t )

=

1) + [ Wj21 ]1 (t )

[Wj21](p 1) (t )

+ [ Wj21 ]p (t )

for the incremental mode of training

(19)

th

Finally, the connecting weight between the j hidden neuron and the output neuron after the tth iteration of BPA using batch training is updated as follows:

Fig. 4. The (a) structural framework of the proposed MLFFNN adopted in the present methodology, (b) flowchart of the purposed CI-based hybrid methodology (GA-tuned MLFFNN).

W 2j1 (t ) = Wj21 (t

(20) Similarly, the connecting weight between ith input neuron (in input layer) and jth hidden neuron (in hidden layer) for the pth training scenario in the tth iteration of BPA is determined as follows:

calculations of the purposed MLFFNN are described below (9)

IOi = Is × IIi

1) + [ Wj21 ]avg (t )(for the batch mode of training )

n

(Wij1 × IOi )

HIj =

[Wij1 ]p (t ) =

(10)

i=1

HOj =

1 1+e

[(Ht ) × (HIj + BHj )]

(for log

sigmoid transfer function in hidden

e[(Ht ) × (HIj + BHj)] e e[(Ht ) × (HIj + BHj)] + e

[(Ht ) × (HIj + BHj )] [(Ht ) × (HIj + BHj )]

function in hidden layer )

(for tan

sigmoid transfer

E Wij1

(12)

(W 2j1 × HOj ) j =1

OO1 =

1 1+e

[(Ot ) × (OI 1+ BO1)]

(13)

E Wij1

(14)

The error in prediction is determined as follows (Pratihar, 2008; Ghosh et al., 2012)

E=

1 (TO1 2

OO1 )2

(21)

HOj HIj E OO1 OI 1 × × × × OO1 OI 1 HOj HIj Wij1

(t ) = p

(t ) p

(TO1

(t ) =

OO1) × {Ot × OO1 × (1

(Wj21) × {Ht × HOj × (1

p

OO1)}×

HOj )} × (IOi )

(t ) p

(23)

And for tan-sigmoid transfer function in the hidden layer, it is calculated as follows:

N

OI 1 =

1)

p

Therefore, for log-sigmoid transfer function in the hidden layer, it is calculated as follows:

HOj =

(t ) + Mf [ Wij1 ]p (t

(22)

(11)

layer )

E Wij1

where,

E Wij1

Lr

(t ) = [ (TO1 p

×

OO1) × {Ot × OO1 × (1

(W j21)

× {Ht × (1

(HOj

)2)}

OO1)} (t )

× (IOi )]p

(24)

The average change of the same connecting weight for the whole scenarios in batch training is calculated as

(15)

[ Wij1 ]avg (t ) =

A Back-Propagation Algorithm (BPA) is implemented to optimize (update) the connecting weights of the networks by minimizing the deviation in predictions, as given below.

1 P

P

[[ Wij1 ]p (t )](for the batch mode of training ) p=1

(25) th

Finally, the updated connecting weight between the i input neuron 8

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K. Samal and S. Ghosh

(in the input layer) and jth hidden neuron (in hidden layer) in the tth iteration of BPA using incremental training is calculated as follows:

[Wij1 ]1 (t ) = [Wij1 ]P (t [Wij1 ]p (t )

=

1) + [ Wij1 ]1 (t )

[Wij1](p 1) (t )

+ [ Wij1 ]p (t ) th

for the incremental mode of training

of the corresponding GA-string. The error in prediction is calculated as the difference between the assigned and MLFFNN-predicted value of the indicator (of the flow-situations). The average percentage error in prediction can be calculated as given below.

(26)

Average percentage error =

th

Finally, the weight between i input neuron and j hidden neuron after the tth iteration of BPA using batch training is updated as follows:

Wij1 (t ) = Wij1 (t

The convergence of the BPA is achieved when the difference in error in prediction between two successive iterations reaches below a sufficiently small magnitude: 10-4. Apart from the connecting weights, performance of the MLFFNN (consequently the accuracy in prediction capability) also depends on some other NN-parameters like: Learning rate (Lr), Momentum factor (Mf), Constant slope in the input layer (Is), Transfer function co-efficient in the hidden (Ht) and output (Ot) layer, and the number of neuron in the hidden layer (N). Therefore, apart from the optimization of connecting weights using BPA, it is also highly required to optimize those above-mentioned NN-parameters (Lr, Mf, Is, Ht, Ot, & N) for an accurate mathematical structure of the MLFFNN to capture the complex nonlinear relation correctly and consequently to predict the flow-conditions/situations (in hot-leg) perfectly. Here, these performance parameters of NN are simultaneously optimized using a GA. The GA started with several (population size) randomly generated initial possible combinations of the performance parameters (of NN) coded in the GAstrings. The first five NN-parameters (Lr, Mf, Is, Ht, and Ot) are coded in the GA-strings using binary code (Ghosh et al., 2012; Ghosh et al., 2013) but the last variable (N) is coded differently (Ghosh et al., 2011). The field size for the first five parameters (Lr, Mf, Is, Ht, and Ot) is chosen as 14 and for the last parameter (N), the field size is chosen as 30. Thus, a particular GA-string will look as follows:

Using the binary coding, the 1st 14 bits are assigned to Lr, 2nd 14 bits are assigned to Mf, 3rd 14 bits are assigned to Is, 4th 14 bits are assigned to Ht, and 5th 14 bits are assigned to Ot. However, the last 30 bits are assigned to N differently. At first, the GA-strings are decoded, and the real-values of the performance parameters are extracted. The first 14 bits, that is, 10110110111110 yield the decoded value (DV) as 1×213 + 0×212 + 1×211 + 1×210 + 0×29 + 1×28 + 1×27 + 0×26 + 1×25 + 1×24 + 1×23 + 1×22 + 1×21 + 0×20 = 11710. The real value of Lr is calculated using the linear mapping rule as shown below.

Lr max Lr min × DV of Lr 2fieldsize 1

P p=1

|TO1p

OO1p |

TO1p

× 100

(29)

Here, P represents the numbers of training or test scenarios. TO1p indicates the target output (the assigned numerical value of the indicator) corresponding to pth scenario and OO1p denotes the network predicted value for the same. After calculating the fitness value of all the GA-strings in this way, the reproduction operation is started. The better solutions from the present population pool of the current generation are chosen for the next generation using the reproduction operator. Reproduction operation is started here using the tournament selection operator (Ghosh et al., 2011, 2012, 2013) to choose the best solutions for the next generation. After reproduction, crossover operation is started using the uniform-crossover operator with 99% probability to exchange qualities between the parent solutions. Uniform crossover is employed here, as it is generally found to perform better than both single as well as twopoint crossovers. To import a certain amount of diversity and to overcome the problem of stacking in the local solution space (or to get a globally optimized solution) mutation operation with a very small probability is performed. After the crossover operation, the calculation moves to the next generation. The above basic operations/steps are repeated for a large number of generations, so that globally optimized solutions can be achieved. The convergence of the GA is assumed to be achieved if the fitness value (error in prediction in %) is not changed for 50 consecutive generations. To get an optimum performance, GAparameters, namely, population size (Ps), tournament size (Ts), mutation probability (Mp), and number of generations (Ng) are also optimized using parametric study for each of the cases (Cases-A–D in sub-

1) + [ Wij1 ]avg (t )(for the batch mode of training ) (27)

Lr = Lr min +

1 P

section 3.3.1 and the case in sub-section 3.3.2) separately. The optimized values of these GA-parameters obtained through the parametric study are mentioned in the result and discussion (sub-section 4.7). Both MLFFNN with BPA and GA are indigenously coded using ‘C’ programming language in the Linux atmosphere. MLFFNN with BPA is coupled with GA in a common monolithic platform using some specially designed system commands using ‘C’ programming language so that the data transfer between GA and MLFFNN with BPA can take place in a fully automatic way without any human interface. The schematic of the combined GA tuned MLFFNN (coupled with BPA) architecture is shown in Fig. 4b.

(28)

4. Results and discussions

where Lrmin and Lrmax are the lower and upper bounds of Lr, respectively. The real values of the next four parameters (Mf, Is, Ht, and Ot) are calculated exactly in the same way. For the last parameter (N), the sum of the digits/bits (Ghosh et al., 2011) assigned to it plus one decides the real integer value of it. In the above GA-string, the assigned bits for the last parameter (N) are 000010111001011000000001011101. Thus, the real integer value of this last parameter (N) is calculated as [0 + 0 + 0 +0+1+0+1+1+1+0+0+1+0+1+1+0+0+0+ 0 + 0 + 0 + 0 + 0 + 1 + 0 + 1 + 1 + 1 + 0 + 1] + 1 = 13. Immediately after getting the real values of these NN-parameters from the GA-string, the network (MLFFNN) is formed and the BPA is started. Ultimately, after a sufficiently large number of BPA-iterations, the network-weights are optimized. Now, the percentage error in prediction (based on the calculated numerical values) using this network with optimized weights is calculated and is assigned as the fitness value

The obtained results using the developed methodologies are categorically presented in this section. Important findings generated through the discussion on obtained results are also mentioned in the respective sub-sections. Before generating any results, at first, proper suitable size of grids is chosen through the grid-independent test. Then the adopted numerical methodology is validated using results available in the literature. After that, detailed hydrodynamics in PWR during LOCA captured through present numerical simulations are discussed. Then the effect of different working conditions (physical situations) namely gas-flowrate, water-flowrate, and stored water-level in HPHL on the flow-condition (occurrence or absence of plugging in the hot-leg during LOCA) are rigorously investigated. At last, performances of the developed computational intelligence-based methodologies to capture 9

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be assumed that the adopted numerical methodology is able to capture the said hydrodynamics. 4.3. Hydrodynamics in the hot-leg during LOCA Several flow-structures may be developed in the hot-leg of PWR for different working conditions (corresponding to different combinations of the ṁa, ṁw, and hw) during LOCA. Mainly three types of flow-structures are found in the hot-leg. Generally, smooth-stratified flow-structures are observed in the hot-leg when hw is very small. It may also occur at a moderate value of hw provided that ṁa is not large. This flow may be found even for a very large value of hw provided that both ṁa and ṁw are not large. For example, when the working condition is chosen as ṁa = 0.268 kg/s, ṁw = 1.0 kg/s, and hw = 0.03 m, smoothstratified flow pattern is observed in the HPHL as can be seen in Fig. 7a. This figure shows the spatial distributions of phases in the flow domain at different instants. These phase distributions may help to visualize and understand the complex hydrodynamics of CCTPF during LOCA in the hot-leg of the PWR. Here, the height of the individual layers remains unchanged along the lengthwise direction of the HPHL. A stable stratified flow-structure (phase distribution does not vary with time) is obtained by this working condition (this combination of ṁa, ṁw, and hw), and consequently PD and VF across the flow domain do not vary with time as can be seen in Fig. 7b and c, respectively. These figures show that the fluctuations/oscillations are almost absent in the time variation of PD and VF across the flow domain. Generally, wavy stratified flow-structures are found in the hot-leg when ṁa is small but ṁw or hw is high. It may also be observed when ṁa is small and hw is moderate. In this situation, flow-structure remains stratified but, the height of the individual fluid layers no longer remains constant along the lengthwise direction of hot-leg. Flow-structure no longer remains stable and becomes wavy stratified. For example, when ṁa = 0.268 kg/s, ṁw = 5.0 kg/s, and hw = 0.1 m, wavy stratified flowstructures are found in the hot-leg. Water from SG, flows more freely and roughly through IPHL and HPHL when gas-flowrate is smaller compared to that in the previous case (during smooth-stratified flow). Gas with low flowrate is not able to suppress the rough behavior of water with relatively high flowrate, subsequently, an undulation at the interface is found (unlike smooth-stratified flow) and ultimately wavy stratified flow-structures are observed in the hot-leg as can be seen in Fig. 8a. As the gas-flowrate is lesser than that in the previous case (during smooth-stratified flow) water coming from SG via IPHL strikes the water in HPHL with relatively higher momentum. Consequently, hydraulic jumps may be observed at the connecting zone as they are not suppressed by the gas with a relatively smaller flowrate (as can be seen in Fig. 8a). This hydraulic jump in the elbow and wavy nature of the interface in the HPHL are absent during smooth-stratified flow due to suppression by high gas-flowrate (as can be seen from Fig. 7a). As the flow-structure is not stable (wavy stratified), the time variation of PD and VF across the flow domain contain a noticeable amount of fluctuations/oscillations as can be seen in Fig. 8b and 8c, respectively. For any value of water-flowrate and stored water-level in HPHL, when gas-flowrate is very high flow-structure no longer remains stratified (smooth-stratified or wavy stratified). Plugging/blocking is observed in the hot-leg for high gas-flowrate. Fig. 9 shows one of these plugging situations in the hot-leg when ṁa = 2.144 kg/s, ṁw = 1.0 kg/ s, and hw = 0.1 m. High gas-flowrate (towards SG) generates waves in the water stored in HPHL and forces the generated waves to move towards IPHL. All these waves will merge together at the elbow and consequently, the effective wave-heights become very high and subsequently the elbow will be flooded by the water. Due to this, the gas flow passage will be restricted, and gas pressure in the elbow will continuously rise. After some time, when the amount of gas pressure reaches a critical limit, gas will be accelerated and blow up the liquid and consequently plugging/blocking occurs in the hot-leg. In this situation, water from the SG unable to flow towards RPV. At this point,

Fig. 5. Variation of the numerically calculated PD and VF across the flow domain at 5 s with grid size (considering ṁa = 0.268 kg/s, ṁw = 1.0 kg/s, and hw = 0.03 m).

the said relationship and to predict the flow-condition are evaluated and presented. 4.1. Grid-independent test A grid-independent test is performed to ensure that the results obtained from the numerical investigations is free from the influence of grid size. Grid-independent results are represented here by considering ṁa as 0.268 kg/s, ṁw as 1.0 kg/s, and hw as 0.03 m. Different sizes of the grid are considered for the numerical investigation. For each size of grids, boundary conditions and initial conditions are kept the same, and numerical calculation is made up to 5 s. At this instance of time (5 s), PD and VF across the flow domain are evaluated for each of the cases (with different grid sizes) and ultimately the variation of PD and VF with grid size are plotted as shown in Fig. 5. From this figure, it is cleared that the variations of PD and VF with grid size (within the chosen range of grid size) are negligible. It indicates that the results are independent of grid size within the chosen range of size of the grid. A grid size of 0.01 m is selected through this grid-independent test to generate numerical results. 4.2. Validation At first, the problem described in Deendarlianto et al. (2011) with ṁw = 0.3 kg/s, ṁa = 0.181 kg/s, is simulated using the presently adopted numerical methodology (as described in sub-section 3.1) and the obtained result (in terms of the spatial distribution of the phases in the hot-leg) is compared with that (experimentally obtained) available in Deendarlianto et al. (2011) as shown in Fig. 6a. A good agreement between them is found. Violent flow-conditions are experimentally found by Deendarlianto et al. (2008) when the mass flow combinations of the phases were chosen as ṁw = 0.3 kg/s, ṁa = 0.34 kg/s or ṁw = 0.3 kg/s, ṁa = 0.32 kg/s. Both these flow-situations are simulated using the presently adopted numerical methodology and compared with the respective experimental results available in Deendarlianto et al. (2008) as shown in Fig. 6(b) and 6(c), respectively. In both the cases, sufficient matching between the results obtained using the present methodology and that available in Deendarlianto et al. (2008), are found. Again, the distribution of water-level inside the hot-leg for the water mass flowrate of 0.3 kg/s and air superficial velocity of 9.71 m/s is evaluated using the presently adopted numerical methodology and the obtained distribution of water-level in hot-leg is compared with that available (numerically evaluated) in Deendarlianto et al. (2011) as shown in Fig. 6(d). Good agreement is found between the results obtained using the present methodology and that available in Deendarlianto et al. (2011) as can be seen in Fig. 6(d). Therefore, it can 10

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K. Samal and S. Ghosh

Fig. 6. Validation of the presently adopted numerical methodology: (a) phase distribution as available in Deendarlianto et al. (2011) and as obtained using present numerical model (ṁw = 0.3 kg/s, ṁa = 0.181 kg/s), (b) phase distribution as available in Deendarlianto et al. (2008) and as obtained using present numerical model (ṁw = 0.3 kg/s, ṁa = 0.34 kg/s), (c) phase distribution as available in Deendarlianto et al. (2008) and as obtained using present numerical model (ṁw = 0.3 kg/s, ṁa = 0.32 kg/s), and (d) the distribution of waterlevel inside the hot-leg as available in Deendarlianto et al. (2011) and as evaluated using the present numerical model.

entrainment of the pressurized gas into water will be started at the elbow, IPHL, and SG. Consequently, air bubbles are formed within the water. In this situation (during plugging), the time variation of PD and VF contains a sufficiently large amount of fluctuations and oscillations due to the above-mentioned activities as can be seen in Fig. 9b and 9c, respectively. Therefore, it is cleared that the occurrence of plugging/ blocking can be identified by the huge oscillations present in the time

variation of PD and VF. Plugging may rarely occur at moderate gas flowrate when the water flowrate and stored water-level both are very high. 4.4. Effect of gas-flowrate The 11

effect

of

gas-flowrate

on

the

said

flow-condition

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Fig. 7. For ṁa = 0.268 kg/s, ṁw = 1.0 kg/s, and hw = 0.03 m, (a) phase-distributions in the flow domain at subsequent time-instants, and the time variation of (b) PD, and (c) VF across the flow domain.

Fig. 8. For ṁa = 0.268 kg/s, ṁw = 5.0 kg/s, and hw = 0.10 m, (a) phase-distributions in the flow domain at subsequent time-instants, and the time variation of (b) PD, and (c) VF across the flow domain.

(hydrodynamics) in hot-leg during LOCA is captured at 9 different fixed combinations of ṁw and hw. The 9 different combinations of ṁw and hw are made by considering 3 different values of ṁw and 3 different values of hw. At each of the 9 fixed combinations of ṁw and hw, numerical

results are extracted for 3 different values of ṁa. Therefore, a total of 27 (3 × 3 × 3) numerical results are considered to evaluate the effect of ṁa. At a fixed combination of ṁw = 5 kg/s and hw = 0.1 m, the effect of ṁa on the hydrodynamics in terms of phase distributions at different

12

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Fig. 9. For ṁa = 2.144 kg/s, ṁw = 1.0 kg/s, and hw = 0.10 m, (a) phase-distributions in the flow domain at subsequent time-instants, and the time variation of (b) PD, and (c) VF across the flow domain.

Fig. 10. Considering ṁw as 5.0 kg/s and hw as 0.10 m, phase-distributions at subsequent time instants when (a) ṁa = 0.268 kg/s, (b) ṁa = 1.072 kg/s, and (c) ṁa = 2.144 kg/s.

instants are shown in Fig. 10. The effect of ṁa in terms of time variation of PD across the flow domain is shown in Fig. 11 for 3 different fixed combinations of ṁw and hw. In this figure, hw is considered as 0.03 m for all the combinations. In the same way, the effect of ṁa for 3 different fixed combinations of ṁw and hw at hw = 0.06 and 0.10 m are shown in Figs. 12 and 13, respectively. In all these figures (Figs. 11–13), the time variation of PD with rapid oscillations indicates the occurrence of plugging and smooth oscillation-free time variation of PD signifies the

absence of plugging (as concluded in the previous sub-section). By analyzing the Figs. 11–13, it is found that the flow-condition (occurrence or absence of plugging) in the hot-leg strongly depends on the gas-flowrate. Hot-leg is generally free from plugging (stratified flow is observed) when gas-flowrate is not large. Plugging may rarely be initiated in moderate gas-flowrate only when water-flowrate and stored water-level, both are very high. For a given condition in terms of ṁw and hw, plugging started when gas-flowrate reaches a threshold limit.

13

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Fig. 11. Time variations of PD across the flow domain for different ṁa (0.268, 1.072, and 2.144 kg/s) when hw = 0.03 m and (a) ṁw = 1.0 kg/s, (b) ṁw = 3.0 kg/s, (c) ṁw = 5.0 kg/s.

Fig. 12. Time variations of PD across the flow domain for different ṁa (0.268, 1.072, and 2.144 kg/s) when hw = 0.06 m and (a) ṁw = 1.0 kg/s, (b) ṁw = 3.0 kg/s, (c) ṁw = 5.0 kg/s.

This threshold limit depends on the operating condition in terms of the combination of ṁw and hw. The nature and behavior of fluctuation and oscillation in the time variation of PD and VF are very much responsive to the gas-flowrate. However, the nature and amount of responsivity strongly depend on the combination of the other two physical parameters: ṁw and hw. When hw is kept fixed at a very small value, the responsivity of PD fluctuation to the gas-flowrate increases with the ṁw (as can be seen in Fig. 11). On the other hand, when the hw is kept fixed at a very high value, the responsivity of PD fluctuation to the gasflowrate decreases with increment in ṁw (as shown in Fig. 13). Here in all these figures, the amount of responsivity of the PD fluctuations to the gas-flowrate is measured in terms of the amplitude of fluctuations. Therefore, it is cleared that the way in which the responsivity of the PD fluctuation (to the gas-flowrate) changes with ṁw is reversed when hw changes from very lower to a higher value. Therefore, a critical value of hw may exist where this responsiveness is reversed. The amount of responsivity of PD fluctuation to the gas-flowrate not much changes with ṁw, when hw is chosen very close to the said critical value (as can be seen in Fig. 12). Almost the same observations are found when the effect of gas-flowrate on the said hydrodynamics in terms of time variation of VF is studied.

4.5. Effect of Water-flowrate The effect of water-flowrate on the flow-condition (hydrodynamics) in hot-leg during LOCA is extracted in the same way at 9 different fixed combinations of ṁa and hw. The effect of water-flowrate is captured by considering 3 different values of ṁw at each of the 9 fixed combinations of ṁa and hw. The 9 different combinations of ṁa and hw are chosen by considering 3 different values of ṁa and 3 different values of hw. Here also, the effect of ṁw is extracted using the time variation of PD and VF across the flow domain. Fig. 14 shows the effect of ṁw for 3 different fixed combinations of ṁa and hw, where hw is considered as 0.03 m for all the three combinations. Similarly, the effect of ṁw for 3 fixed combinations of ṁa and hw, at hw = 0.06 and 0.10 m are shown in Figs. 15 and 16, respectively. From these figures, it can be easily concluded that the PD fluctuation and oscillation not much depend on the water-flowrate when both ṁa and hw are low. At a lower and moderate value of hw (e.g. 0.03 and 0.06 m), plugging occurs only when ṁa is sufficiently large (~2.144 kg/s) and at these situations, PD fluctuations are sufficiently responsive to ṁw (as can be seen in Fig. 14c and 15c). At the higher value of hw (~0.1 m), the PD fluctuations start to become sensitive to ṁw even at moderate value of ṁa (near 1.072 kg/s as can be

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Fig. 13. Time variations of PD across the flow domain for different ṁa (0.268, 1.072, and 2.144 kg/s) when hw = 0.10 m and (a) ṁw = 1.0 kg/s, (b) ṁw = 3.0 kg/s, (c) ṁw = 5.0 kg/s.

Fig. 14. Time variations of PD across the flow domain with different ṁw (1, 3, and 5 kg/s) when hw = 0.03 m and (a) ṁa = 0.268 kg/s, (b) ṁa = 1.072 kg/s, (c) ṁa = 2.144 kg/s.

seen in Fig. 16b) and become very sensitive to ṁw when ṁa is sufficiently large (near to 2.144 kg/s as shown in Fig. 16c). Almost the same observations are found when the effect of ṁw on the said hydrodynamics in terms of time variation of VF is studied.

fluctuation) is almost independent of hw as can be seen in Fig. 18a–b. For a combination of moderate ṁa (~1.072 kg/s) and large ṁw, VF across flow domain contains fluctuation only when hw is sufficiently large and it (fluctuation) is also observed after a long interval from start. Therefore, for a moderate value ṁa, the plugging occurs only when ṁw is sufficiently large and at that situation, flow-condition and VF fluctuations started to become responsive to the hw as can be seen in Fig. 18c. For a very high value of ṁa, plugging is always found in the hot-leg and the time variation of VF is obtained as fluctuating in nature for all values of hw as can be seen in Fig. 19. Almost the same observations are found when the effect of hw on the said hydrodynamics in terms of time variation of PD is deduced. Again, for very high value of ṁa, fluctuation in the time variation of PD and VF is started quickly (at a very smaller instance of time) if hw is very large and it is started with a delay (comparatively at a large instance of time) if hw is very small as can be seen in Fig. 19. That is, for the very high value of ṁa, plugging initiated quickly (rapidly) if hw is very large and it is slowly appeared if hw is very small.

4.6. Effect of stored (initial) water-level at HPHL Using PD and VF fluctuations across the flow domain, the effect of stored water-level (in HPHL) on the said flow-condition during LOCA is acquired under 9 different operating conditions by considering 3 different values of hw at each of the 9 different operating conditions. The 9 operating conditions are found by 9 different fixed combinations of ṁa and ṁw (by considering 3 different values of ṁa and 3 different values of ṁw). The effect of hw for the first 3 fixed combinations of ṁa and ṁw are shown in Fig. 17. In the same way, Figs. 18 and 19 show the said effect of hw for the second 3 and last 3 combinations of ṁa and ṁw. From these figures, it is clearly observed that the time variations of VF are smooth, and plugging does not occur when ṁa is very small. At this smaller value of ṁa (~0.268 kg/s), the VF fluctuations are almost independent of hw as can be seen in Fig. 17. Even, when ṁa reaches a moderate value (~1.072 kg/s), time variations of VF across the flow domain are free from oscillations if ṁw is not so high and flow-conditions (nature of

4.7. Prediction of flow-condition using computational intelligence As described in section ‘3′, the GA (evolutionary algorithm) tuned

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Fig. 15. Time variations of PD across the flow domain with different ṁw (1, 3, and 5 kg/s) when hw = 0.06 m and (a) ṁa = 0.268 kg/s, (b) ṁa = 1.072 kg/s, (c) ṁa = 2.144 kg/s.

Fig. 16. Time variations of PD across the flow domain with different ṁw (1, 3, and 5 kg/s) when hw = 0.10 m and (a) ṁa = 0.268 kg/s, (b) ṁa = 1.072 kg/s, (c) ṁa = 2.144 kg/s.

MLFFNN (neuro biological scheme) is used to predict the flow-condition (occurrence or absence of plugging) in the hot-leg during LOCA. At first, the performance of the developed GA tuned MLFFNN considering the Case-A (as described in section ‘3′) is analyzed. During optimization of the performance parameters of MLFFNN using GA, the variation of the error in prediction (in % based on the calculated numerical value) obtained from the optimized MLFFNN (using BPA optimized weights and GA tuned performance parameters) with the number of GA-generation is shown in Fig. 20a. From this figure, it is clearly observed that the error in prediction decreases with increment in the number of GAgeneration. After a certain number of generations, it is reduced below a sufficiently small value and then remains almost constant in this small value for the rest of the generations. It indicates the achievement of the mature result. After getting the optimized set of performance parameters of MLFFNN (Lr, Mf, Is, Ht, Ot, and N) from GA, the variation of error in prediction (in % based on the calculated numerical value) with BPA-iteration using incremental and batch mode of training (during optimization of weight) is shown in Fig. 20b. This figure shows that the error in prediction continuously decreases with the number of BPiteration. The performance of the developed MLFFNN for Case-A are tested in two ways by considering a set of 4 and 6 number of test data

separately. The Tables 1a, 1b, 1c, 1d, 1e, and 1f show the training data, test data, optimized GA-parameters (obtained through GA-parametric study), optimized NN-parameters (obtained from GA), performance of the developed GA tuned MLFFNN using the incremental training and that using the batch training, respectively, when Case-A with 4 number of test data (Case-A1) are considered. It is already mentioned (in section ‘3′) that to create the training and test data, a value of 0.5 is assigned as the output of the output-neuron to indicate the occurrence of plugging and a value of 1.0 is considered as the output of the same to represent the absence of plugging. Numerical values of the target-outputs for the 4 test data are 1.0, 1.0, 0.5, and 0.5 and the numerical values of the respective predicted (by the GA tuned MLFFNN using the incremental mode of training) outputs are 0.99606, 0.99969, 0.50426, and 0.49804 as shown in Table 1e. Therefore, errors in prediction (in percentage) based on the calculated numerical values are found as 0.39, 0.03, 0.85, and 0.39 for the 1st, 2nd, 3rd, and 4th scenario of test data, respectively as shown in Table 1e. The average error in prediction considering all these 4 test outputs is found as 0.42% as shown in Table 1e. The maximum deviation of the predicted value from the corresponding assigned value of output is obtained for the 3rd test data and that deviation is 0.00426. To convert the predicted value of output (obtained

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Fig. 17. Time variations of VF across the flow domain with different hw (0.03, 0.06, and 0.10 m) when ṁa = 0.268 kg/s and (a) ṁw = 1 kg/s, (b) ṁw = 3 kg/s, (c) ṁw = 5 kg/s.

Fig. 18. Time variations of VF across the flow domain with different hw (0.03, 0.06, and 0.10 m) when ṁa = 1.072 kg/s and (a) ṁw = 1 kg/s, (b) ṁw = 3 kg/s, (c) ṁw = 5 kg/s.

for each scenario of test data from the GA tuned MLFFNN) into the predicted flow-structure, range for the predicted value can be considered as 0.25 to 0.74 for the occurrence of plugging and 0.75 to 1.25 for the absence of plugging. As per that, all the predicted flow-structures absolutely match with the respective target flow-structures and thereby, the error in prediction based on the predicted flow-structure is 0% for each of the 4 outputs as shown in Table 1e. Similarly, Table 1f shows the performance of the GA tuned MLFFNN using the batch mode of training. Exactly in the same way, the training data, test data, optimized GA-parameters (obtained through GA-parametric study), optimized NN-parameters (obtained from GA), and performance of the developed GA tuned MLFFNN using the incremental and batch training are shown in Tables 2a, 2b, 2c, 2d, 2e, and 2f, respectively when Case-A using a test set of 6 data (Case-A2) are considered. From Tables 1a–f and 2a–f, it is observed that the flow-condition (occurrence or absence of plugging) in hot-leg is perfectly reflected by the Me and Sd of VF fluctuation across the flow domain. It can also be concluded that the developed GA tuned MLFFNN is successfully able to predict the flowcondition in the hot-leg by knowing Me and Sd extracted from the time variation of VF across the flow domain. Similarly, Tables 3a–f, 4a–f, and

5a–f show the training data, test data, optimized GA-parameters (obtained through GA-parametric study), optimized NN-parameters (obtained from GA), performance of the developed GA tuned MLFFNN using the incremental training and that using the batch training for the Case-B, Case-C, and Case-D (as described in section ‘3′), respectively. From the Tables 3a–f, it is cleared that the combination of Me and Sd of PD fluctuation perfectly bears the signature of the flow-conditions in hot-leg. These tables also indicate that the relationship between the flow-conditions and those statistical parameters (Me and Sd of PD fluctuation) is perfectly captured by the developed GA tuned MLFFNN. From Tables 4a–f and 5a–f, it is found that the flow-condition in hot-leg during LOCA can be perfectly identified either by 2 statistical parameters (namely Me and Sd) or by 4 statistical parameters (namely Me, Sd, Sk, and Ku) extracted from the time variation of both PD and VF across the flow domain using GA tuned MLFFNN. In the same way, the training data, test data, optimized GA-parameters (obtained through GA-parametric study), optimized NN-parameters (obtained from GA), as well as the performances of said GA tuned MLFFNN considering incremental and batch training using working conditions (in terms of ṁa, ṁw, and hw) are shown in Tables 6a,

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performance parameter, a variable, known as ‘prediction capability’, is defined as:

Prediction capability (in%) = 100

Percentage error in prediction (30)

Using this performance parameter, the said comparison is made by considering incremental and batch mode of training separately as shown in Fig. 21. Performances of all the cases are described in detail in Table 7. Through the incremental mode of training, the best performance is found using Case-D, whereas, the best performance is obtained using Case-C for the batch mode of training. The best performances in terms of the average prediction capability are obtained as 99.96% and 99.87% using incremental and batch mode of training, respectively. 5. Conclusions An attempt is made to characterize and categorize the flow-conditions (flow-structures) in the hot-leg of a PWR during LOCA in terms of fluctuating-nature of the time variations of VF and PD across the flow domain. Dynamics of the flow-structure of CCTPF in the hot-leg of a PWR during LOCA is numerically simulated using the finite volume-based volume of fluid (VOF) model. Effect of individual fluid flowrates and the level of stored (initial) water (in HPHL) on the flow-condition (occurrence or absence of plugging/blocking) in hot-leg during LOCA are extensively investigated under different physical situations or working conditions. Flow-conditions in the hot-leg are parameterized and categorized in terms of the statistical parameters extracted from the time variation of PD and VF across the flow domain. Ultimately, by adopting neuro-biological schemes with the evolutionary-based algorithm, hybrid methodology (GA tuned MLFFNN) using computational intelligence (CI) is developed to predict the flow-condition (plugging or non-plugging) in hot-leg by knowing those statistical parameters. Hybrid methodology (GA tuned MLFFNN) is also developed to correlate the flow-condition in hot-leg with physical situations (working conditions) existing in different parts of the PWR system. To check the versatility of the developed methodologies, their performances are tested by considering different sets of inputs. Important findings obtained through the present numerical and computational study are pointwise mentioned below.

Fig. 19. Time variations of VF across the flow domain with different hw (0.03, 0.06, and 0.10 m) when ṁa = 2.144 kg/s and (a) ṁw = 1 kg/s, (b) ṁw = 3 kg/s, (c) ṁw = 5 kg/s.

• Different CCTPF-structures are found in the hot-leg of the PWR

6b, 6c, 6d, 6e, and 6f, respectively. From Tables 6a–f, it is found that the predicted flow-structures perfectly matches with the target flowstructures. Therefore, the dependency of the flow-condition (in the hotleg during LOCA) on the working condition (in terms of ṁa, ṁw, and hw) is perfectly captured by the developed approaches. The relation between the flow-condition and the working condition is perfectly grasped by the developed GA tuned MLFFNN. As described earlier (section ‘3′), different sets of inputs are used to develop the methodology (GA tuned MLFFNN). The performance of the developed methodologies with different sets of inputs is compared. As a

•

system during LOCA. Smooth-stratified flow-structures are observed when hw is very small. It may also be found at the moderate value of hw provided that ṁa is not large. This flow-structure may occur even for a very high value of hw only if ṁa and ṁw are not large. When ṁa is small but ṁw or hw is high or when ṁa is small and hw is moderate, flow-structure in hot-leg remains stratified but, the height of the individual fluid layers no longer remains constant along the lengthwise direction as well as the flow-structure no longer remains stable. It is called wavy stratified flow-structure. Hydraulic jumps

Fig. 20. Variation of error in prediction (in % based on calculated numerical value) with the (a) number of GA-generation (considering Case-A1 with the batch mode of training) and (b) BPA iteration number (considering Case-A1 with the batch and incremental mode of training).

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Table 1 (a) Training data, (b) test data, (c) optimized GA-parameters, (d) optimized NN-parameters, and performance of the developed GA-tuned MLFFNN using (e) incremental, and (f) batch mode of training considering Case-A1 (in section ‘3′). (a) Training data VF_Me

VF_Sd

Assigned value

Assigned flow-structure

0.92241 0.87764 0.85824 0.83518 0.81207 0.77568 0.75217 0.91353 0.88963 0.84741 0.82283 0.79755 0.78421 0.75910 0.73676 0.89631 0.88477 0.87591 0.87846 0.87336 0.86092 0.84730 0.86328

0.00701 0.03721 0.00912 0.02163 0.03622 0.02269 0.03716 0.00665 0.02221 0.00753 0.02251 0.03890 0.00881 0.02353 0.03709 0.01800 0.02489 0.02777 0.02890 0.02667 0.04367 0.04832 0.05402

1.0 0.5 1.0 1.0 0.5 1.0 0.5 1.0 1.0 1.0 1.0 0.5 1.0 1.0 0.5 1.0 1.0 0.5 1.0 1.0 1.0 0.5 0.5

Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Plugging Plugging

VF_Me

VF_Sd

Assigned value

Assigned flow-structure

0.89991 0.79898 0.86527 0.86484

0.02175 0.01117 0.03793 0.03405

1.0 1.0 0.5 0.5

Non-plugging Non-plugging Plugging Plugging

(b) Test data

(c) Optimized GA-parameters through parametric study

Ps Ts Mp Ng

Incremental training

Batch training

50 2 0.06 150

60 12 0.02 150

Incremental training

Batch training

0.005432 0.134041 2.145736 5.838894 0.609412 16

0.075627 0.187206 2.996814 9.992144 9.692419 10

(d) GA-optimized NN-parameters

Lr Mf Is Ht Ot N (e) Performance using test data (Considering incremental mode) Statistical parameter using numerical simulation

Target values & flow-structures

Predicted output values & flow-structures

VF_Me

Target value

Predicted value

Predicted flow-structure

0.99606 0.99969 0.50426 0.49804

Non-plugging Non-plugging Plugging Plugging

VF_Sd

Target flow-structure

0.89991 0.02175 1.0 Non-plugging 0.79898 0.01117 1.0 Non-plugging 0.86527 0.03793 0.5 Plugging 0.86484 0.03405 0.5 Plugging Average error in prediction Maximum deviation of the predicted numerical values from the assigned value: 0.00426

Error in prediction based on calculated numerical value (%)

Error in prediction based on predicted flow-structure (%)

0.39 0.03 0.85 0.39 0.42

0 0 0 0 0

(f) Performance using test data (Considering batch mode) Statistical parameter using numerical simulation

Target values & flow-structures

Predicted output values & flow-structures

VF_Me

Target value

Predicted value

Predicted flow-structure

0.98726 0.99972 0.50000 0.52546

Non-plugging Non-plugging Plugging Plugging

VF_Sd

Target flow-structure

0.89991 0.02175 1.0 Non-plugging 0.79898 0.01117 1.0 Non-plugging 0.86527 0.03793 0.5 Plugging 0.86484 0.03405 0.5 Plugging Average error in prediction Maximum deviation of the predicted numerical values from the assigned value: 0.02546

19

Error in prediction based on calculated numerical value (%)

Error in prediction based on predicted flow-structure (%)

1.27 0.03 0.00 5.09 1.60

0 0 0 0 0

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Table 2 (a) Training data, (b) test data, (c) optimized GA-parameters, (d) optimized NN-parameters, and performance of the developed GA-tuned MLFFNN using (e) incremental, and (f) batch mode of training considering Case-A2 (in section ‘3′). (a) Training data VF_Me

VF_Sd

Assigned value

Assigned flow-structure

0.92241 0.87764 0.85824 0.83518 0.81207 0.77568 0.75217 0.91353 0.88963 0.84741 0.82283 0.79755 0.78421 0.75910 0.88477 0.87591 0.87846 0.87336 0.86092 0.84730 0.86328

0.00701 0.03721 0.00912 0.02163 0.03622 0.02269 0.03716 0.00665 0.02221 0.00753 0.02251 0.03890 0.00881 0.02353 0.02489 0.02777 0.02890 0.02667 0.04367 0.04832 0.05402

1.0 0.5 1.0 1.0 0.5 1.0 0.5 1.0 1.0 1.0 1.0 0.5 1.0 1.0 1.0 0.5 1.0 1.0 1.0 0.5 0.5

Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Plugging Plugging

VF_Me

VF_Sd

Assigned value

Assigned flow-structure

0.89991 0.79898 0.86527 0.73676 0.89631 0.86484

0.02175 0.01117 0.03793 0.03709 0.01800 0.03405

1.0 1.0 0.5 0.5 1.0 0.5

Non-plugging Non-plugging Plugging Plugging Non-plugging Plugging

(b) Test data

(c) Optimized GA-parameters through parametric study

Ps Ts Mp Ng

Incremental training

Batch training

140 2 0.01 150

150 12 0.05 150

Incremental training

Batch training

0.003296 0.057254 2.931850 9.076048 9.989727 20

0.118538 0.218458 2.957871 9.923256 9.846512 12

(d) GA-optimized NN-parameters

Lr Mf Is Ht Ot N (e) Performance using test data (Considering incremental mode) Statistical parameter using numerical simulation

Target values & flow-structures

Predicted output values & flow-structures

VF_Me

Target value

Predicted value

Predicted flow-structure

0.99875 0.99828 0.49167 0.56459 0.99873 0.51354

Non-plugging Non-plugging Plugging Plugging Non-plugging Plugging

VF_Sd

Target flow-structure

0.89991 0.02175 1.0 Non-plugging 0.79898 0.01117 1.0 Non-plugging 0.86527 0.03793 0.5 Plugging 0.73676 0.03709 0.5 Plugging 0.89631 0.01800 1.0 Non-plugging 0.86484 0.03405 0.5 Plugging Average error in prediction Maximum deviation of the predicted numerical values from the assigned value: 0.06459

Error in prediction based on calculated numerical value (%)

Error in prediction based on predicted flow-structure (%)

0.13 0.17 1.67 12.92 0.13 2.71 2.96

0 0 0 0 0 0 0

Error in prediction based on calculated numerical value (%)

Error in prediction based on predicted flow-structure (%)

1.13 0.02 1.62 0.05 0.23 7.22

0 0 0 0 0 0

1.71

0

(f) Performance using test data (Considering batch mode) Statistical parameter using numerical simulation

Target values & flow-structures

Predicted output values & flow-structures

VF_Me

VF_Sd

Target value

Target flow-structure

Predicted value

Predicted flow-structure

0.89991 0.79898 0.86527 0.73676 0.89631 0.86484

0.02175 0.01117 0.03793 0.03709 0.01800 0.03405

1.0 1.0 0.5 0.5 1.0 0.5

Non-plugging Non-plugging Plugging Plugging Non-plugging Plugging

0.98868 0.99976 0.50809 0.49973 0.99766 0.53610

Non-plugging Non-plugging Plugging Plugging Non-plugging Plugging

Average error in prediction Maximum deviation of the predicted numerical values from the assigned value: 0.03610

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Table 3 (a) Training data, (b) test data, (c) optimized GA-parameters, (d) optimized NN-parameters, and performance of the developed GA-tuned MLFFNN using (e) incremental, and (f) batch mode of training considering Case-B (in section ‘3′). (a) Training Data PD_Me

PD_Sd

Assigned value

Assigned flow-structure

1337.3000 868.41590 1405.3000 1365.6000 592.11810 1360.0000 722.66620 1375.9000 1351.7000 1436.8000 1382.7000 777.36860 1469.7000 1349.4000 370.24540 1409.0000 1374.4000 666.86380 1464.1000 1377.6000 1488.6000 1212.2000 967.62200

11.44910 626.0038 14.41300 8.648500 1692.400 13.01070 1246.400 29.00510 22.42590 23.58120 9.305600 1238.500 26.95940 24.82160 1504.200 43.92490 26.57320 1174.900 31.50060 24.05960 29.25700 332.3991 1222.700

1.0 0.5 1.0 1.0 0.5 1.0 0.5 1.0 1.0 1.0 1.0 0.5 1.0 1.0 0.5 1.0 1.0 0.5 1.0 1.0 1.0 0.5 0.5

Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Plugging Plugging

(b) Test Data PD_Me

PD_Sd

Assigned value

Assigned flow-structure

1318.8000 1446.7000 −1116.000 444.99200

10.0206 23.4526 12004.0 2019.50

1.0 1.0 0.5 0.5

Non-plugging Non-plugging Plugging Plugging

(c) Optimized GA-parameters

Ps Ts Mp Ng

Incremental training

Batch training

80 2 0.07 150

110 4 0.01 150

Incremental training

Batch training

0.012330 0.028139 2.700317 9.140103 7.364109 18

0.236220 0.624245 2.776256 5.310749 8.657279 5

(d) GA-optimized NN-parameters

Lr Mf Is Ht Ot N (e) Performance using test data (Considering incremental mode) Statistical parameter using numerical simulation

Target values & flow-structures

Predicted output values & flow-structures

PD_Me

Target value

Predicted value

Predicted flow-structure

0.95217 0.95454 0.55610 0.48836

Non-plugging Non-plugging Plugging Plugging

PD_Sd

Target flow-structure

1318.8000 10.0206 1.0 Non-plugging 1446.7000 23.4526 1.0 Non-plugging −1116.000 12004.0 0.5 Plugging 444.99200 2019.50 0.5 Plugging Average error in prediction Maximum deviation of the predicted numerical values from the assigned value: 0.05610

Error in prediction based on calculated numerical value (%)

Error in prediction based on predicted flow-structure (%)

4.78 4.55 11.22 2.33 5.72

0 0 0 0 0

Error in prediction based on calculated numerical value (%)

Error in prediction based on predicted flow-structure (%)

9.68 8.00 4.58 5.41 6.92

0 0 0 0 0

(f) Performance using test data (Considering batch mode) Statistical parameter using numerical simulation

Target values & flow-structures

Predicted output values & flow-structures

PD_Me

Target value

Target flow-structure

Predicted value

Predicted flow-structure

1.0 1.0 0.5 0.5

Non-plugging Non-plugging Plugging Plugging

0.90323 0.92000 0.47712 0.47297

Non-plugging Non-plugging Plugging Plugging

PD_Sd

1318.8000 10.0206 1446.7000 23.4526 −1116.000 12004.0 444.99200 2019.50 Average error in prediction Maximum deviation of the predicted

numerical values from the assigned value: 0.09677

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Table 4 (a) Training data, (b) test data, (c) optimized GA-parameters, (d) optimized NN-parameters, and performance of the developed GA-tuned MLFFNN using (e) incremental, and (f) batch mode of training considering Case-C (in section ‘3′). (a) Training Data VF_Me

VF_Sd

PD_Me

PD_Sd

Assigned value

Assigned flow-structure

0.92241 0.87764 0.85824 0.83518 0.81207 0.77568 0.75217 0.91353 0.88963 0.84741 0.82283 0.79755 0.78421 0.75910 0.73676 0.89631 0.88477 0.87591 0.87846 0.87336 0.86092 0.84730 0.86328

0.00701 0.03721 0.00912 0.02163 0.03622 0.02269 0.03716 0.00665 0.02221 0.00753 0.02251 0.03890 0.00881 0.02353 0.03709 0.01800 0.02489 0.02777 0.02890 0.02667 0.04367 0.04832 0.05402

1337.300 868.4159 1405.300 1365.600 592.1181 1360.000 722.6662 1375.900 1351.700 1436.800 1382.700 777.3686 1469.700 1349.400 370.2454 1409.000 1374.400 666.8638 1464.100 1377.600 1488.600 1212.200 967.6220

11.44910 626.0038 14.41300 8.648500 1692.400 13.01070 1246.400 29.00510 22.42590 23.58120 9.305600 1238.500 26.95940 24.82160 1504.200 43.92490 26.57320 1174.900 31.50060 24.05960 29.25700 332.3991 1222.700

1.0 0.5 1.0 1.0 0.5 1.0 0.5 1.0 1.0 1.0 1.0 0.5 1.0 1.0 0.5 1.0 1.0 0.5 1.0 1.0 1.0 0.5 0.5

Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Plugging Plugging

(b) Test Data VF_Me

VF_Sd

PD_Me

PD_Sd

Assigned value

Assigned flow-structure

0.89991 0.79898 0.86527 0.86484

0.02175 0.01117 0.03793 0.03405

1318.800 1446.700 −1116.00 444.9920

10.02060 23.45260 12004.00 2019.500

1.0 1.0 0.5 0.5

Non-plugging Non-plugging Plugging Plugging

(c) Optimized GA-parameters

Ps Ts Mp Ng

Incremental training

Batch training

110 2 0.01 150

130 10 0.01 150

Incremental training

Batch training

0.011170 0.354880 1.681078 9.074235 3.988574 23

0.224867 0.099982 2.998584 10 7.368339 8

(d) GA-optimized NN-parameters

Lr Mf Is Ht Ot N (e) Performance using test data (Considering incremental mode) Statistical parameter using numerical simulation

VF_Me

VF_Sd

PD_Me

PD_Sd

Target values & flow-structures

Predicted output values & flowstructures

Target value

Target flow-structure

Predicted value

Predicted flowstructure

Non-plugging Non-plugging Plugging Plugging

0.98778 0.98987 0.48617 0.49244

Non-plugging Non-plugging Plugging Plugging

0.89991 0.02175 1318.800 10.02060 1.0 0.79898 0.01117 1446.700 23.45260 1.0 0.86527 0.03793 −1116.00 12004.00 0.5 0.86484 0.03405 444.9920 2019.500 0.5 Average error in prediction Maximum deviation of the predicted numerical values from the assigned value: 0.01383

Error in prediction based on calculated numerical value (%)

Error in prediction based on predicted flow-structure (%)

1.22 1.01 2.77 1.51 1.63

0 0 0 0 0

Error in prediction based on calculated numerical value (%)

Error in prediction based on predicted flow-structure (%)

0.19 0.02 0.29 0.00 0.13

0 0 0 0 0

(f) Performance using test data (Considering batch mode) Statistical parameter using numerical simulation

VF_Me

VF_Sd

PD_Me

PD_Sd

Target values & flow-structures

Predicted output values & flowstructures

Target value

Target flow-structure

Predicted value

Predicted flowstructure

Non-plugging Non-plugging Plugging Plugging

0.99809 0.99979 0.50146 0.50000

Non-plugging Non-plugging Plugging Plugging

0.89991 0.02175 1318.800 10.02060 1.0 0.79898 0.01117 1446.700 23.45260 1.0 0.86527 0.03793 −1116.00 12004.00 0.5 0.86484 0.03405 444.9920 2019.500 0.5 Average error in prediction Maximum deviation of the predicted numerical values from the assigned value: 0.00191

22

Nuclear Engineering and Design 359 (2020) 110446

K. Samal and S. Ghosh

Table 5 (a) Training data, (b) test data, (c) optimized GA-parameters, (d) optimized NN-parameters, and performance of the developed GA-tuned MLFFNN using (e) incremental, and (f) batch mode of training considering Case-D (in section ‘3′). (a) Training data VF_Me

VF_Sd

VF_Sk

VF_Ku

PD_Me

PD_Sd

PD_Sk

PD_Ku

Assigned value

Assigned flow-structure

0.92241 0.87764 0.85824 0.83518 0.81207 0.77568 0.75217 0.91353 0.88963 0.84741 0.82283 0.79755 0.78421 0.75910 0.73676 0.89631 0.88477 0.87591 0.87846 0.87336 0.86092 0.84730 0.86328

0.00701 0.03721 0.00912 0.02163 0.03622 0.02269 0.03716 0.00665 0.02221 0.00753 0.02251 0.03890 0.00881 0.02353 0.03709 0.01800 0.02489 0.02777 0.02890 0.02667 0.04367 0.04832 0.05402

0.068691 −0.17541 0.062872 0.010263 −0.10535 0.091746 −0.04828 −0.29816 −0.19143 0.061203 −0.17451 −0.16156 −0.07391 −0.07164 −0.11970 0.264773 0.237596 0.513713 −0.27527 −0.36385 −1.05505 −0.28802 −0.71670

1.796614 1.579716 2.576249 1.835767 1.781790 1.941285 1.831811 1.772808 1.697103 2.030905 1.778459 1.732902 3.012867 1.809677 1.837372 2.718991 4.266486 3.382042 3.293710 3.125558 3.707990 2.368647 2.522934

1337.3000 868.41590 1405.3000 1365.6000 592.11810 1360.0000 722.66620 1375.9000 1351.7000 1436.8000 1382.7000 777.36860 1469.7000 1349.4000 370.24540 1409.0000 1374.4000 666.86380 1464.1000 1377.6000 1488.6000 1212.2000 967.62200

11.44910 626.0038 14.41300 8.648500 1692.400 13.01070 1246.400 29.00510 22.42590 23.58120 9.305600 1238.500 26.95940 24.82160 1504.200 43.92490 26.57320 1174.900 31.50060 24.05960 29.25700 332.3991 1222.700

−0.3360 −2.5950 −0.6081 −1.6525 −2.8566 −2.1185 1.0082 −0.1745 −0.5011 −0.5080 −1.2380 −3.0813 −0.5983 −0.1600 −2.2417 −0.2374 −0.9404 −3.3282 −0.5312 −1.0118 −0.5437 −3.0916 −1.0463

1.80490 9.31260 2.46540 5.26200 15.2333 9.01830 11.2753 1.74150 2.18470 2.00180 3.80470 17.6208 2.52410 1.95090 9.63480 1.66370 2.75740 −3.3282 2.04050 3.16990 2.49910 13.2903 7.12640

1.0 0.5 1.0 1.0 0.5 1.0 0.5 1.0 1.0 1.0 1.0 0.5 1.0 1.0 0.5 1.0 1.0 0.5 1.0 1.0 1.0 0.5 0.5

Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Plugging Plugging

(b) Test Data VF_Me

VF_Sd

VF_Sk

VF_Ku

PD_Me

PD_Sd

PD_Sk

PD_Ku

Assigned value

Assigned flow-structure

0.89991 0.79898 0.86527 0.86484

0.02175 0.01117 0.03793 0.03405

−0.20484 −0.46966 −0.13645 0.366882

1.619710 4.609580 1.721184 2.920658

1318.800 1446.700 −1116.00 444.9920

10.020600 23.452600 12004.000 2019.5000

−0.3862 −0.6337 −7.3507 −3.8884

1.92450 2.86470 55.9634 20.1226

1.0 1.0 0.5 0.5

Non-plugging Non-plugging Plugging Plugging

(c) Optimized GA-parameters

Ps Ts Mp Ng

Incremental training

Batch training

140 4 0.10 100

80 2 0.04 100

Incremental training

Batch training

0.033632 0.630226 1.243856 3.301502 5.156656 11

0.771287 0.168589 1.905176 5.455777 0.862003 18

(d) GA-optimized NN-parameters

Lr Mf Is Ht Ot N (e) Performance using test data (Considering incremental mode) Statistical parameter using numerical simulation

Target values & flowstructures

Predicted output values & flow-structures

VF_Me

VF_Sd

VF_Sk

VF_Ku

PD_Me

PD_Sd

PD_Sk

PD_Ku

Target value

Target flowstructure

Predicted value

Predicted flowstructure

0.89991 0.79898 0.86527 0.86484

0.02175 0.01117 0.03793 0.03405

−0.20484 −0.46966 −0.13645 0.366882

1.619710 4.609580 1.721184 2.920658

1318.800 1446.700 −1116.00 444.9920

10.020600 23.452600 12004.000 2019.5000

−0.3862 −0.6337 −7.3507 −3.8884

1.92450 2.86470 55.9634 20.1226

1.0 1.0 0.5 0.5

Non-plugging Non-plugging Plugging Plugging

0.99968 0.99975 0.50030 0.50023

Non-plugging Non-plugging Plugging Plugging

Average error in prediction

Error in prediction based on calculated numerical value (%)

Error in prediction based on predicted flow-structure (%)

0.03 0.03 0.06 0.05

0 0 0 0

0.04

0

Maximum deviation of the predicted numerical values from the assigned value: 0.00032 (f) Performance using test data (Considering batch mode) Statistical parameter using numerical simulation

Target values & flowstructures

Predicted output values & flow-structures

VF_Me

VF_Sd

VF_Sk

VF_Ku

PD_Me

PD_Sd

PD_Sk

PD_Ku

Target value

Target flowstructure

Predicted value

Predicted flow-structure

0.89991 0.79898 0.86527 0.86484

0.02175 0.01117 0.03793 0.03405

−0.20484 −0.46966 −0.13645 0.366882

1.619710 4.609580 1.721184 2.920658

1318.800 1446.700 −1116.00 444.9920

10.020 600 23.452 600 12004.000 2019.5000

−0.3862 −0.6337 −7.3507 −3.8884

1.92450 2.86470 55.9634 20.1226

1.0 1.0 0.5 0.5

Non-plugging Non-plugging Plugging Plugging

0.98932 0.98842 0.50053 0.49578

Non-plugging Non-plugging Plugging Plugging

Average error in prediction Maximum deviation of the predicted numerical values from the assigned value: 0.01158

23

Error in prediction based on calculated numerical value (%)

Error in prediction based on predicted flow-structure (%)

1.07 1.16 0.11 0.84

0 0 0 0

0.80

0

Nuclear Engineering and Design 359 (2020) 110446

K. Samal and S. Ghosh

Table 6 (a) Training data, (b) test data, (c) optimized GA-parameters, (d) optimized NN-parameters, and performance of the developed GA-tuned MLFFNN using (e) incremental, and (f) batch mode of training considering working conditions as inputs with 4 test data. (a) Training data ṁa

ṁw

hw

Assigned value

Assigned flow-structure

0.268 0.268 0.268 0.268 0.268 0.268 0.268 1.072 1.072 1.072 1.072 1.072 1.072 1.072 1.072 2.144 2.144 2.144 2.144 2.144 2.144 2.144 2.144

1 5 1 3 5 3 5 1 3 1 3 5 1 3 5 1 3 5 1 3 1 3 5

0.03 0.03 0.06 0.06 0.06 0.10 0.10 0.03 0.03 0.06 0.06 0.06 0.10 0.10 0.10 0.03 0.03 0.03 0.06 0.06 0.10 0.10 0.10

1.0 0.5 1.0 1.0 0.5 1.0 0.5 1.0 1.0 1.0 1.0 0.5 1.0 1.0 0.5 1.0 1.0 0.5 1.0 1.0 1.0 0.5 0.5

Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Plugging Non-plugging Non-plugging Non-plugging Plugging Plugging

ṁa

ṁw

hw

Assigned value

Assigned flow-structure

0.268 0.268 1.072 2.144

3 1 5 5

0.03 0.10 0.03 0.06

1.0 1.0 0.5 0.5

Non-plugging Non-plugging Plugging Plugging

(b) Test Data

(c) Optimized GA-parameters

Ps Ts Mp Ng

Incremental training

Batch training

120 10 0.10 100

120 8 0.08 100

Incremental training

Batch training

0.001648 0.386620 0.284447 6.573704 9.095990 12

0.257035 0.166331 2.213532 1.578081 9.510529 12

(d) GA-optimized NN-parameters

Lr Mf Is Ht Ot N (e) Performance using test data (Considering incremental mode) Statistical parameter using numerical simulation

Target values & flow-structures

Predicted output values & flowstructures

ṁa

ṁw

hw

Target value

Target flow-structure

Predicted value

Predicted flow-structure

0.268 0.268 1.072 2.144

3 1 5 5

0.03 0.10 0.03 0.06

1.0 1.0 0.5 0.5

Non-plugging Non-plugging Plugging Plugging

0.98487 0.99939 0.49544 0.50228

Non-plugging Non-plugging Plugging Plugging

Average error in prediction

Error in prediction based on calculated numerical value (%)

Error in prediction based on predicted flow-structure (%)

1.51 0.06 0.91 0.46

0 0 0 0

0.74

0

Maximum deviation of the predicted numerical values from the assigned value: 0.01513

(continued on next page) 24

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K. Samal and S. Ghosh

Table 6 (continued) (f) Performance using test data (Considering batch mode) Statistical parameter using numerical simulation

Target values & flow-structures

Predicted output values & flowstructures Predicted flowstructure

Error in prediction based on calculated numerical value (%)

Error in prediction based on predicted flow-structure (%)

ṁa

ṁw

hw

Target value

Target flowstructure

Predicted value

0.268 0.268 1.072 2.144

3 1 5 5

0.03 0.10 0.03 0.06

1.0 1.0 0.5 0.5

Non-plugging Non-plugging Plugging Plugging

0.86789 0.99902 0.50702 0.50133

Non-plugging Non-plugging Plugging Plugging

13.21 0.10 1.40 0.27

0 0 0 0

3.75

0

Average error in prediction Maximum deviation of the predicted numerical values from the assigned value: 0.13211

Fig. 21. Overall comparison between the performances of the developed approaches.

may be observed frequently at the elbow in this situation.

avoid plugging in hot-leg during LOCA.

• Plugging generally occurs at higher gas-flowrates. It may also occur •

• • •

• The occurrence of plugging is very much responsive to the gas-

at moderate gas-flowrates when both water-flowrates and stored water-levels are very high. During plugging, entrainment of pressurized gas into water is started and thus, air bubbles are generated at the elbow, IPHL, and SG. Smooth time variations of PD and VF across the flow domain (almost free from oscillations) are obtained during smooth-stratified flow in the hot-leg. For the wavy stratified flow, the time variation of PD and VF contain small fluctuations. Large fluctuations and oscillations are observed in the time variation of PD and VF when plugging occurs. Gas mass flowrate is the key parameter to avoid plugging. Plugging/ blocking never found at low gas mass flowrate. Low gas mass flowrate is desirable to avoid plugging during LOCA. During LOCA, the time required to start plugging depends on the stored level of water in the HPHL. It is started promptly as the waterlevel increases. Apparently, water-flowrate is not much responsible for the occurrence of plugging. However, water-flowrate indirectly influences the rate of increment of stored water in HPHL. The height of this stored water increases with the increment of water-flowrate. Therefore, it is one of the important parameters which should be taken care of to

•

• • 25

flowrate. Plugging may occur at any physical situation when ṁa is sufficiently large. At the same time, it may also be noted that plugging can occur even at a moderate value of gas-flowrate when hw is sufficiently large and greater than a threshold value. Therefore, most important factors, which are required to control to avoid plugging during LOCA, are gas-flowrate and stored water-level in HPHL. The developed GA tuned MLFFNN is able to perfectly predict (with 100% accuracy) the flow-conditions (plugging or non-plugging) in the hot-leg by knowing the said statistical parameters (Me, Sd, Sk, and Ku) extracted from time variation of PD and VF across the flow domain. Therefore, it can be concluded that the flow-condition in hot-leg can be perfectly parameterized using the said statistical parameters of PD and VF fluctuations (nature of fluctuation present in the time variation of PD and VF) across the flow domain. As per the present investigation (by comparing the performance of Case A and Case B), the time-series fluctuation of volume fraction provides better prediction results compare to that of pressure drop when the statistical parameters Me and Sd are only considered to make the model (hybrid GA tuned MLFFNN). Data-wise, time-series fluctuation of volume fraction is dominant

Nuclear Engineering and Design 359 (2020) 110446

Processor, 2 GB RAM, 512 GB

Processor, 2 GB RAM, 512 GB

Processor, 4 GB RAM, 512 GB

Processor, 4 GB RAM, 512 GB

Processor, 2 GB RAM, 512 GB

• •

9 96.25

6

12 99.20

9.5

10 99.87

8

8 93.08

6

9 98.29

6

9 98.40

6

Incremental mode Batch mode

Batch mode

i7 3rd Gen hard disk i7 3rd Gen hard disk i5 3rd Gen hard disk i5 3rd Gen hard disk i7 3rd Gen hard disk i7 3rd Gen hard disk

•

• •

99.96

99.26

Case-D

Working conditions as inputs

and parameter-wise, the combination of Me, Sd, Sk, and Ku are dominant. Some of the objective signatures of the flow are arrested by the timeseries fluctuation of volume fraction and some of the objective signatures are arrested by the time-series fluctuation of pressure drop. Naturally, the total objective signatures arrested by the combination of them (considering both time-series fluctuation) will be more compared to those arrested by any one of them. Consequently, the performances of the model are increased when statistical parameters extracted from both the time-series fluctuations (Cases C and D) are used. Sometime the model may be confused by considering too many statistical parameters as the inputs. The relation between the flow-conditions (occurrence or absence of plugging) in hot-leg and the working conditions (in terms of gasflowrate, water-flowrate, and stored water-level in HPHL) is perfectly captured (with 100% capability) by the developed GA tuned MLFFNN with BPA. In actual PWR, it is difficult to observe the flow patterns inside the hot-leg. Therefore, the possible solution to detect the flow-conditions during LOCA, is to continuously record the PD and VF fluctuations across the flow domain with time. Gas-flowrate and water-level in HPHL should be properly controlled as they have important roles in the occurrence or absence of plugging in hot-leg. Arrangement of water removal should be provided with the PWR to maintain the water-level in HPHL below a critical level during LOCA. Proper control of fission-reaction is essential and quick leak detection (as early as possible) is mandatory to maintain the gas-flowrate below a critical limit to avoid the onset of plugging in hot-leg. Model uncertainties within the CFD-code may have an impact on the performance of the proposed CI-based prediction model. Uncertainties of the proposed model using experimental objective data may be less compared to that using numerically obtained objective data. However, the main developing procedure of the proposed prediction model is independent of the fact that whether the objective data are collected through experimentations or CFD simulations. Here, the main thrust is given on the process of developing a methodology using the online objective signatures of the flow for the automatic prediction of plugging/non-plugging situations in the hot-leg during LOCA. As the adopted CFD-code is extensively validated with experimental results, numerically generated objective data maybe not so far from the experimental data and one can rely on it.

Acknowledgments This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. But, the work was performed and supported by the high-speed computational recourses of the National Institute of Technology, Rourkela, India.

Input: VF_Me, VF_Sd, VF_Sk, VF_Ku, PD_Me, PD_Sd, PD_Sk, PD_Ku Training data: 23 & Test data: 4 Input: ṁa, ṁw, hw Training data: 23 & Test data: 4

98.37 Case-C Input: VF_Me, VF_Sd, PD_Me, PD_Sd Training data: 23 & Test data: 4

94.28 Case-B Input: PD_Me, PD_Sd Training data: 23 & Test data: 4

97.04 Case-A2 Input: VF_Me, VF_Sd Training data: 21 & Test data: 6

99.58 Case-A1

Incremental mode

•

Input: VF_Me, VF_Sd Training data: 23 & Test data: 4

Cases Different sets of inputs

Table 7 Performances of the developed GA-tuned MLFFNN with different sets of inputs.

Average prediction capability (% based on calculated numerical value)

Computing Time (hour)

System Configuration

Processor, 2 GB RAM, 512 GB

K. Samal and S. Ghosh

References Ardron, K.H., Banerjee, S., 1986. Flooding in an elbow between a vertical and a horizontal or near horizontal pipe: part II: theory. Int. J. Multiphase Flow. 12, 543–558. https://doi.org/ 10.1016/0301-9322(86)90059-5. Deendarlianto,, Höhne, T., Lucas, D., Vallée, C., Zabala, G.A.M., 2011. CFD studies on the phenomena around counter-current flow limitations of gas/liquid two-phase flow in a model of a PWR hot leg. Nucl. Eng. Des. 241, 5138–5148. https://doi.org/10.1016/j. nucengdes.2011.08.071. Deendarlianto, Höhne, T., Lucas, D., Vierow, K., 2012. Gas-liquid countercurrent two-phase flow in a PWR hot leg: a comprehensive research review. Nucl. Eng. Des. 243, 214–233. https://doi.org/10.1016/j.nucengdes.2011.11.015. Deendarlianto,, Vallée, C., Lucas, D., Beyer, M., Pietruske, H., Carl, H., 2008. Experimental study on the air/water counter-current flow limitation in a model of the hot leg of a pressurized water reactor. Nucl. Eng. Des. 238, 3389–3402. https://doi.org/10.1016/j. nucengdes.2008.08.003. Geffraye, G., Bazin, P., Pichon, P., 1995. CCFL in hot legs and steam generators and its prediction with the CATHARE code. In: Proceeding of the 7th International Meeting on

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