# Numerical Simulations and Design Optimization of  the PHT Loop of Natural Circulation BWR

Authors: G. V. Durga Prasad; G. Gopa Kishor; Manmohan Pandey; Uday S. Dixit
Journal: Science and Technology of Nuclear Installations
 (2008)

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## Abstract

Mathematical modeling and numerical simulation of natural circulation boiling water reactor (NCBWR) are very important in order to study its performance for different designs and various off-design conditions and for design optimization. In the present work, parametric studies of the primary heat transport loop of NCBWR have been performed using lumped parameter models and RELAP5/MOD3.4 code. The lumped parameter models are based on the drift flux model and homogeneous equilibrium mixture (HEM) model of two-phase flow. Numerical simulations are performed with both models. Compared to the results obtained from the HEM model, those obtained from the drift flux model are closer to RELAP5. The variations of critical heat flux with various geometric parameters and operating conditions are thoroughly investigated. The material required to construct the primary heat transport (PHT) loop of NCBWR has been minimized using sequential quadratic programming. The stability of NCBWR has also been verified at the optimum point.


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## Body

## 1. Introduction

Water-cooled nuclear power plants (light and heavy water reactors)
represent 94% of the global nuclear power capacity, and advanced designs are
being developed to meet future energy needs. Common goals for advanced designs
are high reliability and competitive economics while meeting stringent safety
requirements. Passive safety systems based on natural circulation are the key
to several evolutionary water-cooled designs and many revolutionary
water-cooled reactor designs. In natural circulation boiling water reactor
(NCBWR), the heat removal from the core takes place by natural circulation
during startup, power rising, and accidental conditions in addition to the
rated full power operating conditions. The thermal hydraulic analysis of NCBWR
is very important for better insight into the process that governs the
in-vessel physics. Parametric studies of NCBWR will guide the designer to
choose the best possible design. Numerical simulations and parametric studies
can be performed using thermal hydraulic codes such as RELAP5. This code is
based on governing equations of thermal hydraulics and takes considerable
computational time. Extensive parametric studies and design optimization can be
done with the help of computationally inexpensive mathematical model, such as
lumped parameter model (LPM).Aybar et al. [1] have investigated the performance of Ohio State University inherently safe reactor (OSU-ISR)
as a function of several design variables, to know the interdependency among
the several system variables. They simulated the steady-state performance of
OSU-ISR with dynamic simulation for nuclear power
plants (DSNP) code and validated the results with RELAP5. Jafari et al. [2]
have analyzed the effect of design parameters on natural circulation loop and
have done stability analysis using RELAP5 thermal hydraulic code. Rao et al. [3]
have studied the performance of natural circulation loop using homogeneous
equilibrium model (HEM) and drift flux model through one-dimensional approach.
Kok and van der Hagen [4] developed a steady-state mathematical model to simulate DESIRE
test facility and conducted parametric studies. Garg et al. [5] did parametric
studies of NCBWR using artificial neural network (ANN) model, which was trained
using the results of RELAP5/MOD3.2 simulation. They have illustrated a
methodology of training ANN with RELAP, which could then be used for parametric
studies and design optimization.In the present study, steady-state lumped
parameter models (LPMs) are derived for the primary heat transport (PHT) loop
of NCBWR. The same loop is also modeled in RELAP5/MOD3.4 code. Parametric
trends are computed with both models (LPM and RELAP), and the results are
compared. The variations of mass flow rate and void fraction with geometric
properties and with the power are studied. After that the critical heat flux
(CHF) phenomenon is investigated. The variation of CHF with various geometric
and operating conditions is studied. Subsequently, design optimization of NCBWR
is done to minimize the volume of material required to construct the PHT loop
(referred to as the “minimum material condition” in this paper) subject to
constraints based on safety considerations. The stability of the NCBWR is
checked at minimum material condition, and alternative designs are suggested
based on stability considerations.


## 2. Numerical Simulations

### 2.1. Description of the System

In
natural circulation, the primary pumps are eliminated, and the necessary flow
rates are achieved by locating the steam drums at a suitable height from the center
of the core, taking advantage of the reactor building height. A schematic view
of a natural circulation boiling water reactor (NCBWR) is shown in Figure1.
The input data are listed in Tables 1 and 2. These data are for a typical
design of NCBWR, taken from literature [6]. The PHT loop consists of core,
tail pipes (risers), steam drum, downcomer, inlet header, and inlet feed water
pipes. Constant heat is supplied in the active core region. The steam water
mixture from each coolant channel is led through tail pipes to the steam drum,
which is located at certain elevation with respect to inlet header (coolant
channel bottom). In the present work, 113 coolant channels are taken for the
analysis.Table 1
Input parameters.
ParameterValuePressure70 barPower250 MWFeed water inlet enthalpy552 kJ/kgHydraulic diameter of core0.0081 mLength of core bottom4.9 mFlow area of single channel in core bottom0.00274 m2Length of the core3.5 mFlow area of single channel in core0.0048 m2Length of core top3.87 mFlow area of single channel in core top0.00325 m2Length of horizontal tail pipe 13.51 mFlow area of single channel in horizontal tail pipe 10.00325 m2Length of horizontal tail pipe 219.8 mFlow area of single channel in horizontal tail pipe 20.01171 m2Length of vertical tail pipe25.7 mFlow area of single channel in vertical tail pipe0.01171 m2Length of the downcomer26.8 mArea of the downcomer0.2621 m2Area of inlet header5.7 m2Length of inlet feeder17.0 mFlow area of inlet feeder0.0074 m2Roughness value of the PHT loop50.0e-6 mTable 2
Loss coefficients.
Loss coefficientCore bottom2.8Core bottom to active core8.02Core channel junction0.5Core extension to tail pipes2.5Inlet header6.5Figure 1
Schematic diagram of NCBWR.The
two-phase mixture is present in the core and riser sections, and the single
phase liquid is in the downcomer section. Due to density difference between
these two sections natural circulation will develop. The natural circulation
flow rate is decided by the density difference of the fluid between the downcomer
(cold leg) and that of the riser (hot leg), and the flow resistances. The
enthalpy of the coolant at the core inlet depends on the feed water flow rate,
its temperature, and the recirculation flow rate.


### 2.2. RELAP Modeling

The
PHT loop is modeled using RELAP5/MOD3.4. It is a time domain code, which is
based on two-fluid formulation. The nodalization scheme for the PHT loop is
shown in Figure2. The PHT loop components are modeled in RELAP using pipe
components. The boundary conditions, that is, steam outlet from steam drum and
the feed water inlet are defined using time dependent volumes. The pipe
component P1 is divided into 12 volumes. The first two volumes model the feed
water inlet pipe, the third volume models the bottom of the core followed by 5
volumes of active core, one volume of core top and three volumes for tail
pipes. The steam drum (pipe component P2) is divided into 10 volumes. The
saturated water from steam drum and the feed water are mixed in single volume
S1. The downcomer is divided into 2 volumes, and the inlet
header is modeled as a single volume. All the components are connected using
single junctions. The heat supply to the active core is modeled using a heat
structure component divided into 5 nodes (equivalent to 5 volumes of active
core), and constant power option (no neutronics) is used.Figure 2
Nodalization diagram of NCBWR.


## 2.1. Description of the System

In
natural circulation, the primary pumps are eliminated, and the necessary flow
rates are achieved by locating the steam drums at a suitable height from the center
of the core, taking advantage of the reactor building height. A schematic view
of a natural circulation boiling water reactor (NCBWR) is shown in Figure1.
The input data are listed in Tables 1 and 2. These data are for a typical
design of NCBWR, taken from literature [6]. The PHT loop consists of core,
tail pipes (risers), steam drum, downcomer, inlet header, and inlet feed water
pipes. Constant heat is supplied in the active core region. The steam water
mixture from each coolant channel is led through tail pipes to the steam drum,
which is located at certain elevation with respect to inlet header (coolant
channel bottom). In the present work, 113 coolant channels are taken for the
analysis.Table 1
Input parameters.
ParameterValuePressure70 barPower250 MWFeed water inlet enthalpy552 kJ/kgHydraulic diameter of core0.0081 mLength of core bottom4.9 mFlow area of single channel in core bottom0.00274 m2Length of the core3.5 mFlow area of single channel in core0.0048 m2Length of core top3.87 mFlow area of single channel in core top0.00325 m2Length of horizontal tail pipe 13.51 mFlow area of single channel in horizontal tail pipe 10.00325 m2Length of horizontal tail pipe 219.8 mFlow area of single channel in horizontal tail pipe 20.01171 m2Length of vertical tail pipe25.7 mFlow area of single channel in vertical tail pipe0.01171 m2Length of the downcomer26.8 mArea of the downcomer0.2621 m2Area of inlet header5.7 m2Length of inlet feeder17.0 mFlow area of inlet feeder0.0074 m2Roughness value of the PHT loop50.0e-6 mTable 2
Loss coefficients.
Loss coefficientCore bottom2.8Core bottom to active core8.02Core channel junction0.5Core extension to tail pipes2.5Inlet header6.5Figure 1
Schematic diagram of NCBWR.The
two-phase mixture is present in the core and riser sections, and the single
phase liquid is in the downcomer section. Due to density difference between
these two sections natural circulation will develop. The natural circulation
flow rate is decided by the density difference of the fluid between the downcomer
(cold leg) and that of the riser (hot leg), and the flow resistances. The
enthalpy of the coolant at the core inlet depends on the feed water flow rate,
its temperature, and the recirculation flow rate.


## 2.2. RELAP Modeling

The
PHT loop is modeled using RELAP5/MOD3.4. It is a time domain code, which is
based on two-fluid formulation. The nodalization scheme for the PHT loop is
shown in Figure2. The PHT loop components are modeled in RELAP using pipe
components. The boundary conditions, that is, steam outlet from steam drum and
the feed water inlet are defined using time dependent volumes. The pipe
component P1 is divided into 12 volumes. The first two volumes model the feed
water inlet pipe, the third volume models the bottom of the core followed by 5
volumes of active core, one volume of core top and three volumes for tail
pipes. The steam drum (pipe component P2) is divided into 10 volumes. The
saturated water from steam drum and the feed water are mixed in single volume
S1. The downcomer is divided into 2 volumes, and the inlet
header is modeled as a single volume. All the components are connected using
single junctions. The heat supply to the active core is modeled using a heat
structure component divided into 5 nodes (equivalent to 5 volumes of active
core), and constant power option (no neutronics) is used.Figure 2
Nodalization diagram of NCBWR.


## 3. Mathematical Modeling

### 3.1. Steady-State Modeling

Two lumped
parameter models are derived using drift flux and homogeneous equilibrium model
(HEM) from the one-dimensional steady-state equations of mass, momentum, and
energy balances. The following assumptions are made while deriving the
equations. These assumptions are used to reduce the complexity of the system at
the expense of some accuracy and to give a better insight into the system.(1)
The thermodynamic properties over the entire loop are
evaluated at the system pressure.(2)
The liquid and vapor phases are in thermodynamic
equilibrium with each other.(3)
The processes of subcooled boiling and flashing are
not considered.(4)
Viscous dissipation and changes in kinetic energy,
potential energy, and flow work are neglected in the energy equation.(5)
The distribution of linear power of the fuel in the
axial direction is uniform.(6)
The coolant in the downcomer is in the liquid phase
only. The present
models are similar to an earlier lumped parameter dynamic model of natural
circulation loop derived by van Bragt and van der Hagen [7]. The core is
divided into single-phase region (from core inlet to boiling boundary) and
two-phase region (from boiling boundary to the core exit). The one-dimensional
unsteady-state continuity, momentum, and energy equations are given below:
(1)Continuity:A∂ρ∂t+∂m∂z=0,momentum:1A∂m∂t+1A2∂(m2ν)∂z+gρ+f(m2ν)2DA2+∂p∂z=0,energy:ρA∂h∂t+m∂h∂z=q'. The steady-state
equations are obtained by equating the time derivative terms to zero in the
above three equations. The boiling boundary equation is derived by integrating
the energy equation from the core inlet to the boiling boundary:
(2)zbb=GciLcAc(hl-hci)q. The expression
for the exit steam quality is derived by integrating the energy equation from
boiling boundary to exit of the core:
(3)xce=(Lc-zbb)qAcLc(hg-hl)Gci. Under
steady-state conditions, the core exit mass flux density is equal to core inlet
mass flux density. Applying steady-state mass and energy balance at the
junction of steam drum, feedwater and downcomer, the core inlet enthalpy can be
expressed in terms of feedwater inlet enthalpy as
(4)hci=hl−xce(hl−hFi). The relation
between volumetric quality (β) and
void fraction (α) is given by Zuber and Findlay
[8] as follows:
(5)α=[βC0+Vgj/j], where C0 is the flow distribution
parameter, Vgj is drift
velocity, and j is the volumetric flux. The volumetric quality β can be expressed in terms of core
exit quality as
(6)β=[xcexce+(ρg/ρl)(1−xce)]. For the drift
flux model, numerous empirical correlations are available for C0 and Vgj in literature. Coddington and Macian [9]
discussed 13 correlations and their ranges of operation. They found that the
void fraction predictions with Maier and Coddington correlation are better than those for the other correlations. Therefore, in
the present study, Maier and Coddington correlation has been used. This
correlation was purely based on empirical approach to the evaluation of C0 and Vgj of the drift-flux
equation. The distribution parameter is a function of pressure, and drift
velocity is a function of pressure and mass flux density. Under steady-state
and constant pressure conditions and with constant linear power, the axial
variation of mass quality in the core will be linear. Using the above relations
for void fraction α and volumetric
quality β, the expression for the
core average void fraction is obtained as
(7)〈α〉c=[xceb+ab2log(aa+bxce)](Lc-zbbLcxce), where
(8)a=C0(ρgρl)+(ρgVgjGci),b=C0(1-(ρgρl)). Under
steady-state and constant pressure conditions, the riser void fraction is equal
to the exit void fraction of the core.The momentum
equation is integrated over the natural circulation loop, including the core,
the riser, and the downcomer. Since the loop is closed, the total pressure drop
in the natural circulation loop must be zero. There are various pressure drops
in different sections of the loop, arising due to gravitational, frictional,
acceleration, and minor losses. These pressure drop terms are derived by
integrating the respective terms of the one-dimensional momentum equation along
each component of the boiling system. The friction factor is calculated by Zigrang-Sylvester correlation [10]. The
steady-state momentum balance of the closed loop can be expressed as
(9)Δpfr+Δpgr+Δpa+Δpm=0. The expressions
for all the pressure drops in HEM model and drift flux model (except
acceleration pressure drop in drift flux model) and the expression for average
core void fraction in the HEM model are taken from van Bragt and van der Hagen
[7]. The expression for acceleration pressure drop in the drift flux model is
taken from Kok and van der Hagen [4]:
(10)ΔPa=G2[xceαceρg+(1−xce)2(1−αce)ρl−1ρl]. The above set of
equations is converted into explicit boiling boundary equation and momentum
equations (2) and (9). These two equations are solved using the FSOLVE function in
MATLAB.


### 3.2. Formulation for Stability

The
governing equations for stability analysis are derived from one-dimensional unsteady-state
equations of mass, momentum, and energy balances (1). These equations are
derived using HEM model as discussed in [7, 11] . The core is divided into single-phase and two-phase regions as
explained in Section 3.1. Integrating
the energy equation from core inlet to boiling boundary (Zbb), the state equation for the boiling boundary is
(11)dZbb(t)dt=2[GC,i(t)ρl−Zbb(t)q′(t)ρlAC(hl−hci)]. By integrating the continuity
equation, we get the state equation of core average void dynamics given by
(12)d〈α〉cdt=Gce(t)−Gci(t)(ρl−ρg)Lc.The relation between core inlet mass flux density and the exit mass flux
density as given below is obtained by integrating the energy equation in the
two-phase region:(13)Gci(t)=Gce(t)[1+ρl−ρgρgxce(t)]−(ρl−ρg)ρg[LC−Zbb(t)]q′(t)AC(hl−hg).Assuming linear variation of void fraction along the core, the relation
between the core average void fraction and the core exit quality is given by(14)〈α〉C(t)=H−Zbb(t)Hρlρl−ρg−{1−ρg(ρl−ρg)xce(t)ln[1+ρl−ρgρgxce(t)]}.Riser is an adiabatic section, divided intoNR nodes, of equal length. Integrating the differential
continuity equation along the riser gives the riser void dynamics equation as
(15)d〈α〉R,n(t)dt=GR,n(t)−GR,n−1(t)ρl−ρgNRLR.Integrating the energy equation along each node in riser gives the
relation between the mass flux densities of adjacent nodes:(16)GR,n−1(t)[1+ρl−ρgρgxR,n−1(t)]=GR,n(t)[1+ρl−ρgρgxR,n(t)]. The dynamic pressure is continuous
along the closed natural circulation loop and can be expressed as
(17)ΔPC(t)+ΔPR(t)+ΔPD(t)=0.Each term in (17) is the summation of inertial, gravitational, frictional,
acceleration, and minor losses terms. The detailed expressions of these terms
are given in [7] The total dynamic
pressure drop (summation of inertial, gravitational, acceleration, friction)
can be evaluated by integrating the differential momentum equation along each
component of the boiling system considered. The friction factor “f” in core and riser is evaluated using
the Darcy-Weisbach friction factor formula. It is assumed that the feed
water flow rate is equal to the steam flow rate to make the drum dynamics
steady. The core top is considered as a single node, and the riser is divided
into two nodes of equal length. The resulting mathematical model is a set of
ordinary differential equations with state variables as the Gci,zbb,ac,αr1,αr2. The system is linearized at the
fixed point (obtained from steady-state analysis), and the Jacobian matrix is
evaluated numerically. The nature of the eigenvalues determines the stability of
the system. If all the eigenvalues lie in the left half of the complex plane
then the system is stable.


## 3.1. Steady-State Modeling

Two lumped
parameter models are derived using drift flux and homogeneous equilibrium model
(HEM) from the one-dimensional steady-state equations of mass, momentum, and
energy balances. The following assumptions are made while deriving the
equations. These assumptions are used to reduce the complexity of the system at
the expense of some accuracy and to give a better insight into the system.(1)
The thermodynamic properties over the entire loop are
evaluated at the system pressure.(2)
The liquid and vapor phases are in thermodynamic
equilibrium with each other.(3)
The processes of subcooled boiling and flashing are
not considered.(4)
Viscous dissipation and changes in kinetic energy,
potential energy, and flow work are neglected in the energy equation.(5)
The distribution of linear power of the fuel in the
axial direction is uniform.(6)
The coolant in the downcomer is in the liquid phase
only. The present
models are similar to an earlier lumped parameter dynamic model of natural
circulation loop derived by van Bragt and van der Hagen [7]. The core is
divided into single-phase region (from core inlet to boiling boundary) and
two-phase region (from boiling boundary to the core exit). The one-dimensional
unsteady-state continuity, momentum, and energy equations are given below:
(1)Continuity:A∂ρ∂t+∂m∂z=0,momentum:1A∂m∂t+1A2∂(m2ν)∂z+gρ+f(m2ν)2DA2+∂p∂z=0,energy:ρA∂h∂t+m∂h∂z=q'. The steady-state
equations are obtained by equating the time derivative terms to zero in the
above three equations. The boiling boundary equation is derived by integrating
the energy equation from the core inlet to the boiling boundary:
(2)zbb=GciLcAc(hl-hci)q. The expression
for the exit steam quality is derived by integrating the energy equation from
boiling boundary to exit of the core:
(3)xce=(Lc-zbb)qAcLc(hg-hl)Gci. Under
steady-state conditions, the core exit mass flux density is equal to core inlet
mass flux density. Applying steady-state mass and energy balance at the
junction of steam drum, feedwater and downcomer, the core inlet enthalpy can be
expressed in terms of feedwater inlet enthalpy as
(4)hci=hl−xce(hl−hFi). The relation
between volumetric quality (β) and
void fraction (α) is given by Zuber and Findlay
[8] as follows:
(5)α=[βC0+Vgj/j], where C0 is the flow distribution
parameter, Vgj is drift
velocity, and j is the volumetric flux. The volumetric quality β can be expressed in terms of core
exit quality as
(6)β=[xcexce+(ρg/ρl)(1−xce)]. For the drift
flux model, numerous empirical correlations are available for C0 and Vgj in literature. Coddington and Macian [9]
discussed 13 correlations and their ranges of operation. They found that the
void fraction predictions with Maier and Coddington correlation are better than those for the other correlations. Therefore, in
the present study, Maier and Coddington correlation has been used. This
correlation was purely based on empirical approach to the evaluation of C0 and Vgj of the drift-flux
equation. The distribution parameter is a function of pressure, and drift
velocity is a function of pressure and mass flux density. Under steady-state
and constant pressure conditions and with constant linear power, the axial
variation of mass quality in the core will be linear. Using the above relations
for void fraction α and volumetric
quality β, the expression for the
core average void fraction is obtained as
(7)〈α〉c=[xceb+ab2log(aa+bxce)](Lc-zbbLcxce), where
(8)a=C0(ρgρl)+(ρgVgjGci),b=C0(1-(ρgρl)). Under
steady-state and constant pressure conditions, the riser void fraction is equal
to the exit void fraction of the core.The momentum
equation is integrated over the natural circulation loop, including the core,
the riser, and the downcomer. Since the loop is closed, the total pressure drop
in the natural circulation loop must be zero. There are various pressure drops
in different sections of the loop, arising due to gravitational, frictional,
acceleration, and minor losses. These pressure drop terms are derived by
integrating the respective terms of the one-dimensional momentum equation along
each component of the boiling system. The friction factor is calculated by Zigrang-Sylvester correlation [10]. The
steady-state momentum balance of the closed loop can be expressed as
(9)Δpfr+Δpgr+Δpa+Δpm=0. The expressions
for all the pressure drops in HEM model and drift flux model (except
acceleration pressure drop in drift flux model) and the expression for average
core void fraction in the HEM model are taken from van Bragt and van der Hagen
[7]. The expression for acceleration pressure drop in the drift flux model is
taken from Kok and van der Hagen [4]:
(10)ΔPa=G2[xceαceρg+(1−xce)2(1−αce)ρl−1ρl]. The above set of
equations is converted into explicit boiling boundary equation and momentum
equations (2) and (9). These two equations are solved using the FSOLVE function in
MATLAB.


## 3.2. Formulation for Stability

The
governing equations for stability analysis are derived from one-dimensional unsteady-state
equations of mass, momentum, and energy balances (1). These equations are
derived using HEM model as discussed in [7, 11] . The core is divided into single-phase and two-phase regions as
explained in Section 3.1. Integrating
the energy equation from core inlet to boiling boundary (Zbb), the state equation for the boiling boundary is
(11)dZbb(t)dt=2[GC,i(t)ρl−Zbb(t)q′(t)ρlAC(hl−hci)]. By integrating the continuity
equation, we get the state equation of core average void dynamics given by
(12)d〈α〉cdt=Gce(t)−Gci(t)(ρl−ρg)Lc.The relation between core inlet mass flux density and the exit mass flux
density as given below is obtained by integrating the energy equation in the
two-phase region:(13)Gci(t)=Gce(t)[1+ρl−ρgρgxce(t)]−(ρl−ρg)ρg[LC−Zbb(t)]q′(t)AC(hl−hg).Assuming linear variation of void fraction along the core, the relation
between the core average void fraction and the core exit quality is given by(14)〈α〉C(t)=H−Zbb(t)Hρlρl−ρg−{1−ρg(ρl−ρg)xce(t)ln[1+ρl−ρgρgxce(t)]}.Riser is an adiabatic section, divided intoNR nodes, of equal length. Integrating the differential
continuity equation along the riser gives the riser void dynamics equation as
(15)d〈α〉R,n(t)dt=GR,n(t)−GR,n−1(t)ρl−ρgNRLR.Integrating the energy equation along each node in riser gives the
relation between the mass flux densities of adjacent nodes:(16)GR,n−1(t)[1+ρl−ρgρgxR,n−1(t)]=GR,n(t)[1+ρl−ρgρgxR,n(t)]. The dynamic pressure is continuous
along the closed natural circulation loop and can be expressed as
(17)ΔPC(t)+ΔPR(t)+ΔPD(t)=0.Each term in (17) is the summation of inertial, gravitational, frictional,
acceleration, and minor losses terms. The detailed expressions of these terms
are given in [7] The total dynamic
pressure drop (summation of inertial, gravitational, acceleration, friction)
can be evaluated by integrating the differential momentum equation along each
component of the boiling system considered. The friction factor “f” in core and riser is evaluated using
the Darcy-Weisbach friction factor formula. It is assumed that the feed
water flow rate is equal to the steam flow rate to make the drum dynamics
steady. The core top is considered as a single node, and the riser is divided
into two nodes of equal length. The resulting mathematical model is a set of
ordinary differential equations with state variables as the Gci,zbb,ac,αr1,αr2. The system is linearized at the
fixed point (obtained from steady-state analysis), and the Jacobian matrix is
evaluated numerically. The nature of the eigenvalues determines the stability of
the system. If all the eigenvalues lie in the left half of the complex plane
then the system is stable.


## 4. Parametric Studies

### 4.1. Effects of Operating and Geometric Parameters

Numerical simulations of
NCBWR were performed by the lumped parameter model as well as the RELAP5 model,
and the results were compared. The effects of various operating parameters like
pressure, power, feedwater inlet enthalpy (FIE), and geometric parameters like
riser length, core length, hydraulic diameter of core, etc were
studied. While performing the parametric studies, certain parameters were
varied, and other parameters were fixed. The input data are listed in Tables1
and 2. The input parameters are chosen based on the information available in
the literature [6] and suitable assumptions. The trends obtained are discussed
in the following paragraphs.At
constant pressure and FIE, as power increases, the mass flow rate decreases (see
Figure3), except for the low power range, in which the mass flow rate
increases with power. As power increases, the steam quality should also
increase. Due to the presence of more vapor, two effects will occur. Firstly,
with an increase in the steam quality, the density difference between the riser
section and the downcomer section will increase. This, in turn, will tend to
increase the mass flow rate. Secondly, the velocity of the coolant will
increase as the steam quality increases. This will increase the frictional
resistance and hence tends to reduce the mass flow rate. In most of the power
range considered in the present study, the frictional resistance seems to be
more dominant, so the mass flow rate decreases as shown in Figure 3.Figure 3
Variation of mass flow rate with power.At
constant power and FIE, with an increase in the riser length, the mass flow
rate increases (see Figure4). As the riser length increases, two effects will
occur. Due to longer riser length, the frictional resistance to the flow
increases that tends to decrease the mass flow rate in the system. However,
there is a gain in the buoyancy force due to the increased elevation. This will
increase the flow rate in the system. The latter effect seems to be dominant.
Hence, there is a notable rise in the mass flow rate of the system as the
length of the riser increases. As the pressure and total power are kept constant,
the mass flow rate decreases with an increase in core length (see Figure 5). As
the core length increases, the mass flow rate may increase due to increased
elevation The mass flow rate may
decrease due to an increase in frictional resistance because of longer core.
The flow area of the core is very small compared to the riser. Because of this,
the frictional resistance is dominant; hence the mass flow rate decreases with
an increase in core length. As the hydraulic diameter of the core increases,
the frictional resistance should decrease due to an increase in the flow area,
decrease in the aspect ratio, and decrease in the friction coefficient (due to
the decrease in relative roughness). Therefore, the mass flow rate increases
with hydraulic diameter of the core (see Figure 6).Figure 4
Variation of mass flow rate with riser length.Figure 5
Variation of mass flow rate with core length.Figure 6
Variation of mass flow rate with hydraulic diameter of the core.From
Figures3–6, it is observed
that the values of mass flow rate predicted by the drift flux model are closer
to those obtained with RELAP compared to the homogeneous equilibrium model.
Therefore, drift flux model is the more accurate of the two lumped parameter
models. For further analysis, lumped parameter model using the drift flux
formulation is used. This is denoted as “LPM” in this paper.As
power and pressure are kept constant and FIE increases, the core inlet enthalpy
(CIE) should also increase, as shown in Figure7. Due to this, more sensible
heat is available to the coolant, so the vapor generation rate increases.
Consequently, the steam quality in the core and the riser should also increase
and hence, the density of the mixture in the two-phase region should decrease.
This, in turn, increases the frictional resistance in the two-phase region. The
reduction in density will also increase the driving pressure difference due to
buoyancy effects, which may result in an increased mass flow rate. The
frictional resistance seems to be dominant. Therefore, as FIE increases, the
mass flow rate decreases (see Figure 8).Figure 7
Variation of CIE with FIE.Figure 8
Variation of mass flow rate with FIE.As
power and pressure are kept constant and FIE increases, the void fraction
should also increase as shown in Figure9. This is because of the decrease in
mass flow rate as the FIE is increased (see Figure 8). This will increase the
heat removed per unit mass of the coolant. Due to this, the steam quality
increases; hence the void fraction also increases. At the same FIE and power,
the void fraction decreases with an increase in pressure (see Figure 9).Figure 9
Variation of void fraction with FIE.


### 4.2. Variations in Critical Heat Flux (CHF)

CHF
occurs during the transition from nucleate boiling to film boiling. Typically,
this transition is accompanied by a large temperature jump on the surface at
which boiling occurs. This drastic increase in temperature can cause burnout or
failure of equipment. Therefore, it is very important to know about the CHF,
and when it will occur. In the present study, CHF is studied at the exit of the
core. The chances for the occurrence of CHF are more at the exit of the core
where the steam quality is high (at constant linear power). The presence of
more vapor will impede the liquid flow back to the heated surface, and the liquid
cannot rewet the heated surface.In the present analysis,
the PG-CHF correlation [12] in the basic form is used. This basic form uses the
local equilibrium quality and the local heat flux. In numerical simulations,
the PG-CHF correlation is used as an alternative for the “CHF table look-up”
method. This correlation uses the direct substitution method, which makes use
of the available thermal-hydraulic and geometry information, and predicts the
critical heat flux ratio (CHFR) at each point along the channel based on input
heat flux at that point.If
the mass flux is high, the chances for the coolant to wet the heater wall are
more. So, the transition from nucleate boiling to film boiling will occur at a
relatively higher heat flux. If the steam quality is high, the vapor will
surround the heater wall. So, the chances for coolant to wet the heater wall
will be less. Therefore, the transition from nucleate boiling to film boiling
will occur at a relatively lower heat flux.In
this analysis, the axial power profile is taken as uniform along the length of
the core. From Figure10, at fixed inlet conditions, with an increase in the
power, CHFR (which is equal to the ratio of the critical heat flux to local
heat flux) decreases. This is because, as power increases, the core exit steam
quality increases (see Figure 10(a)), and the mass flux decreases (see Figure 10(b)),
hence the transition from nucleate boiling to film boiling occurs quickly. So,
the CHFR decreases with an increase in the power (see Figure 10(c)).The variations of (a) steam quality, (b) mass flux, and (c) CHFR with power.
(a)(b)(c)As
shown in Figure11, with an increase in the core inlet enthalpy (CIE), the
critical heat flux value first increases and then decreases. This is because,
as CIE increases, the mass flux first increases and then decreases, and steam
quality increases monotonically (see Figures 11(a) and 11(b)). At lower core
inlet enthalpy, the mass flux increases, and steam quality also increases. But
an increase in mass flux has the more influence on CHF than the increase in
steam quality. So, the CHF value increases at lower core inlet enthalpy (see Figure
11(c)). At higher CIE, the CHF decreases with an increase in CIE. This is
because, at higher CIE, mass flux decreases, and steam quality increases with
an increase in CIE. Both effects contribute towards lower CHF as CIE increase.The variations of (a) steam quality, (b) mass flux, and (c) CHF with CIE.
(a)(b)(c)From
Figure12(b), the CHF decreases with the core length. As the core length
increases, the mass flow rate decreases (see Figure 5), and steam quality value
increases (see Figure 12(a)). Because of both of these effects, the transition from
nucleate boiling to film boiling becomes faster. So, CHF will occur at lower
heat flux (see Figure 12(b)). With an increase in the hydraulic diameter of the
active core, the CHF will increase as shown in Figure 12(b). As the hydraulic
diameter increases, the mass flow rate also increases (see Figure 6). The
quality decreases with an increase in the hydraulic diameter (see Figure 12(a)).
Variation of both of these parameters will increase the CHF (see Figure 12(b)).The variations of (a) steam quality and (b) CHF with core length.
(a)(b)From
Figure13, as the riser length increases, the CHF also increases. This is
because, as the length of the riser increases, the mass flux increases (see Figure
4), and steam quality decreases (see Figure 13(a)). This will delay the
transition from nucleate boiling to film boiling. So, the CHF will occur at
relatively higher heat flux (see Figure 13(b)).The variations of (a) steam quality and (b) CHF with core length.
(a)(b)


## 4.1. Effects of Operating and Geometric Parameters

Numerical simulations of
NCBWR were performed by the lumped parameter model as well as the RELAP5 model,
and the results were compared. The effects of various operating parameters like
pressure, power, feedwater inlet enthalpy (FIE), and geometric parameters like
riser length, core length, hydraulic diameter of core, etc were
studied. While performing the parametric studies, certain parameters were
varied, and other parameters were fixed. The input data are listed in Tables1
and 2. The input parameters are chosen based on the information available in
the literature [6] and suitable assumptions. The trends obtained are discussed
in the following paragraphs.At
constant pressure and FIE, as power increases, the mass flow rate decreases (see
Figure3), except for the low power range, in which the mass flow rate
increases with power. As power increases, the steam quality should also
increase. Due to the presence of more vapor, two effects will occur. Firstly,
with an increase in the steam quality, the density difference between the riser
section and the downcomer section will increase. This, in turn, will tend to
increase the mass flow rate. Secondly, the velocity of the coolant will
increase as the steam quality increases. This will increase the frictional
resistance and hence tends to reduce the mass flow rate. In most of the power
range considered in the present study, the frictional resistance seems to be
more dominant, so the mass flow rate decreases as shown in Figure 3.Figure 3
Variation of mass flow rate with power.At
constant power and FIE, with an increase in the riser length, the mass flow
rate increases (see Figure4). As the riser length increases, two effects will
occur. Due to longer riser length, the frictional resistance to the flow
increases that tends to decrease the mass flow rate in the system. However,
there is a gain in the buoyancy force due to the increased elevation. This will
increase the flow rate in the system. The latter effect seems to be dominant.
Hence, there is a notable rise in the mass flow rate of the system as the
length of the riser increases. As the pressure and total power are kept constant,
the mass flow rate decreases with an increase in core length (see Figure 5). As
the core length increases, the mass flow rate may increase due to increased
elevation The mass flow rate may
decrease due to an increase in frictional resistance because of longer core.
The flow area of the core is very small compared to the riser. Because of this,
the frictional resistance is dominant; hence the mass flow rate decreases with
an increase in core length. As the hydraulic diameter of the core increases,
the frictional resistance should decrease due to an increase in the flow area,
decrease in the aspect ratio, and decrease in the friction coefficient (due to
the decrease in relative roughness). Therefore, the mass flow rate increases
with hydraulic diameter of the core (see Figure 6).Figure 4
Variation of mass flow rate with riser length.Figure 5
Variation of mass flow rate with core length.Figure 6
Variation of mass flow rate with hydraulic diameter of the core.From
Figures3–6, it is observed
that the values of mass flow rate predicted by the drift flux model are closer
to those obtained with RELAP compared to the homogeneous equilibrium model.
Therefore, drift flux model is the more accurate of the two lumped parameter
models. For further analysis, lumped parameter model using the drift flux
formulation is used. This is denoted as “LPM” in this paper.As
power and pressure are kept constant and FIE increases, the core inlet enthalpy
(CIE) should also increase, as shown in Figure7. Due to this, more sensible
heat is available to the coolant, so the vapor generation rate increases.
Consequently, the steam quality in the core and the riser should also increase
and hence, the density of the mixture in the two-phase region should decrease.
This, in turn, increases the frictional resistance in the two-phase region. The
reduction in density will also increase the driving pressure difference due to
buoyancy effects, which may result in an increased mass flow rate. The
frictional resistance seems to be dominant. Therefore, as FIE increases, the
mass flow rate decreases (see Figure 8).Figure 7
Variation of CIE with FIE.Figure 8
Variation of mass flow rate with FIE.As
power and pressure are kept constant and FIE increases, the void fraction
should also increase as shown in Figure9. This is because of the decrease in
mass flow rate as the FIE is increased (see Figure 8). This will increase the
heat removed per unit mass of the coolant. Due to this, the steam quality
increases; hence the void fraction also increases. At the same FIE and power,
the void fraction decreases with an increase in pressure (see Figure 9).Figure 9
Variation of void fraction with FIE.


## 4.2. Variations in Critical Heat Flux (CHF)

CHF
occurs during the transition from nucleate boiling to film boiling. Typically,
this transition is accompanied by a large temperature jump on the surface at
which boiling occurs. This drastic increase in temperature can cause burnout or
failure of equipment. Therefore, it is very important to know about the CHF,
and when it will occur. In the present study, CHF is studied at the exit of the
core. The chances for the occurrence of CHF are more at the exit of the core
where the steam quality is high (at constant linear power). The presence of
more vapor will impede the liquid flow back to the heated surface, and the liquid
cannot rewet the heated surface.In the present analysis,
the PG-CHF correlation [12] in the basic form is used. This basic form uses the
local equilibrium quality and the local heat flux. In numerical simulations,
the PG-CHF correlation is used as an alternative for the “CHF table look-up”
method. This correlation uses the direct substitution method, which makes use
of the available thermal-hydraulic and geometry information, and predicts the
critical heat flux ratio (CHFR) at each point along the channel based on input
heat flux at that point.If
the mass flux is high, the chances for the coolant to wet the heater wall are
more. So, the transition from nucleate boiling to film boiling will occur at a
relatively higher heat flux. If the steam quality is high, the vapor will
surround the heater wall. So, the chances for coolant to wet the heater wall
will be less. Therefore, the transition from nucleate boiling to film boiling
will occur at a relatively lower heat flux.In
this analysis, the axial power profile is taken as uniform along the length of
the core. From Figure10, at fixed inlet conditions, with an increase in the
power, CHFR (which is equal to the ratio of the critical heat flux to local
heat flux) decreases. This is because, as power increases, the core exit steam
quality increases (see Figure 10(a)), and the mass flux decreases (see Figure 10(b)),
hence the transition from nucleate boiling to film boiling occurs quickly. So,
the CHFR decreases with an increase in the power (see Figure 10(c)).The variations of (a) steam quality, (b) mass flux, and (c) CHFR with power.
(a)(b)(c)As
shown in Figure11, with an increase in the core inlet enthalpy (CIE), the
critical heat flux value first increases and then decreases. This is because,
as CIE increases, the mass flux first increases and then decreases, and steam
quality increases monotonically (see Figures 11(a) and 11(b)). At lower core
inlet enthalpy, the mass flux increases, and steam quality also increases. But
an increase in mass flux has the more influence on CHF than the increase in
steam quality. So, the CHF value increases at lower core inlet enthalpy (see Figure
11(c)). At higher CIE, the CHF decreases with an increase in CIE. This is
because, at higher CIE, mass flux decreases, and steam quality increases with
an increase in CIE. Both effects contribute towards lower CHF as CIE increase.The variations of (a) steam quality, (b) mass flux, and (c) CHF with CIE.
(a)(b)(c)From
Figure12(b), the CHF decreases with the core length. As the core length
increases, the mass flow rate decreases (see Figure 5), and steam quality value
increases (see Figure 12(a)). Because of both of these effects, the transition from
nucleate boiling to film boiling becomes faster. So, CHF will occur at lower
heat flux (see Figure 12(b)). With an increase in the hydraulic diameter of the
active core, the CHF will increase as shown in Figure 12(b). As the hydraulic
diameter increases, the mass flow rate also increases (see Figure 6). The
quality decreases with an increase in the hydraulic diameter (see Figure 12(a)).
Variation of both of these parameters will increase the CHF (see Figure 12(b)).The variations of (a) steam quality and (b) CHF with core length.
(a)(b)From
Figure13, as the riser length increases, the CHF also increases. This is
because, as the length of the riser increases, the mass flux increases (see Figure
4), and steam quality decreases (see Figure 13(a)). This will delay the
transition from nucleate boiling to film boiling. So, the CHF will occur at
relatively higher heat flux (see Figure 13(b)).The variations of (a) steam quality and (b) CHF with core length.
(a)(b)


## 5. Optimization of PHT Loop

Optimizing
the material required to construct the PHT loop is very important from the
economic point of view. In the present study, sequential quadratic programming
(SQP) is adopted for the optimization procedure [13]. The SQP algorithm has
been one of the most successful general methods for solving nonlinear
constrained optimization problems. Broadly defined, the SQP method is a
procedure that generates iterates converging to a solution of the problem by
solving a quadratic problem that is an approximation to the nonlinear problem.
This type of method has been shown to be a very useful tool for solving
nonlinear problems, especially where a significant degree of nonlinearity is
present.


### 5.1. Formulation for Optimization

The
optimization procedure adopted here was to minimize the volume of the material
required to construct the PHT loop of NCBWR. The design parameters for
optimization technique are chosen as geometric parameters like length of the
core, diameter of the coolant channel, length of the riser, diameter of the
riser, and the cross sectional area of the downcomer. Appropriate upper and
lower bounds were imposed on these design variables. The steady-state lumped
parameter model (2))–(9) was used for
the calculations in the optimization procedure.The
heat flux to the coolant in the active core region should be less than the
allowable heat flux (AHF) based on CHF considerations. If the applied heat flux
crosses AHF, a stable vapor film may form on the surface of the core. If the
heat flux to the core increases further, the fuel rod temperature will become
very high. So, considering the critical heat flux constraint for optimization
is important from safety point of view.The system should be stable
for the given operating conditions and at the minimum material condition. So,
the dynamic stability of the system should be checked at the minimum material
condition. The riser length is the most influencing parameter for natural
circulation mass flow rate (see Figure4). At the minimum material condition,
the optimization code will adjust the length of the riser to the minimum
possible value. At this situation, the mass flow rate of the system is very
low. At this lower mass flow rate, the critical heat flux constraint will be
satisfied, but the system may be unstable. To avoid the possibility of
instabilities, a novel approach adopted here is to impose a minimum mass flux
constraint in the optimization procedure. The mass flux in the active core
region should be greater than or equal to a certain allowable mass flux (AMF)
value. The value of AMF has to be chosen by trial and error, depending on the
reactor power and operating conditions. If the resulting minimum material
condition turns out to be unstable, AMF can be increased and another minimum
material condition can be obtained.Considering a
typical steel material, the formula for the thickness of wall of the heater is
taken as [14]
(18)T=((P+17.26)D1240+0.99)/1000. Here, pressure is in MPa, diameter
is in mm, and thickness in meters. The volume of the material required for a
pipe is
(19)V=π4[(D+2T)2−D2]L. Here, lengths, diameters, and
thicknesses are in meters, volume is in m3.The optimization
problem is(20)minimizef(X)=VC+Vr+Vdc,subjectedto:ϕ≤AHF,G≥AMF,Xmin≤X≤Xmax, where Vc is the total volume of the
core, Vr is the total
volume of the riser, Vdc is
the total downcomer volume. Here, AHF is the allowable heat flux AHF=CHF/FOS, (FOS means the factor of safety), and X is a vector of design variables
(geometric parameters). Xmin and Xmax are lower and upper bounds imposed on the
design variables. AMF is the minimum allowable mass flux based on stability
considerations.


### 5.2. Optimization Results

Figure14 shows the plot of minimum volume of the material required to fabricate the
PHT loop versus power, at FOS of 1.4, and mass flux at 1000 kg/m2-s.
It is seen that generally the volume of the material required increases with
power. The variation of the material volume with the power is almost linear.
The real part of the least stable eigenvalue of the Jacobian matrix has been
indicated for minimum material condition at different powers in Figure 14.
Since the least stable eigenmode lies in the left half plane (LHP), the system
is stable for the given range of powers, at the minimum material condition.Figure 14
Variation of minimum volume of the material required with power.Yan et al. [15] have studied the effect of mass flow rate on the instabilities of
natural circulation boiling water reactors. The mass flow rate in natural
circulation depends on the geometric parameters and power. Therefore, the power
and geometric parameters were taken as main governing factors for stability
analysis. From Figure 15, it can be seen that, at the power of 220 MW, and for
the minimum material condition, the system is stable at high mass fluxes, and
unstable for the mass flux below 400 kg/m2-s. From this figure, it
can be inferred that the system is unstable at low mass fluxes, if designed for
the minimum material condition. Therefore, the use of the minimum mass flux
constraint in the optimization procedure is justified. The system has largest
stability margin for the mass flux of 800 kg/m2-s.Figure 15
Variation of minimum volume of the material required with mass flux.In Figure16, the material required versus
factor of safety has been plotted at power of 220 MW and mass flux of 1000 kg/m2-s.
At lower values of the FOS, the volume of the material required is almost
constant. At higher values of FOS, the volume of the material required is very
sensitive to FOS. The volume of the material required increases drastically at
higher FOS. The system is stable for the range of FOS considered.Figure 16
Variation of minimum volume of the material required with FOS.The
factor of safety should be chosen to provide sufficient safety margin in the
presence of uncertainty in CHF prediction and other model uncertainties If FOS is chosen as 1.5, then the resulting
design will be reasonably safe from the point of view of CHF as well as
stability margin. However, it will be less economical compared to the design
based on a lower FOS, for example,
1.4.Table3 shows parameters corresponding to 3 designs. Design 1 corresponds to FOS of
1.4 and mass flux of 1000 kg/m2-s. Design 2 corresponds to FOS of
1.4 and mass flux of 800 kg/m2-s. Design 3 corresponds to FOS of 1.5
and mass flux of 1000 kg/m2-s. It is seen that design 2 and design 3
are better than design 1, based on stability margin criterion (see Table 3).
Out of designs 2 and 3, from the point of view of economy, design 2 is better,
whilst from the point of view of safety and stability margin, design 3 is
better. Thus, the optimization problem requires higher level decisions based on
Pareto-optimality [16].Table 3
Design parameters at minimum material condition.
ParameterDesign 1Design 2Design 3Pressure70 bar70 bar70 barPower220 MW220 MW220 MWCore inlet
enthalpy1204.8 kJ/kg1204.8 kJ/kg1204.8 kJ/kgFactor of
safety1.41.41.5Mass flux1000 kg/m2-s800 kg/m2-s1000 kg/m2-sCoolant
channel diameter0.1076 m0.1128 m0.1105 mActive core
length3.8262 m3.389 m6.1964 mRiser diameter0.1018 m0.1063 m0.1277 mRiser length27.5323 m20.0 m28.3428 mArea of the
downcomer0.1007 m20.1011 m20.1381 m2Diameter of
the fuel rod0.011 m0.011 m0.011 mNumber of
coolant channels113113113Number of fuel
rods545454Volume of
material4.0 m33.1138 m35.9441 m3Core
exit void fraction0.80340.80250.778Stability
margin0.0902s−10.1069s−10.1481s−1


## 5.1. Formulation for Optimization

The
optimization procedure adopted here was to minimize the volume of the material
required to construct the PHT loop of NCBWR. The design parameters for
optimization technique are chosen as geometric parameters like length of the
core, diameter of the coolant channel, length of the riser, diameter of the
riser, and the cross sectional area of the downcomer. Appropriate upper and
lower bounds were imposed on these design variables. The steady-state lumped
parameter model (2))–(9) was used for
the calculations in the optimization procedure.The
heat flux to the coolant in the active core region should be less than the
allowable heat flux (AHF) based on CHF considerations. If the applied heat flux
crosses AHF, a stable vapor film may form on the surface of the core. If the
heat flux to the core increases further, the fuel rod temperature will become
very high. So, considering the critical heat flux constraint for optimization
is important from safety point of view.The system should be stable
for the given operating conditions and at the minimum material condition. So,
the dynamic stability of the system should be checked at the minimum material
condition. The riser length is the most influencing parameter for natural
circulation mass flow rate (see Figure4). At the minimum material condition,
the optimization code will adjust the length of the riser to the minimum
possible value. At this situation, the mass flow rate of the system is very
low. At this lower mass flow rate, the critical heat flux constraint will be
satisfied, but the system may be unstable. To avoid the possibility of
instabilities, a novel approach adopted here is to impose a minimum mass flux
constraint in the optimization procedure. The mass flux in the active core
region should be greater than or equal to a certain allowable mass flux (AMF)
value. The value of AMF has to be chosen by trial and error, depending on the
reactor power and operating conditions. If the resulting minimum material
condition turns out to be unstable, AMF can be increased and another minimum
material condition can be obtained.Considering a
typical steel material, the formula for the thickness of wall of the heater is
taken as [14]
(18)T=((P+17.26)D1240+0.99)/1000. Here, pressure is in MPa, diameter
is in mm, and thickness in meters. The volume of the material required for a
pipe is
(19)V=π4[(D+2T)2−D2]L. Here, lengths, diameters, and
thicknesses are in meters, volume is in m3.The optimization
problem is(20)minimizef(X)=VC+Vr+Vdc,subjectedto:ϕ≤AHF,G≥AMF,Xmin≤X≤Xmax, where Vc is the total volume of the
core, Vr is the total
volume of the riser, Vdc is
the total downcomer volume. Here, AHF is the allowable heat flux AHF=CHF/FOS, (FOS means the factor of safety), and X is a vector of design variables
(geometric parameters). Xmin and Xmax are lower and upper bounds imposed on the
design variables. AMF is the minimum allowable mass flux based on stability
considerations.


## 5.2. Optimization Results

Figure14 shows the plot of minimum volume of the material required to fabricate the
PHT loop versus power, at FOS of 1.4, and mass flux at 1000 kg/m2-s.
It is seen that generally the volume of the material required increases with
power. The variation of the material volume with the power is almost linear.
The real part of the least stable eigenvalue of the Jacobian matrix has been
indicated for minimum material condition at different powers in Figure 14.
Since the least stable eigenmode lies in the left half plane (LHP), the system
is stable for the given range of powers, at the minimum material condition.Figure 14
Variation of minimum volume of the material required with power.Yan et al. [15] have studied the effect of mass flow rate on the instabilities of
natural circulation boiling water reactors. The mass flow rate in natural
circulation depends on the geometric parameters and power. Therefore, the power
and geometric parameters were taken as main governing factors for stability
analysis. From Figure 15, it can be seen that, at the power of 220 MW, and for
the minimum material condition, the system is stable at high mass fluxes, and
unstable for the mass flux below 400 kg/m2-s. From this figure, it
can be inferred that the system is unstable at low mass fluxes, if designed for
the minimum material condition. Therefore, the use of the minimum mass flux
constraint in the optimization procedure is justified. The system has largest
stability margin for the mass flux of 800 kg/m2-s.Figure 15
Variation of minimum volume of the material required with mass flux.In Figure16, the material required versus
factor of safety has been plotted at power of 220 MW and mass flux of 1000 kg/m2-s.
At lower values of the FOS, the volume of the material required is almost
constant. At higher values of FOS, the volume of the material required is very
sensitive to FOS. The volume of the material required increases drastically at
higher FOS. The system is stable for the range of FOS considered.Figure 16
Variation of minimum volume of the material required with FOS.The
factor of safety should be chosen to provide sufficient safety margin in the
presence of uncertainty in CHF prediction and other model uncertainties If FOS is chosen as 1.5, then the resulting
design will be reasonably safe from the point of view of CHF as well as
stability margin. However, it will be less economical compared to the design
based on a lower FOS, for example,
1.4.Table3 shows parameters corresponding to 3 designs. Design 1 corresponds to FOS of
1.4 and mass flux of 1000 kg/m2-s. Design 2 corresponds to FOS of
1.4 and mass flux of 800 kg/m2-s. Design 3 corresponds to FOS of 1.5
and mass flux of 1000 kg/m2-s. It is seen that design 2 and design 3
are better than design 1, based on stability margin criterion (see Table 3).
Out of designs 2 and 3, from the point of view of economy, design 2 is better,
whilst from the point of view of safety and stability margin, design 3 is
better. Thus, the optimization problem requires higher level decisions based on
Pareto-optimality [16].Table 3
Design parameters at minimum material condition.
ParameterDesign 1Design 2Design 3Pressure70 bar70 bar70 barPower220 MW220 MW220 MWCore inlet
enthalpy1204.8 kJ/kg1204.8 kJ/kg1204.8 kJ/kgFactor of
safety1.41.41.5Mass flux1000 kg/m2-s800 kg/m2-s1000 kg/m2-sCoolant
channel diameter0.1076 m0.1128 m0.1105 mActive core
length3.8262 m3.389 m6.1964 mRiser diameter0.1018 m0.1063 m0.1277 mRiser length27.5323 m20.0 m28.3428 mArea of the
downcomer0.1007 m20.1011 m20.1381 m2Diameter of
the fuel rod0.011 m0.011 m0.011 mNumber of
coolant channels113113113Number of fuel
rods545454Volume of
material4.0 m33.1138 m35.9441 m3Core
exit void fraction0.80340.80250.778Stability
margin0.0902s−10.1069s−10.1481s−1


## 6. Conclusions

Two lumped
parameter steady-state models of NCBWR have been developed based on integral
balance equations of mass, momentum, and energy, using the drift flux model and
the HEM model for two-phase flow. Numerical simulations have been carried out
with these models as well as RELAP5/MOD3.4 code. The parametric trends obtained from the lumped
parameter models are found to be similar to those calculated with RELAP5. Of
the two lumped models, the one based on the drift flux model predicts values
closer to those computed with RELAP5.The
variation of CHF with different geometric and operating conditions is
investigated. The following parametric trends are observed.(i)
With an increase in power, the steam quality in(ii)
With an increase in CIE, the steam quality increases, the
mass flux first increases and then decreases, and CHF also first increases and
then decreases.(iii)
With an increase in the core length, the steam quality
increases and the mass flow rate decreases, hence CHF decreases.(iv)
With an increase in the hydraulic diameter of the core, the
mass flow rate increases and the steam quality decreases, hence CHF increases.(v)
With an increase in the riser length, the mass flow rate
increases and the steam quality decreases, hence CHF increases.The optimum design of the
PHT loop is obtained using the SQP algorithm based on safety and stability
constraints. The system shows instabilities at lower mass fluxes for the
minimum material condition. Three alternative designs are obtained for the
minimum material condition, subjected to different values of the constraints.



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*Source: 690357-2008-05-26.xml*