A Numerical Algorithm for the Solution of

A Numerical Algorithm for the Solution of Simultaneous Nonlinear Equations to Simulate Instability in Nuclear Reactor and Its Analysis Goutam Dutta1 and Jagdeep B. Doshi2 1

Indian Institute of Information Technology Design and Manufacturing Jabalpur, Dumna Airport Road, Jabalpur: 482 005, Madhya Pradesh, India [email protected] http://www.iiitdmj.ac.in/Faculty/goutam.html 2 Indian Institute of Technology Bombay, Powai, Mumbai: 400 076, Maharastra, India [email protected] http://www.me.iitb.ac.in/~ doshi/

Abstract. The paper demonstrates the development of a numerical algorithm to solve a set of nonlinear hyperbolic equations in time domain in Eulerian frame of reference using a characteristics based finite difference implicit scheme to analyze density wave oscillations in boiling water nuclear reactor. The present algorithm removes the uncertainties existing in literature over the treatment of boundary conditions while simulating parallel channel instability of a reactor core by providing requisite mathematical support. The model is used to simulate parallel channel instability of a boiling water reactor core undergoing in-phase and outof-phase modes of oscillations for both forced and natural circulation systems and numerical investigation confirms the existence of type-I and type-II instabilities in appropriate conditions. Next, the numerical simulations are conducted to evaluate the relative dominance of in-phase and out-of-phase modes of oscillations under various operational regime. Keywords: Parallel channel instability, numerical model development and boundary conditions, in-phase and out-of-phase modes of oscillations, forced and natural circulation nuclear reactor systems.

1

Introduction

The boiling water reactor (BWR) core, consisting of large number of vertical channels through which coolant water flows vertically upwards between two large inlet and exit plena while extracting heat from surrounding nuclear fuel pins, is subjected to density wave oscillations (DWOs) which can trigger single and parallel channel instabilities under various operating conditions. DWO being a natural phenomena, evident from reviews by Boure et al. [1] and Leuba et al. [2] and from Leuba and Blakeman [3], can attract severe consequences unless proper safety measures are ensured and it demands extensive and accurate investigation B. Murgante et al. (Eds.): ICCSA 2011, Part II, LNCS 6783, pp. 695–710, 2011. c Springer-Verlag Berlin Heidelberg 2011

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G. Dutta and J.B. Doshi

to predict and preclude the potential instabilities, or at least to detect and take countermeasures in order to mitigate such accidental conditions, in the operational regime of the reactor. Numerical methodologies based on time-domain approaches, because of its capability of providing good accuracy and deep insight into the analysis of oscillatory instabilities, are preferred and thus proposed by many research works (Boure et al. [1], Leuba et al. [2]) in analyzing the DWOs in BWR, but found inadequate in removing the uncertainties over the boundary conditions (BCs) to be imposed while simulating parallel channel instabilities in the reactor core with such models (Chatoorgoon [4], Bragt and Hagen [5], [6], Nayak et al. [7], Lin and Pan [8], Aritomi et al. [9], Guido et al.[10]). The paper presents the development of a numerical model for the solution of simultaneous nonlinear partial differential equations (PDEs) and provides the mathematical support to define the BCs to be imposed while using such model to analyze two-phase flow boiling and to simulate parallel channel instabilities in BWR core. The governing equations, physically obtained from mass, momentum and energy conservation equations, are first identified as hyperbolic set of equations and then transformed into primitive and then into characteristic form and finally discretized to solve in time-domain in Eulerian reference frame using a characteristics based implicit and backward finite difference scheme. Various steady state and transient models are developed and integrated together and the proper BCs in correct sequence at the respective places are imposed to simulate the parallel channel instabilities in the reactor core undergoing in-phase and outof-phase modes of oscillations. The capability of TH model developed to analyze flow boiling is validated against the numerical benchmark, provided by Hancox and Banerjee [11], [12], and then its capability to simulate DWOs is validated against the available experimental results in earlier research work by Dutta and Doshi [13], [14]. Next, extensive parametric studies are carried out to find the marginal stability boundaries (MSBs) for forced circulation (FC) and natural circulation (NC) BWR systems in order to evaluate the relative dominance of in-phase and out-of-phase modes of oscillations of the respective reactors under various operating conditions.

2

Mathematical Formulation for Transient Solution

A thermal-hydraulic (TH) model, similar to a model proposed by Hancox and Banerjee [11], [12], is developed to simulate two-phase flow boiling in BWR and then extended to simulate the channel instabilities in the reactor core. The model is based on mass, momentum and energy conservation equations. The fundamental one-dimensional conservation equations to simulate BWR transients for homogeneous two-phase fluid flow model are as follows: ∂ ∂ (ρA)+ (ρAu)=0 ∂t ∂z ∂ ∂ ∂p dH ( ρAu )+ ρ A u2 = −A − τw Pw − ρ A g ∂t ∂z ∂z dz

(1) (2)

A Numerical Algorithm for the Solution

∂ ∂ (ρAe)+ ( ρ A u ef ) = qw PH ∂t ∂z

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(3)

2

where e = ef − ρp and ef = h + u2 + gH. These equations can be written in compact form as follows: ∂ ∂ R + S = T ∂t ∂z where

(4)

⎡

⎤ ⎡ ⎤ ⎡ ⎤ 0 ρA ρAu dA dH R = ⎣ ρAu ⎦ , S = ⎣ A ρu2 + p ⎦ , T = ⎣ p dz − τw Pw − ρAg dz ⎦ ρAe ρAu ef qw PH

These conservative form of governing equations are first converted into the following primitive form: ∂ ∂ U + A (U) U = D (U) (5) ∂t ∂z T

where U is a vector of unknown dependent variables [ W, h, p ] , A is a square matrix of coefficients which are functions of U, and D is a vector containing allowances for mass, momentum, and energy transfer across the system boundaries and between phases and the values of the A matrix and D vector are described by Dutta and Doshi [13], [14]. Equation of state, ρ = ρ(p, h) is used and it is to be noted that all the thermodynamic properties of the present model are subjected to changes as the pressure drops along the axial direction of the channels. The eigenvalues of matrix A determine the mathematical class of Eqs. (5) and it is found that all eigenvalues of A are real (u, u + a, u − a); and hence they are classified as hyperbolic equations. Next, the compatibility equations are obtained by transforming the PDEs into ordinary differential equations (ODEs) along the d ∂ corresponding characteristics using substantial derivatives (i.e., dt ≡ ∂t +V • ∇) and one can use these equations to work in the Lagrangian reference frame where spatial and temporal discretizations are not predefined. Characteristics equations along with corresponding compatibility equations are to be solved simultaneously to obtain all field variables defining flow, as well as spatial and temporal grid structure. The method known as MECA (Hancox and Banerjee [11], [12]) is accepted as benchmark solution, since it follows the physics of the problem very closely and thus, produces accurate results. 2.1

Transformation of Governing Equations into Characteristic Form

The system of equations (5) is hyperbolic if and only if A is diagonalizable. In other words, the system of equations (5) is hyperbolic if and only if Ψ−1 A Ψ = Λ

(6)

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where Λ is diagonal matrix whose diagonal elements λi are eigenvalues of A and Ψ is a matrix whose columns are right eigenvectors of A which indicates that A−λI Ψ =0 (7) Now, the Eq. (6) can be converted into Ψ−1 A = Λ Ψ−1

(8)

Next, premultiplying both sides of Equation (5) by Ψ−1 , one can obtain ∂ ∂ Ψ−1 U + Ψ−1 A U = Ψ−1 D (9) ∂t ∂z It also can be written in following form with the help of Eq. (8): ∂ ∂ Ψ−1 U + Λ Ψ−1 U = Ψ−1 D ∂t ∂z

(10)

The Eqs. (9) and (10) are of characteristic form of the set of Eqs. (5). Now, taking transpose of both sides of Eq. (8), it can be proved that T AT − λ I Ψ−1 =0 (11) Comparing Eqs. (7) and (11), it can be concluded that Ψ−1 is a matrix whose rows are left eigenvectors of matrix A. Now let us define B ≡ Ψ−1 and the Eq. (10) is written as ∂ ∂ B U +Λ B U = C (12) ∂t ∂z where B D=C. In spite of its accuracy, MECA requires much computation time and cannot be easily applied on geometrical complex problems because of problem dependence of spatial and temporal discretizations which also need to be determined in addition to the determination of unknown field variables using both characteristics and compatibility equations. MECA also demands the determinination of exact location of boiling boundary (BB) to take into account the drastic change of slope of charateristics while transition of liquid phase to two-phase takes place. The method adopted at present uses predefined spatial and temporal grid distributions, is based on implicit and backward finite difference method depending on the sign and slope of the charateristics and the determination of BB is optional for the overall solution procedure with the present method. Therefore, present solution procedure provides more flexiblity, is computationally faster and reasonably accurate in spite of inclusion of numerical diffusion with the adopted method. The compatibility equations, applicable to Eulerian frame of reference, are considered and used for further analysis. These compatibility equations are represented by set of Eqs. (12) where the columns of B−1 are the eigenvectors of A, Λ is a diagonal matrix of eigenvalues of A and C = BD. The values of B, Λ and C are described by Dutta [15].


3

699

Characteristic and Compatibility Equations

A similar system of equations comparable to Eq. (12) can be obtained from the fundamental Eq. (4) for different set of independent variables which will be characteristic form of governing equations in Eulerian frame of reference. These PDEs can be converted into ODEs in Lagrangian coordinate system along the characteristic directions and the corresponding transformed equations for T primitive variables U= u, h, p are shown below: Along

the

Co

since

Along

the

since

3.1

characteristic,

the

compatibility

equation

dp du + ρa = D3 + ρaD1 dt dt

∂p ∂p ∂p ∂p + (u + a) + ρa + (u + a) = D3 + ρaD1 ∂t ∂z ∂t ∂z

since

Along

C+

the

C−

characteristic,

the

compatibility

equation

dh 1 dp D3 − = D2 − dt ρ dt ρ

∂h ∂h 1 ∂p ∂p D3 +u − +u = D2 − ∂t ∂z ρ ∂t ∂z ρ characteristic,

the

compatibility

equation

dp du − ρa = D3 − ρaD1 dt dt

∂p ∂p ∂p ∂p + (u − a) + ρa − (u − a) = D3 − ρaD1 ∂t ∂z ∂t ∂z

is (13)

is (14)

is (15)

Importance of Boundary Conditions

The present discussion is aimed to stress upon the fact that judicious treatment of BCs are mandatory while solving the TH conservation equations for the present case of simulation. The PDEs used for the TH modeling are of hyperbolic in nature and three set of field variables (u, h and p) can be used to define the two-phase flow dynamics completely. A detailed discussion about the BCs, used in Euler’s equations for an aerodynamics application, was presented by Laney [16]. Similarly we, at present, will provide a brief description on it, as it is relevant for the problem of our interest. In general, the number of flow quantities required to be specified at the boundaries when there are three sets of field variables are given in the Table 1. It is already noted that all three eigenvalues of the PDEs are real, among which two are positive (Λ11 = u + a and Λ22 = u) and one is negative (Λ33 = u − a) since we are dealing with a subsonic

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G. Dutta and J.B. Doshi Table 1. Number of flow quantities required at the boundary Flow direction Flow type Number Incoming Supersonic 3 Subsonic 2 Outgoing Supersonic 0 Subsonic 1

t

dz/dt = u

dz/dt = u o

o

C _ C

_ C

+ C dz/dt = u + a

dz/dt = u −a

C

+ C dz/dt = u + a

dz/dt = u −a

Exterior

Interior

Exterior z

Subsonic inflow must specify two quantities ( 0 , + )

Subsonic flow implies u < a

Subsonic outflow must specify one quantity ( )

Fig. 1. An illustration of the number of flow quantities required to specify boundary conditions

flow (u < a) situation. One example, similar to the present case, is illustrated in Figure 1. The characteristic variables (κo , κ+ , κ− ) and the primitive variables (u, h, p) are related by the following differential equations (from the compatibility Eqs. (13), (14) and (15)): dh 1 dp D3 − = D2 − dt ρ dt ρ ⇒

⇐ C o Compatibility equation along

dκo ≡ Ko dt = dp − ρ dh = (D3 − ρ D2 ) dt

dz =u dt (16)

dp du dz + ρa = D3 + ρaD1 ⇐ C + Compatibility equation along =u+a dt dt dt dκ+ and dκ− ⇒

dκ+ ≡ K+ dt = dp + ρ a du = (D3 + ρ a D1 ) dt

(17)


701

dp du dz − ρa = D3 − ρaD1 ⇐ C − Compatibility equation along = u−a dt dt dt ⇒

dκ− ≡ K− dt = dp − ρ a du = (D3 − ρ a D1 ) dt

(18)

The characteristic variables (dκo , dκ+ ) are required to be specified for the inlet BCs. It can be noticed that the specification of a single primitive variable such as dp (or du or dh) would lead to underspecification, whereas the specification of all three primitive variables (du, dh, dp) would lead to overspecification. Therefore, the possibility remaining is to specify two primitive variables and the possible combinations are (du, dp), (dh, dp), and (du, dh). The pair (du, dp) can specify two characteristics variables, namely, dκ+ or dκ− completely, where the partial specification of dκo is only possible by this pair. Therefore, this pair can’t specify the inlet BCs. In a similar fashion, the pair (dh, dp) can specify dκo completely, but the full specification of dκ+ or dκ− is not possible by the pair. Again, it is observed that the pair (du, dh) does not even specify any of the characteristic variables completely. Therefore, it can be concluded that the any of the pairs (du, dp), (dh, dp), and (du, dh) alone can’t specify the characteristic variables (dκo , dκ+ ) completely and thus, the inlet BCs. However, it is to be noticed that dκ− is known from the interior flow, and therefore (du, dh, dκ− ) or (dh, dp, dκ− ) completely specifies (dκo , dκ+ ) as required. The problem is in the pair (du, dp) since it specifies dκ− completely - which is wrong and actually, dκ− should be determined by the interior flow. Thus, to summarize, for subsonic inflow, one can specify enthalpy and pressure, or enthalpy and velocity, but the specification of pressure and velocity simultaneously is not possible; and for subsonic outflow, one can specify any one among velocity, enthalpy and pressure. Supersonic boundaries are relatively simple as they either require all of the primitive variables or nothing, hence does not cause any conflict. 3.2

Discretized Equations

After coefficient and source terms linearizations, the system of Eqs.(5) with characteristics based finite difference scheme can be approximated by following left and right difference equations: B nk

U n+1 − U n+1 U n+1 − U nk k−1 k + Λ nk B nk k = C nk δt Δzk−1

(19)

B nk

U n+1 − U n+1 U n+1 − U nk k k + Λ nk B nk k+1 = C nk δt Δzk

(20)

Whether the difference form Eq. (19) or Eq. (20) is used depends on the flow character at mesh point k. Spatial derivatives are always approximated by backward differences when the characteristic is positive and forward differences when the characteristic is negative. For example, in case of supersonic flow from left to right, all three equations use spatial difference operators which point upstream

702


of mesh point k. For the present case of subsonic flow (u < a), the spatial derivatives for C + (i.e., Λ11 = u + a) and C o (i.e., Λ22 = u) characteristic equations, are approximated by backward difference equations, whereas the C − (i.e., Λ33 = u − a) characteristic equations, are approximated by forward difference equations. This choice of finite differencing is necessary as it facilitates a natural way of treating the BCs, such as the velocity and enthalpy or the pressure and enthalpy at the channel inlet for subsonic flow and the pressure (or the enthalpy or velocity – any one of these) at the channel outlet. 3.3

Solution Methodology

To model the boiling channel, it has been divided into K nodes, where the first node kmin = 0 and the last node kmax = K comprise the boundaries in axial direction. For k = 2 to k = K − 2 mesh points, the set of Eqs. (12) are used without any changes. For mesh point k = 1, matrix M 1,0 and source vector N 1 have been adjusted to take care of appropriate BCs depending on the problem. Similarly, matrix M K−1,K and source vector N K−1 have been modified for mesh point K − 1. To complete the adjustments for BCs, the compatibility equation, corresponding to C − characteristics, is used with the help of system of Eqs. (20) for k = 0 mesh point and next, the compatibility equations, corresponding to C + and C o characteristics, are used with the help of set of Eqs. (19) for k = K mesh point. Thus, the scheme ensures that the BCs are same as the original problem. It is to be noted that for applying BCs in our case of subsonic flow, we need three BCs, two at inlet and one at outlet. Finally, an appropriate matrix inversion technique is used to solve the 3 × K simultaneous equations for the same number of unknown variables.

4

Mathematical Formulation for Steady State Solution

To get the steady state equations, Eqs. (1), (2) and (3) are used and all terms ∂ containing ∂t have been neglected. Discretized equations used for steady state solution are as follows: ρi+1 Ai+1 ui+1 = ρi Ai ui (21a)

1 1 pi − pi+1 + ρAu2 i+1 − ρAu2 i Ai Ai+1

1 + ρ (F + g) + ρ (F + g) (zi+1 − zi ) 2 i i+1 1 q˙w q˙w (ef )i+1 − (ef )i = + (zi+1 − zi ) 2 ρAu ρAu 1 = 2

i

(21b)

(21c)

i+1

For FC BWRs, inlet velocity, pressure and enthalpy are specified at time t = 0 and then, the steady state equations along with the thermodynamic equation of state are solved to find all the variables throughout the channel. The results


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obtained from steady state model are used as initial conditions for transient analysis. Inlet and exit pressures obtained from steady state model are used as BCs to analyze in-phase mode of oscillation of the boiling reactor core. For the NC BWRs, inlet velocity is determined employing a shooting method to satisfy the specified inlet enthalpy and pressure drop BCs across the channels. If a large diameter downcomer is used in NC BWRs, the fixed pressure drop boundary condition can be calculated with the hydrostatic pressure head difference across the downcomer.

5

Model Development to Analyze Parallel Channel Instability

The BWR core consisting of several vertical channels can undergo in-phase or out-of-phase modes of oscillations depending on the operational loading conditions. Numerical models, available in the literature, used to simulate in-phase and out-of-phase modes of oscillations of BWR core in absence of neutronic feedbacks, consider hottest channels which are subjected to constant pressure drop BCs without mentioning the other BCs which are important when compressibility effect can’t be ignored because of initiation of boiling. 5.1

Boundary Conditions to Model In-phase Mode of Oscillations

It is observed that the reactor core while undergoing in-phase mode of oscillations is subjected to constant and same pressure drop across the boiling channels. Therefore, to follow the physics of the problem and to satisfy the mathematical constraints obtained from section 3.1, the in-phase oscillations are simulated with specified pressure and enthalpy as inlet BCs and specified pressure as the exit BC and the same set of BCs are used for all the channels lying in the BWR core. Details are shown in earlier research work by Dutta and Doshi [17]. 5.2

Boundary Conditions to Model Out-of-Phase Mode of Oscillations

In addition to constant pressure drop BC for the boiling channels lying in the reactor core, there is an additional constraint in the form of constant total inlet mass flow rate, is observed for the channels undergoing out-of-phase mode of oscillations during operational transients. Therefore, to follow the physics of the problem and to avoid over-specification of BCs, out-of-phase oscillations are to be simulated with specified pressure and enthalpy as inlet BCs and with specified pressure as exit BC only for the channels lying azimuthally in first half of the core. Transient solution provides the inlet mass flow rate of these channels. Total inlet mass flow rate minus the inlet mass flow rate of the channels lying in the first half of the core provides the inlet mass flow rate for the channels lying in the other half of the core and it acts as first BC. Specified enthalpy is considered to be the other inlet BC and outlet pressure is to be specified as exit BC. For the

704


next time step, the channels lying in the second half of the core are subjected to constant inlet pressure as BC, whereas the channels lying in first half of the core are subjected to specified flow rate as inlet BC. For subsequent time levels, the above mentioned steps are repeated alternately and it ensures that the channels undergoing out-of-phase mode of oscillations are subjected to constant total inlet mass flow rate and approximately constant pressure drop boundary conditions. Details along with the algorithm are shown in earlier research work by Dutta and Doshi [17]. Munoz-Cobo et al. [18], [19], [20] with their numerical model simulated the out-of-phase modes of oscillations with same inference of approximately constant pressure drop BC for two halves of the reactor, but without going into the depth of mathematical complexities to justify the procedure adopted by them. Analogy obtained through section 3.1 in the present paper provides the adequate mathematical support in deriving the required conclusion to impose the mathematically compatible BCs to follow the physics of the problem while capturing out-of-phase modes of oscillations in BWR core with the numerical TH model developed.

6

Validation of the Model

The capability of model developed to simulate boiling flow dynamics is tested against the numerical benchmark and its capability to analyze DWOs is verified comparing with available experimental results. 0.45 Discretization details : Number of nodes = 80 , dt = 1.0 ( ms ) Present model at axial location z = 0.000 ( m ) Present model at axial location z = 7.316 ( m) MECA at axial location z = 0.000 ( m ) MECA at axial location z = 7.316 ( m)

0.4

Mass flow rate ( kg/s )

0.35

0.3

0.25

0.2

0.15

0.1

0.05 0

1

2

3

4

5

Time ( s )

Fig. 2. Variation of mass flow rate with time

6

7


6.1

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Comparison with MECA Results

The TH model developed is validated against the numerical benchmark, provided by Hancox and Banerjee [11], [12], with a figure 2 which shows the comparison of variation of mass flow rate with time obtained by the present model and that obtained by Hancox and Banerjee [11], [12]. The close proximity between the results of two separate models establishes the successful development of the present model for the intended purpose. 6.2

Comparison with Solberg’s Experiments

The TH model is then extended to analyze DWOs for the reactor channels and its capability to simulate DWOs is confirmed by close agreement between MSBs (shown in Fig. 3) obtained by present model and that obtained through the experimental works by Solberg as quoted in Ph.D. thesis [21].

7

Results and Discussions: Analysis on Parallel Channel Instability of BWR

Various parametric studies are carried out to evaluate the performance of water cooled BWRs under FC and NC heat transfer cooling systems. BWR/6 [22] has been taken as the representative of FC channels and Lin and Pan [8] provides the information for NC water cooling channels. Details of the relevant input data used for FC and NC boiling channels are listed in Dutta and Doshi [14]. 12 Discretization details : Number of nodes = 120 , dt = 1.0 ( ms ) Present model Solberg’s experiment

Subcooling number ( Nsub )

10

8

6

Stable zone

4 Type--II unstable zone 2

0 10

12

14

16

18 20 22 Phase change number ( Npch )

24

Fig. 3. Comparison with Solberg’s experiments

26

28

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G. Dutta and J.B. Doshi 0.2826

Operating conditions with uniform heat source : LR = 0.0 , ki = 5.0, ke = 0.0 Subcooling number ( Nsub ) = 4.0, Phase Change number ( Npch ) = 12.0 Channel number 1 representing one half of the core Channel number 2 representing remaining half of the core

0.2824

0.2823

0.2822

0.2821

0

5

10

15

20

25

Time ( s )

Fig. 4. Variation of inlet mass flow rate with time 7

Operating conditions : LR =0.0, ki = 5.0 and ke = 0.0

6

In-phase mode of oscillation Out-of-phase mode of oscillation

5 Subcooling number ( Nsub )

Inlet mass flow rate ( kg/s )

0.2825

4

Stable zone

3

Type--II unstable zone

2

1

0

8

10

12 Phase change number ( Npch )

14

Fig. 5. In-phase and out-of-phase instabilities for BWR/6

16

A Numerical Algorithm for the Solution 9

707

Operating conditions: LR = 0.0, ki = 15.0, ke = 0.0

8

In-phase mode of oscillation Out-of-phase mode of oscillation

Subcooling number ( Nsub )

7

6

5

Type--I unstable zone

4

Stable zone

3

Type--II unstable zone

2

1

0

1

2

3 4 Phase change number ( Npch )

5

6

7

Fig. 6. In-phase and out-of-phase instabilities for natural circulation loop

Figure 4 shows the capability of the thermal hydraulics solver to simulate limit cycle oscillations during out-of-phase mode of instabilities for the two channels lying in the opposite halves of BWR/6 core at symmetric heat input conditions. Figure 5 shows the in-phase and out-of-phase marginal stability boundaries (MSBs) which divide the parameter space into stable and unstable zones for BWR/6 FC systems. It is to be noted that type-I instability does not exist for both the modes of oscillations in case of FC systems. Stability maps for NC systems undergoing in-phase and out-of-phase modes of oscillations are shown in Fig. 6. It confirms the existence of both type-I and type-II instabilities for all the parallel channel instability modes. Moreover, both the Figures 5 and 6 show that the in-phase mode of instability occurs at lower heat input than that of the out-of-phase mode for same subcooling number as far as type-II instability is concerned. Figure 6 also shows that there is not much difference in MSBs for these two modes of oscillations when type-I instability is under consideration. The trend obtained with the present results are contradictory with the results obtained by Nayak et al. [7]. They observed that type-II instability due to out-of-phase mode of oscillations took place at lower power than in-phase oscillations. Bragt et al. [6] also reported the relative dominance of out-of-phase modes of instabilities over the other. It is to be noticed that both of these works were on NC loops. To model the in-phase oscillations, they considered the whole recirculation loop where various frictional losses were taken into account and which makes the reactor core subjected to variable pressure drop boundary

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conditions, liable to change as per the loop dynamics. In our case, we have modeled the NC loop without any frictional pressure drops and local losses. It allows us to assume that the channels lying in the core are subjected to constant pressure drop boundary condition, calculated from hydrostatic head difference across the large diameter down comer which enables us to ignore inertia effects. It is already known that the channels subjected to constant (with respect to time) pressure drop boundary condition are likely to be more unstable while the others can be in stable condition. Therefore, it can be concluded that the type-II MSB for in-phase mode of oscillations will shift rightwards (i.e., at higher power levels) in Npch and Nsub plane if the loss terms and inertia effects are included in recirculation loop model.

8

Summary and Conclusions

A numerical algorithm is developed to solve a set of nonlinear hyperbolic equations in time-domain using a characteristics based implicit finite difference scheme to analyze two-phase flow boiling in the vertical channels of a BWR core and then it is extended to capture the parallel channel instabilities in the reactor core. The paper goes into the depth of mathematical complexities and provides the required mathematical support to define the compatible BCs to be imposed in the numerical model in order to simulate in-phase and out-of-phase modes of oscillations. The model treats the BCs naturally. Next, the model is used to analyze FC and NC BWR systems. Both in-phase and out-of-phase modes of oscillations are investigated for each of the FC and NC BWR. Extensive parametric studies are carried out to find the dominance of in-phase modes of oscillations over the out-of-phase modes of oscillations at the operating conditions taking into consideration for FC BWR as well as for NC BWR. Relative dominance of in-phase and out-of-phase modes of oscillations is required to be investigated when ex-core components will be included in the TH model for the sake of more realistic conclusion and will be taken up for investigation in future research work.

9

Scope of Future Work

The TH model can be extended to analyze the ex-core components of nuclear reactors and to find the effect of subcooled boiling and various boiling regimes on the stability of the reactors. Because of the capability of the model to take into account the compressibility effect of two-phase flow dynamics, it can be used to analyze loss of coolant accident problems and flashing induced instabilities. The model can handle the variation of axial heat flux conditions and therefore, it has been successfully applied to take into account the neutronic feedback in addition to TH flow feedback by Dutta and Doshi [23]. The numerical algorithm, if extended, can contribute to the worldwide research program on generation-iv supercritical water reactor and it can be applied to advanced heavy water reactor which is being developed in India.


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