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Research into flow instability at both subcritical and supercritical pressures has attracted attention in recent years because of its potential of occurrence in industrial heat transfer systems. Flow instability has the potential to affect the safety of design and operation of heat transfer equipment. Flow instability is therefore undesirable and should be avoid ed in the design and operation of industrial equipment. Rahman et al. reviewed studies on supercritical water heat transfer with the aim of providing references for SCWR researchers. It was found out that most of the CFD studies and experimental studies were performed with single tube geometry due to the complexity of parallel channel geometry. Because studies performed with parallel channel geometry could provide detailed information to the design of the SCWR core, they called for more studies in parallel channel geometry at supercritical pressures in the future. In order to help understand how flow instability investigations are carried out and also highlight the need to understand flow instability phenomenon and equip the designers and operators of industrial heat transfer equipment with the needed knowledge on flow instability, this study carried out a review of flow instability in parallel channels with water at supercritical pressures.

The growing demand for clean energy for domestic, industrial and other uses for socio-economic gains cannot be over-emphasized. Because of this growing demand for energy, several studies have been devoted to Generation IV reactors including Supercritical Water Cooled Reactor SCWR proposed purposely for power or electricity generation in the near future. Though SCWR system has a potential of increasing thermal efficiency, issues such as materials to withstand high temperature and pressure conditions for design and construction, heat transfer and flow instability related problems have to be dealt with before its deployment for energy generation [

Studies have shown that SCWR has a potential of experiencing flow instability similar to instability that occurs in the two-phase flow systems [

As studies have shown that the efficiency of light water reactors can be improved considerably from 33% at subcritical pressures to 45% at supercritical pressures, stability of the operation of SCWR at the supercritical conditions has become a major concern to the nuclear engineers worldwide, especially around the pseudo-critical point where dramatic change of the fluid properties is experienced. Instability is undesirable as high amplitude sustained flow oscillations beyond uncontrollable limits may cause forced mechanical vibration of components, and also disturb control systems and cause operational problems in nuclear reactors [

A system is considered to be stable if it goes back to the original steady state following a perturbation or a disturbance in one form or another. In other words, the original steady state of a system is the solution of the system if the

system is disturbed and is producing slight perturbations that damp out to produce the original steady state. A system is neutrally stable if the system continues to oscillate with the same amplitude. A system is said to be unstable if it stabilizes to a new steady state or if the system oscillation continues with growing amplitude following a perturbation or a disturbance. However, it should be noted that the oscillations amplitudes cannot continue growing indefinitely for a system which is unstable. The oscillations usually form repetition patterns in forms of limit cycle oscillations. These cycle oscillations that are eventually established in the unstable system could be periodic or chaotic because of nonlinearities of the system. Because of growing nature of the oscillation amplitudes, it becomes necessary to quantify some percentage value of the oscillation amplitude below which the system stability is stable and above which the system stability is unstable based on the steady state value. This quantification of flow oscillation is needed to make it possible for numerical and experimental studies to be able to determine whether the system flow is stable or unstable. Some authors recommend that amplitude values more than ±10% or ±30% should indicate that the system is unstable [

From theoretical and experimental studies, it is well known that for dynamical systems where two phase flow occurs like BWRs there are operational points (OP) in which unstable behavior is observed. Instabilities of such systems can be subdivided into two main classes. These are:

1) Static instabilities (mostly thermal-hydraulic oscillations) and

2) Dynamic instabilities (mostly thermal-hydraulic and neutron kinetic-thermal hydraulic coupled oscillations).

A static instability is the instability type of a system having its original operating conditions moving towards new operating conditions which is not the same as the initial original operation conditions if the flow of the system is disturbed. A dynamic instability is the instability type which occurs as a result of sufficient interaction and delayed feedback between the inertia of flow and compressibility of the two-phase mixture or occurs as a result of multiple feedbacks between flow rate, pressure-drop and the change in density due to generation of vapor in the boiling system. These two static and dynamic instability types are normally classified as thermal-hydraulic instability. But flow instability is also caused by multiple feedback interactions that involve neutron flux fluctuations normally referred to as void-reactivity feedback. The dynamic instability as a result of void-reactivity feedback is normally referred to as nuclear coupled instability. Dynamic instabilities are characterised by either self-sustained periodic or diverging oscillations of the state variables. Examples of dynamic instabilities are density wave oscillations, pressure-drop oscillations, acoustic instabilities, thermal oscillations, condensation-induced instabilities (appearing in thermal-hydraulic-systems) and power oscillations (neutron kinetic-thermal hydraulic coupled oscillations).

In the context of the nonlinear BWR stability analysis, dynamic instabilities, in particular power oscillations of coupled Thermal-Hydraulic-neutron kinetic systems are of paramount interest. The coupled neutronic and thermal-hydraulic power oscillation can be categorized into the global instability (Core-wide instability) and into the regional instability. In the first mode, the global core power oscillates in-phase, while in the regional oscillating mode, the power in a half core oscillates in an out-of-phase mode with respect to the other half [

In the stability diagram (power flow map) associated with BWRs, the unstable flow or power oscillations occur in the low-flow high power region. For safety of design and operations of systems prone to flow or power excursions, this region should be avoided during normal operation. There is possibility of safety limit values including critical power ratio being exceeded which could lead to failure of system design materials giving rise to various flow or power excursions events especially when the amplitudes of flow or power oscillations become too large. These situations could lead to failure of monitoring systems [

Generally, the three similar dimensionless parameters based on 1D model used to describe flow instability boundary are provided by Gómez et al. [

Trans-pseudo-criticalnumber : N T P C = β P C C P , P C Q t M t (1)

STATIC INSTABILITY | DYNAMIC INSTABILITY |
---|---|

1) Ledinegg (flow excursion) instability | 1) Acoustic oscillations |

2) Thermal (boiling crisis) instability | 2) Density wave oscillations |

3) Flow pattern (regime) transition instability | 3) Thermal oscillations |

4) Interfacial instabilities | 4) Boiling water reactor (BWR) instability |

5) Burnout and Quenching instability | 5) Parallel channel instability |

6) Unstable vapour formation (bumping, geysering, vapor burst) | 6) Condensation oscillation |

7) Condensation Chugging | 7) Pressure drop oscillation |

8) Flashing instability | 8) Channel instability |

9) Non-equilibrium-state instability | 9) Core-wide instability |

10) Flow distribution instability | 10) Regional instability |

Date | Plant | Location | Event (as described by the operator) |
---|---|---|---|

30.06.82 | Caorso | Italy | Core instability during plant start up |

01.10.83 | Caorso | Italy | Core instability during special tests |

13.01.84 | Caorso | Italy | Instability after pump trip |

17.10.84 | S. Maria de Garona | Spain | Power oscillations during operation |

23.02.87 | TVO 1 | Finland | Power oscillations during plant start up |

09.03.88 | La Salle 2 | USA | Core instability with scram caused by neutron flux oscillation |

29.10.88 | Vermont Yankee | USA | Power oscillations |

26.10.89 | Ringhals 1 | Sweden | Instability during power ascent after refueling |

08.01.89 | Oskarshamn | Sweden | Power oscillations |

29.01.91 | Cofrentes | Spain | Power oscillations due to inadvertent entry in the reactor power-core flow map instability zone “B” |

03.07.91 | Isar 1 | Germany | Scram due to power oscillations |

15.08.92 | WNP | USA | Power oscillations |

09.07.93 | Perry | USA | Entry into a region of core instability |

01.1995 | Laguna Verde | Spain | Power oscillations during start-up |

17.07.96 | Forsmark 1 | Sweden | Local oscillations due to a bad seated fuel assembly |

08.02.98 | Oskarshamn 3 | Sweden | Power oscillations due to a bad combination of core design and control-rod pattern during start up |

25.02.99 | Oskarshamn 2 | Sweden | Power oscillations after a turbine trip with pump runback |

--11.01 | Philippsburg-1 | Germany | In-phase power oscillation |

Subcoolingpseudo-criticalnumber : N S P C = β P C C P , P C ( h P C − h i n ) (2)

The one by Gómez is represented as:

Phasechangenumber : N P C H = υ f g q ″ P H L H h f g A x - s υ f G (3)

Subcoolingnumber : N S U B = ( υ L H − υ i n ) υ i n ( h λ − h i n ) ( h L H − h i n ) (4)

And the one by Zhao is represented as:

Expansion number : N e x p = R P C p q ″ P h A c L u i n (5)

Pseudo Subcooling number : N p s u b = ( h A − h i n ) h A B ρ A − ρ B ρ B (6)

β_{pc} (1/K), C_{p}_{,pc} (J/(kg K)) and h_{pc} (J/kg) are respectively volume expansivity, specific heat and enthalpy at pseudo-critical point; and Q_{t}_{ }(W), M_{t} (kg/s) and h_{in} are respectively total heating power, total mass flow rate and inlet enthalpy of the coolant. A_{x-s} is the cross-sectional flow area (m^{2}), A_{c} is the fuel assembly cross sectional flow area (m^{2}), C_{p} is the specific heat capacity (J/(kg K)), G is mass flux (kg/m^{2 }s), h is enthalpy (J/kg), h L H is enthalpy at the exit of the heated length (J/kg), h λ is reference enthalpy (J/kg), h_{f}_{ }is the enthalpy of saturated liquid (J/kg), h_{fg}_{ }is the latent heat (J/kg), N is the characteristic non-dimensional number, N_{PCH} is the phase change number, N_{SPC} is the sub- pseudo-critical number, N_{SUB} subcooling number, N_{TPC} is the true trans-pseudo- critical number, p is the system pressure (Pa), P_{H }is the heated perimeter (m), P_{h} is the fuel rods outside perimeter per fuel assembly (m), q ″ is the uniform axial heat flux (W/m^{2}), R is the ideal gas constant (J/(mol K)), v is the specific volume (m^{3}/kg), v_{f} is the specific volume of saturated liquid (m^{3}/kg), v_{fg} is the difference between v_{g} and v_{f} (m^{3}/kg), v_{g} is the specific volume of saturated vapor (m^{3}/kg), and υ L H is the specific volume at the exit of the heated length (m^{3}/kg).

These 1D dimensionless parameters cannot be adopted to describe flow instability boundary in 3D analysis because of assumptions made in their derivations including the frictional pressure drop coefficient ξ is thought to be constant which is different from reality [

conditions to the right of the instability boundary curves are referred to as “Unstable region” to operate a system. The trends of flow instability results obtained and described in stability diagrams are almost linear or curves in most cases.

Figures 4(a)-(c) respectively show Schematic diagram of a parallel-channel test facility, Inconnel 625 pipe in the experiment and Experimental channels/ pipes in the test section of the flow instability experiment carried out by Xi et al. [

Investigation of flow instability can be carried out by three different approaches including theoretical analysis with frequency domain method (FDM); time domain method (TDM) with one dimensional (1D) and three dimensional (3D) codes; and by experiment. Because of the high temperature and pressure conditions that are associated with experiments at supercritical pressures, there

are few supercritical flow instability experiments for flow instability investigations. Most of the investigations at supercritical pressures are based on FDM and TDM [

Frequency-domain analysis is based on the linearization of nonlinear equations by perturbing the governing equations around a steady-state point. Once the linear model has been converted from time domain to a frequency domain, exact analytical solutions can be obtained. As a result, marginal stability boundaries (MSBs) in a parameter space can be determined and the space is divided into stable and unstable regions. In order to obtain stability boundaries in Time- domain analysis, the nonlinear time domain approach relies on a digital numerical simulation of nonlinear partial differential equations (PDEs) by means of finite- difference techniques [

Koshizuka et al.; Yi et al.; Jain and Corradini; and Zhao et al. performed various studies analyzing flow instability in the SCWR based on FDM. Their findings include flow instability will not occur if the inlet pressure loss coefficient is big enough; SCWR will be stable when operated under normal operation condition, but could be unstable when operated under low power condition such as start-up phase; flow instability is obtained in natural circulation loops; specific heat capacity of fuel rod and existence of water rod will favor the stability of SCWR; and parameters such as core height, axial power shape, inlet mass flow rate and density feedback have less influence on flow instability [

There are several research activities that were carried out addressing flow instability at supercritical pressures using time domain method (TDM) with one dimensional (1D) and/or three dimensional (3D or CFD) codes. In most commercial CFD codes, CFD approach adopts the fundamental governing conservation equations and these equations are solved using Finite Volume method whilst system codes (ID codes) adopt the lump parameter approach [

Dutta et al. [

Several types of studies have been made to understand and address flow instability at supercritical pressures by considering four structure types of the studied fluids. These types of investigations include studies involving single-channel stabilities, parallel-channel stabilities, reactor core flow instabilities and natural circulation or closed-loop system stabilities [

Hou et al. [

A single-phase one-dimensional model in the time domain was developed also for non-linear analysis. The results of the non-linear analysis agree quite well with that of frequency-domain analyses (

Xiong et al. [

modeling of model C. They observed also that the variation of inlet temperature with the threshold power is not linear and the threshold power is more or less proportional to the total mass flow rate irrespective of whether the flow distribution in the parallel channels is symmetrical or not (

Su et al. [_{SPC} (Pseudo-subcooling number) and N_{TPC} (pseudo-phase change number) and

dimensional numbers, heat flux and mass flow rate or inlet temperature. Their parametric investigations show that the system stability increases with increasing pressure (

subcooling pseudo-critical number region, i.e., flow stability increases in the low subcooling pseudo-critical number region (high inlet temperature region) and decreases in the high subcooling pseudo-critical number region (low inlet temperature region) with increase of inlet temperature (

Jingjing et al. [

In fact, all the above works confirmed the occurrence of instability phenomena in heated channels with supercritical fluids and much attention was paid to the 1-D dimensionless numbers adopted to describe supercritical instability boundary. There are few numerical studies, to my best of knowledge that described supercritical instability boundary using dimensional numbers, coolant inlet temperature and the ratio of critical or threshold power to mass flow rate, rather than using dimensionless numbers. These studies were performed by Xi [

Xi et al. [

Shitsi et al. [

temperatures and axial power shapes using STAR-CCM+ CFD code. They found out that the system parameters have significant effect on the amplitude of the mass flow oscillation and maximum temperature of the heated outlet temperature oscillation but have little effect on the period of the mass flow oscillation (Tables 3-6). A system with larger amplitude of flow oscillation is more unstable. The results of Shitsi et al. and experimental data used for comparison show

Power shape | Amplitude, kg/h | Periods, s | Outlet temperature, ˚C |
---|---|---|---|

Constant axial | 2.2 | 0.65 | 392 |

Uniform axial | 23.0 | 0.87 | 427 |

Axially Decreased | 9.0 | 0.83 | 404 |

Axially Increased | 14.7 | 1.0 | 408 |

Pressure, MPa | Amplitude, kg/h | Periods, s | Outlet temperature, ˚C |
---|---|---|---|

23 | 5.8 | 0.8 | 391 |

25 | 5.5 | 0.7 | 405 |

Mass flow rate, kg/h | Amplitude, kg/h | Periods, s | Outlet temperature, ˚C |
---|---|---|---|

125 | 17.0 | 0.8 | 394 |

145 | 11.8 | 1.0 | 390 |

Gravity | Amplitude, kg/h | Periods, s | Outlet temperature, ˚C |
---|---|---|---|

With gravity | 6.0 | 0.7 | 389 |

Without gravity | 5.0 | 0.7 | 388 |

that flow stability for some operating parameters decreases with coolant inlet temperature without any point of inflection. For some operating parameters, there is point of inflection below which flow instability decreases and above which stability increases with coolant inlet temperature (

section, and for homogeneous axial power shape HAPS constant heat flux is applied to the heated section. It was observed that the heating regime adopted in heating the walls of heated sections of parallel channels has significant effects on flow instability and system with HAPS is more stable than the system with ADPS.

To my best of knowledge, there are three experiments that were performed on flow instability in parallel channels with water at supercritical pressures. These experiments were performed by Xi et al. [

Xi et al. [

Xiong et al. [

only the sustained out-of-phase oscillations accompanied by evident amplitude during the experiment. They obtained stability boundaries in a two-dimensional plane using two different approaches: two ID dimensionless parameters proposed for supercritical flow (

system pressure or inlet temperature and threshold heat flux (

Zhang et al. [

two ID dimensionless parameters proposed for supercritical flow (

oscillation (2 - 5 s). Based on stability map drawn using Ambrosini’s non-dimensional parameters (

power (

increased. For Type II instability, the total inlet mass flow rate and pressure are almost constant and the mass flow rate between two channels are 180˚C out of phase (

Research into flow instability at both subcritical and supercritical pressures has attracted attention in recent years because of its potential of occurrence in industrial heat transfer systems. Flow instability has the potential to affect the safety of design and operation of heat transfer equipment. Flow instability is therefore undesirable and should be avoid in the design and operation of industrial equipment.

Rahman et al. reviewed studies on supercritical water heat transfer with the aim of providing references for SCWR researchers. It was found out that most of the CFD studies and experimental studies were performed with single tube geometry due to the complexity of parallel channel geometry. Because studies performed with parallel channel geometry could provide detailed information to the design of the SCWR core, they called for more studies in parallel channel geometry at supercritical pressures in the future. In order to help understand how flow instability investigations are carried out and also highlight the need to understand flow instability phenomenon and equip the designers and operators of industrial heat transfer equipment with the needed knowledge on flow instability, this study carried out a review of flow instability in parallel channels with water at supercritical pressures. The following are the major findings obtained as a result of this review:

・ Flow stability for some operating parameters decreases with coolant inlet temperature without any point of inflection. For some operating parameters, there is point of inflection below which flow stability decreases and above which flow stability increases with coolant inlet temperature.

・ Flow stability is influenced by operating parameters and the type of axial power shape adopted in heating the walls of the heated sections of the parallel channels.

・ An out-of-phase mass flow oscillation is observed in parallel channels when the flow distribution in the channels is no more symmetrical as a result of continuous power perturbation beyond Threshold or Critical or Boundary power of flow instability.

・ The entrance and riser sections are important to numerical modeling of flow instability in parallel channels and cannot be eliminated.

・ Amplitude of flow oscillation and stability map/diagram developed in terms of dimensionless or dimensional parameters are two main approaches used to show whether a system is stable or unstable.

・ Two types of dynamic instabilities can occur in parallel channels. Type I instability occurs at low heating powers with long period of oscillation (20 - 300 s) whereas type II instability occurs at high heating powers with short period of oscillation (2 - 5 s).

・ Increase in frictional pressure drop enhances flow stability in parallel channels.

・ The lower the power density of the hottest channel, the more stable the system will be.

・ More experimental data on flow instability should be provided to help in validation of numerical studies. The design of these experimental studies is helpful in designing similar numerical studies.

Shitsi, E., Debrah, S.K., Agbodemegbe, V.Y. and Ampomah-Amoako, E. (2018) Flow Instability in Parallel Channels with Water at Supercritical Pressure: A Review. World Journal of Engineering and Technology, 6, 128-160. https://doi.org/10.4236/wjet.2018.61008