Stability Tests and Analysis of a Low-Pressure Natural Circulation Loop with Flashing Instability

Abstract

Natural circulation is employed in new designs of light water reactors to enhance passive safety by maintaining flow and heat removal without pumps. Under low-pressure and low-flow-rate conditions, natural circulation is susceptible to two-phase instabilities leading to undesirable flow oscillations and operational difficulties. Flashing instability is one of the most widely reported low-pressure natural circulation instabilities, related to saturated vaporization triggered by a hydrostatic pressure drop in an adiabatic riser above a heated section. While existing studies have reported flashing instability experiments, modeling, and simulations including successes in matching numerical results and experimental data, solid yet clear analytical explanations for many of its qualitative features are still rare. To enhance the physical understanding beyond stability boundary prediction, the current work develops, validates, and analyzes a linear stability model of flashing instability. This model adopts a one-dimensional Drift-Flux Model simplified by physical assumptions and approximations, and it includes optional component models to match an actual facility for validation. Stability tests are performed on a 5-m-tall natural circulation loop, providing comprehensive benchmark data covering stability boundaries, one-dimensional transient signals, and periodic mean waveforms from local measurements. Validation confirms acceptable predictions of steady states, stability boundaries, and oscillation periods. The tractable model formulation leads to a closed-form characteristic function facilitating analytical manipulations and physical interpretations, based on which dominant pressure drop responses to inlet flow rate are extracted. The major instability mechanism is identified as a strong response of the two-phase driving force to the inlet flow rate that is delayed by enthalpy transportation through a long single-phase distance and can become an overwhelmingly destabilizing positive feedback under low-frequency perturbations. Experimentally reported qualitative features, including stability changes, timescale relations, and oscillation patterns, are analytically predicted and physically explained with clarity. In general, this study enriches experimental resources of flashing instability with a comprehensive dataset and provides a simple yet realistic analytical basis for physically understanding flashing instability beyond predicting stability boundaries.

Keywords:

• Instability
• natural circulation
• two-phase flow
• flashing

Nomenclature

$A$	=	= flow area (m2)
$B$	=	= oscillation magnitude
C0	=	= distribution factor in drift-flux correlations
cg	=	= mass fraction of vapor, also known as the vapor static quality, x
$c_p$	=	= isobaric specific heat (J/kg‧K)
$D_h$	=	= hydraulic diameter (m)
$\mathrm{E}\mathrm{u}$	=	= Euler number
$\mathrm{F}\mathrm{r}$	=	= Froude number
$f$	=	= Darcy friction factor
$f_{HX}$	=	= normalized steady-state flow quality profile in the heat exchanger
$f_j$	=	= dimensionless steady-state volumetric flux profile
$f_{\epsilon }$	=	= dimensionless steady-state specific volume profile
G	=	= mass flux (kg/m2‧s)
$g$	=	= gravitational acceleration (m2/s)
$h$	=	= specific enthalpy (J/kg)
$I$	=	= inertia-related coefficient
$i$	=	= imaginary unit
$j$	=	= volumetric flux (m/s)
$K$	=	= minor loss coefficient
$L$	=	= length of the flow channel (m)
$L_A$	=	= length of the adiabatic chimney (m)
$L_c$	=	= geometric dimensionless number for the cold leg
$L_H$	=	= length of the heated section (m)
$L_h$	=	= geometric dimensionless number for the hot leg
$\mathcal{L}\left[y(t)\right]$	=	= Laplace transformation of $y$
M	=	= number of roots on the right-half complex plane
$N_f$	=	= flashing number
$N_{pch}$	=	= phase change number
$N_s$	=	= subcooling number
$P_H$	=	= heated perimeter (m)
$p$	=	= pressure (Pa)
Q	=	= heating power (kW)
$q^{\prime \prime }$	=	= heat flux (W/m2)
$q^{\prime \prime \prime }$	=	= volumetric heat source (W/m3)
$\mathrm{R}\mathrm{i}$	=	= Richardson number
Ri	=	= inner radius of annular channel (m)
Ro	=	= outer radius of annular channel (m)
r	=	= radial location (m)
$s$	=	= complex frequency (s–1)
$t$	=	= time (s)
$t_b$	=	= reference time for nondimensionalization (s)
$t_0$	=	= time label in the Lagrange representation (s)
$u_0$	=	= reference velocity for nondimensionalization (m/s)
Vgj	=	= mean drift velocity (m/s)
$⟨⟨V_{gj}⟩⟩$	=	= void-weighted drift velocity (m/s)
v	=	= velocity (m/s)
$\tilde{v}$	=	= dimensionless steady-state inlet velocity
w	=	= mass flow rate (kg/s)
w†	=	=empirical function representing the heat exchanger cooling ability (kg/s)
Y	=	= coefficient related to the mixture density–related pressure drop
$\bar{y}$	=	= steady-state value of $y$
$z$	=	= axial location along the main-stream direction (m)

Greek

α	=	= void fraction
$\beta$	=	= isobaric thermal expansion coefficient (K–1)
Γg	=	= volumetric vapor generation rate [kg/(m3‧s)]
$\gamma$	=	= phase shift
$\mathrm{\Delta }p$	=	= pressure drop (Pa)
Δpa	=	= pressure drop contribution from acceleration (Pa)
Δpg	=	= pressure drop contribution from gravity (Pa)
Δpf	=	= pressure drop contribution from friction loss (Pa)
ΔpK	=	= pressure drop contribution from minor loss (Pa)
Δpβ	=	=gravitational pressure drop contribution from thermal expansion (Pa)
$\mathrm{\Delta }T$	=	= subcooling temperature (oC)
Δz	=	= elevation change (m)
$\delta V$	=	= Laplace transformation of $\delta v_{in}$ (m)
$\delta y$	=	= infinite small perturbation of $y$
ε	=	= specific volume (m3/kg)
ς	=	= dummy variable appearing in an integral
Θ	=	= normalized temperature profile
$\theta$	=	= incline angle of the flow direction
$\mathrm{\Lambda }$	=	= friction number
$\lambda$	=	= length of the single-phase region in the hot leg (m)
ξk	=	= local axial coordinate measured from the inlet of Region k (m)
ρ	=	= density (kg/m3)
σ	=	= area ratio
$\tau$	=	= residence time (s)
${\varphi }^2$	=	= two-phase multiplier for friction loss
$\psi$	=	= characteristic function
${\psi }_s$	=	= simplified characteristic function
$\omega$	=	= oscillation angular frequency (s–1)

Subscripts

$c$	=	= cold leg
$f$	=	= liquid phase
$fg$	=	= phase change related quantity
$g$	=	= gas phase
$h$	=	= hot leg
$k$	=	= index for sections or regions
$in$	=	= heated section inlet, or regional inlet if specified with an index
out	=	= regional outlet
valve	=	= inlet globe valve
$\beta$	=	= single-phase thermal expansion
1 through 7	=	= indices of the regions in the modeled loop configuration
$1\varphi$	=	= single-phase
$2\varphi$	=	= two-phase

Superscripts

*

=

= dimensionless quantity

Disclosure Statement

No potential conflict of interest was reported by the author(s).