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Impact of system pressure on the characteristics of stability boundary for a single-channel flow boiling system

  • Deepraj Paul
  • Suneet SinghEmail author
  • Surendra Mishra
Original Paper
  • 24 Downloads

Abstract

The two-phase flow boiling channel as a nonlinear dynamical system has very interesting features. Parametric stability analysis of a flow boiling thermal hydraulic system has been extensively studied in the past. Linear stability analysis provides a stability boundary which is a demarcation between stable and unstable region. The studies on the impact of system pressure on the stability boundary are limited. Moreover, the variation of subcritical and supercritical Hopf region on stability boundary with respect to pressure is not available in the literature. Nonlinear dynamics is used here to characterize the stability boundary(Hopf curve) for a single-channel flow boiling thermal hydraulic system. In the current work, focus is on the effect of the system pressure on the location of the generalized Hopf (GH) point over the Hopf curve which divides subcritical and supercritical Hopf region. The disappearance of the GH point with increasing system pressure, such that the stability boundary is completely subcritical, in the operating parameter range is one of the interesting features observed. The complete subcritical boundary is identified by noting that the first Lyapunov coefficient is always positive on the stability boundary for higher system pressures. The subcritical Hopf bifurcation indicates the presence of unstable limit cycle in the (linearly) stable region which is of prime concern as finite perturbation in this region leads to system instability. Numerical simulations are carried out around the stability boundary to verify its characteristics.

Keywords

Bifurcation Subcritical Hopf Supercritical Hopf Generalized Hopf Thermal hydraulics Mathematical modelling 

List of symbols

Greek symbols

\(\delta \)

Dirac delta function

\(\lambda \)

Boiling boundary (m)

\(\lambda ^+\)

Non-dimensional boiling boundary \((\frac{\lambda }{L_H})\)

\(\Lambda _{1\phi }\)

Single-phase friction number \(\left( \frac{f_{1\phi }L_H}{2D_H}\right) \)

\(\Lambda _{2\phi }\)

Two-phase friction number \(\left( \frac{f_{2\phi }L_H}{2D_H}\right) r\)

\(\rho \)

Density \(({\hbox {kg/m}}^3)\)

\(\rho ^+\)

Non-dimensional density \(\left( \frac{\rho }{\rho _f}\right) \)

Subscripts

0

Steady state

e

Channel exit

f

Liquid

g

Vapour

H

Heated channel

i

Channel inlet

n

nth node in single-phase region

Other symbols

\(\Delta P\)

Pressure drop (Pa)

\(\Delta P^+\)

Non-dimensional pressure drop \(\left( \frac{\Delta P}{\rho _f u_\mathrm{s}^2}\right) \)

A

Area \(({\hbox {m}}^2)\)

\(C_{pf}\)

Liquid constant pressure specific heat \(({\hbox {J/kg \,K}})\)

D

Diameter (m)

f

Friction factor

\(f_{1\phi }\)

Single-phase friction factor

\(f_{2\phi }\)

Two-phase friction factor

\(\textit{Fr}\)

Froude number \(\left( \frac{u_\mathrm{s}^2}{g L_H}\right) \)

g

Gravity acceleration \(({\hbox {m}}^2/{\hbox {s}})\)

h

Enthalpy \(({\hbox {kJ/kg}})\)

\(h^+\)

Non-dimensional enthalpy \(\left( \frac{h - h_f}{h_f}\right) \)

k

Resistance coefficient

L

Length (m)

\(L^+\)

Non-dimensional length \((L/L_H)\)

M

Mass (kg)

\(M^+\)

Non-dimensional mass \(\left( \frac{M}{\rho A_H L_H}\right) \)

\(N_\mathrm{s}\)

No. of nodes in single-phase region

\(N_\mathrm{exp}\)

Thermal expansion number \(\left( \frac{\beta h_{fg} \rho _g}{C_{pf}\rho _{fg}}\right) \)

\(N_\mathrm{pch}\)

Phase change number \(\left( \frac{Q}{\rho _f A_H u_\mathrm{s}}\frac{\rho _{fg}}{h_{fg}\rho _g}\right) \)

P

Pressure (Pa)

Q

Heating power (W)

\(q^{''}\)

Steady-state heat flux \(({\hbox {W/m}}^2)\)

t

Time (s)

u

Velocity (m/s)

\(u^+\)

Non-dimensional velocity \((u/u_\mathrm{s})\)

\(u_\mathrm{s}\)

Steady-state velocity (m/s)

z

Axial coordinate (m)

\(z^+\)

Non-dimensional axial coordinate \(\left( \frac{z}{L_H}\right) \)

x

Equilibrium quality

\(t^+\)

Non-dimensional time \(\left( \frac{t}{t_\mathrm{ref}}\right) \)

\(N_\mathrm{sub}\)

Subcooling number \(\left( \frac{h_i - h_f}{h_{fg}\rho _g}\rho _{fg}\right) \)

\(t_\mathrm{ref}\)

Timescale \(\left( \frac{L_H}{u_\mathrm{s}}\right) \)

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Nuclear Power Corp. of India LtdMumbaiIndia
  2. 2.Indian Institute of Technology-BombayMumbaiIndia

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