Linear stability analysis of a supercritical loop

1. Linear stability analysis of a supercritical loop C. T’Joen, M. Rohde*, M. De Paepe * Delft University of Technology Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University – UGent

2. Introduction: supercritical fluids • Raising cycle temperature and pressure increases the thermal efficiency (Carnot efficiency) • T and p above critical condition: ‘supercritical fluid’ • Current applications: supercritical water boilers (up to 320 bar) for coal fired plants, supercritical extraction, dyeing of fabrics… • Future target applications: • Supercritical Water Nuclear Reactor (SCWR) • Supercritical Organic Rankine Cycle: heat recovery • Transcritical cooling cycles (CO2) Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

3. Supercritical fluid properties • Very strong variation of fluid properties close to the critical point • Behaviour ranges from liquidlike at low temperatures to gaslike at high temperatures. • Strong peak of specific heat capacity: large enthalpy raisewith small temperature increase • Large impact on the fluid flow:onset of buoyancy, mixed convection conditions… • Large density difference: potential for natural circulation? Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

4. Natural circulation loops • Extensive research has been published on ‘subcritical’ natural circulation loops: single and two-phase loops • Instability can occur: • Static: Ledinegg excursions, flow excursion • Dynamic: ‘density wave oscillations’: triggered by the density differences in the loop and the interaction with the pressure drop • Limited research available on supercritical natural circulation loops: • Do these phenomena also occur? Instabilities? • Numerical research conducted here Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

5. Numerical rectangular testloop • Rectangular test loop: 2m high 0.5m wide • Uniform flux heating (bottom) and cooling (top) • 1D time dependent conservation equations • Equation of state Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

6. Numerical rectangular testloop • Fluid: supercritical R23 (CHF3), scaling fluid for H2O • Pressure: 5.7 MPa • Pseudo-critical temperature: 33°C • Friction modeling: • Bends: K-factor 0.5 • Wall friction: Haaland equation (surface roughness) • Non-dimensional properties: • Subcooling number • Pseudo phase change number Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

7. Numerical implementation • Comsol multiphysics software is used: finite element solver • The conservation equations are recast into G (mass flux), P (total pressure) and h (enthalpy) • Equation of state implemented as a series of splines (based on REFPROP), care is needed to define proper derivatives from tabular data • Natural circulation: through boundary condition: static pressure inlet = static pressure exit Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

8. Numerical implementation and verification • Stability: studied through eigenvalue analysis of the linearised system • Solve the steady state problem (UMFPACK methods) • Linearise the matrix around this solution • Determine the eigenvalues (LAPACK methods) • if any have a real part > 0 unstable system • Grid independence verified: good agreement between 74, 102, 208 and 500 cells for steady state and stability predictions • Convergence criterion: 1e-8 Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

9. Model validation • Comparison with steady state data from open literature: • Jain and Uddin (supercritical CO2), Chatoorgoon (supercritical water) • Good agreement found for both systems • Stability: only 5 points by Jain and Uddin: good agreement Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

10. Results: steady state flow • Strong impact of the heater inlet temperature: a lower temperature increases the driving force • Two regimes: • Gravity dominated: increasing the heater power raises the flow rate (left side) as the driving forceincreases more than the friction • Friction dominated regime: further increases of the powerresult in a net reduction of the flow rate due to a stronger raiseof the friction Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

11. Results: stability map • Similar trends to the stability boundary of a boiling system: 2 types of instabilities (low and high frequency), but also differences! • Grid independent stability map • Clear ‘bump’ in the map at low subcooling numbers (highinlet temperature) and at highNPCH (high power): potentialinteresting operating point • Red zone: undefined properties Impossible to reach for a boiling system Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

12. Results: stability map • Similar trends to the stability boundary of a boiling system: 2 types of instabilities (low and high frequency), but also differences! • Grid independent stability map • Clear ‘bump’ in the map at low subcooling numbers (highinlet temperature) and at highNPCH (high power): potentialinteresting operating point • Red zone: undefined properties Impossible to reach for a boiling system Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

13. Results: stability map • Similar trends to the stability boundary of a boiling system: 2 types of instabilities (low and high frequency), but also differences! • Frequency plot shows suddenjumps as one follows the neutralstability boundary? • Stability plane is build up fromdifferent modes each with anotherfrequency spectrum! Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

14. Results: stability map: mode analysis • First mode: low frequencies, forming the entire left branch of the stability plot Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

15. Results: stability map: mode analysis • First mode: low frequencies, forming the entire left branch of the stability plot • Second mode: higher frequencycuts off the tip of the bump Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

16. Results: stability map: mode analysis • First mode: low frequencies, forming the entire left branch of the stability plot • Second mode: higher frequencycuts off the tip of the bump • Third mode: forms the highfrequency branch of the neutralstability boundary Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent

17. Conclusions • The stability of a natural circulation loop with a supercritical fluid (R23) was investigated • A numerical tool was developed in Comsol, and validated based on existing numerical data for steady state and stability behaviour • Results indicate similarities between the boiling loop behaviour (well known) and that of a supercritical natural circulation loop: multimodal behaviour • 3 modes detected with varying frequency that build up the stability boundary Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent