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

2. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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 (CO 2 )

3. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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 raise with small temperature increase Large impact on the fluid flow: onset of buoyancy, mixed convection conditions… Large density difference: potential for natural circulation?

4. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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

5. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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

6. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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

7. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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

8. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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

9. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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

10. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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 force increases more than the friction ‣ Friction dominated regime: further increases of the power result in a net reduction of the flow rate due to a stronger raise of the friction

11. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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 (high inlet temperature) and at high NPCH (high power): potential interesting operating point Red zone: undefined properties Impossible to reach for a boiling system

12. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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 (high inlet temperature) and at high NPCH (high power): potential interesting operating point Red zone: undefined properties Impossible to reach for a boiling system

13. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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 sudden jumps as one follows the neutral stability boundary? Stability plane is build up from different modes each with another frequency spectrum!

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

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

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

17. Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.be Ghent University-UGent 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