Linear stability analysis of a supercritical loop

Department of Flow, Heat and Combustion Mechanics – www.FloHeaCom.UGent.beGhent University – UGent

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.beGhent 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 (CO2)


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?


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


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

0

z

M

tA

jj zzKA

fP

A

MAg

z

pA

M

zt

M

42

22

t

pAq

z

Mh

t

hA

'

hf


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 pcin

hPCH hAG

LPqN

"

pc

inpcSUB h

hhN


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


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


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


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


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




• 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




• Frequency plot shows suddenjumps as one follows the neutralstability boundary?

• Stability plane is build up fromdifferent modes each with anotherfrequency spectrum!


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




• Second mode: higher frequencycuts off the tip of the bump

• Third mode: forms the highfrequency branch of the neutralstability boundary


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

Linear stability analysis of a supercritical loop

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