The gravitational effect in the linear and non-linear dynamics of boiling channels
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ORIGINAL
The gravitational effect in the linear and non-linear dynamics of boiling channels A. Coleff 1,2 & C. P. Marcel 1,2,3 & D. F. Delmastro 1,2 Received: 1 June 2019 / Accepted: 25 March 2020 # Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The effect of gravity on the dynamics of flow boiling channels is studied using linear and non-linear stability analysis tools. The developed tools are used to predict experimental results showing good agreement. The linear stability analysis shows that gravity enhances the density wave instability mechanism, enlarging the linear unstable region. Non-linear effects are then investigated by using Hopf bifurcation characterization, allowing identifying meta-stable operating conditions, associated with subcritical bifurcations. In addition, different dynamic basins have been found numerically, which correspond to sub and supercritical Hopf bifurcations. From the results obtained with the model presented here, it is concluded the gravitational effect is found to enhance the density wave instability mechanism, for both the linear and the non-linear domain. This result indicates neglecting gravity in stability studies may lead to nonconservative results. Furthermore, the gravitational effect tends to expand the region where the system exhibits a subcritical behavior. Keywords Boiling channels stability . Gravitational effects . Hopf bifurcations
NomenclatureNormal alphabet A Cross sectional area (m2) k Local pressure drop coefficient (−) L Length (m) f Frictional factor (−) g Gravitational acceleration (m s−2) h Enthalpy (J kg−1) M Mass flow in the channel (kg) p Pressure (kg m−1 s−2) q Volumetric power transferred from fuel to coolant (J s−1 m−3) s Variable in the Laplace domain (s−1)
* C. P. Marcel [email protected] A. Coleff [email protected] D. F. Delmastro [email protected] 1
Instituto Balseiro, S. C. de Bariloche, Argentina
2
Centro Atómico Bariloche, S. C. de Bariloche, Argentina
3
Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Buenos Aires, Argentina
t u U z
Time (s) Coolant velocity (m s−1) Laplace transformation of e u (−) Axial variable (m)
Non-dimensional variables NEu Euler number N Eu ¼ ρ ΔP ∙u2 f
ref
u 2 N Fr ≡gLrefref
NFr
Froude number
Ng
Gravitational number N g ≡ u
NPCH
Phase change number N PCH ≡AChqutoti ρ hfg ρlρ−ρv
Nsub
Subcooling number N sub ≡
gLref
ref 2 ðk e þ1Þ l
hl;sat −hl;inlet ρl −ρv hfg ρv
v
=Ω · ν
Greek letters δi Dirac’s delta function (−) ε Small perturbation parameter (−) λ Boiling boundary location (m) ν Single-phase residence time of a fluid particle (s) Ω Characteristic phase change frequency (s−1) υ Specific volume (kg−1 m3) ρ Coolant density (kg m−3) τ Transport delay (s) ω Angular linear frequency (rad−1) ϕ Angular phase delay between Ui and Λ (o) Λ Laplace transformation of e λ (−)
Heat Mass Transfer
Subscripts and superscripts 0 characteristic Ch Channel DW Density wave e Channel exit f fluid fg Difference between saturated vapor
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