A Comparative Assessment on Different Aspects of the Non-Linear Instability Dynamics of Supercritical Fluid in Parallel Channel Systems
<p>Schematic view of the parallel channel system under supercritical pressure conditions.</p> "> Figure 2
<p>Stability threshold comparison at different working fluids for single and parallel channel systems.</p> "> Figure 3
<p>Stability threshold in the <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">N</mi> <mrow> <mi>tpc</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">N</mi> <mrow> <mi>spc</mi> </mrow> </msub> </mrow> </semantics></math> space at different <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">h</mi> <mrow> <mi>fd</mi> </mrow> </msub> </mrow> </semantics></math> values. (<b>a</b>) Supercritical water <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">H</mi> <mn>2</mn> </msub> <mi mathvariant="normal">O</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. (<b>b</b>) Supercritical carbon dioxide <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow> <mi>CO</mi> </mrow> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> "> Figure 4
<p>Schematic view of the inclined parallel heated channel system under supercritical pressure conditions.</p> "> Figure 5
<p>Stability threshold in the <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">N</mi> <mrow> <mi>tpc</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">N</mi> <mrow> <mi>spc</mi> </mrow> </msub> </mrow> </semantics></math> space at different <math display="inline"><semantics> <mrow> <mi>θ</mi> <mtext> </mtext> </mrow> </semantics></math> values. (<b>a</b>) Supercritical water <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">H</mi> <mn>2</mn> </msub> <mi mathvariant="normal">O</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. (<b>b</b>) Supercritical carbon dioxide <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow> <mi>CO</mi> </mrow> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> "> Figure 6
<p>Stability threshold in the <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">N</mi> <mrow> <mi>tpc</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">N</mi> <mrow> <mi>spc</mi> </mrow> </msub> </mrow> </semantics></math> space under different flow rate conditions. (<b>a</b>) Supercritical water <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">H</mi> <mn>2</mn> </msub> <mi mathvariant="normal">O</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. (<b>b</b>) Supercritical carbon dioxide <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow> <mi>CO</mi> </mrow> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> "> Figure 7
<p>Comparison stability threshold in the <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">N</mi> <mrow> <mi>tpc</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">N</mi> <mrow> <mi>spc</mi> </mrow> </msub> </mrow> </semantics></math> space with different literature data.</p> "> Figure 8
<p>Stability threshold in the <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">N</mi> <mrow> <mi>tpc</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">N</mi> <mrow> <mi>spc</mi> </mrow> </msub> </mrow> </semantics></math> space for supercritical water at <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>h</mi> <mrow> <mi>f</mi> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>.</p> "> Figure 9
<p>Stability threshold in the <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">N</mi> <mrow> <mi>tpc</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">N</mi> <mrow> <mi>spc</mi> </mrow> </msub> </mrow> </semantics></math> space for supercritical carbon dioxide at <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>h</mi> <mrow> <mi>f</mi> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>.</p> "> Figure 10
<p>Numerical simulation of the inlet velocity around different locations corresponding to <a href="#energies-15-03652-f008" class="html-fig">Figure 8</a>. (<b>a</b>) on the stable side; (<b>b</b>) on the unstable side; (<b>c</b>) on the stability boundary.</p> "> Figure 11
<p>(<b>a</b>) The nature of the first Lyapunov coefficient corresponding to <a href="#energies-15-03652-f008" class="html-fig">Figure 8</a>. (<b>b</b>) The nature of the first Lyapunov coefficient corresponding to <a href="#energies-15-03652-f009" class="html-fig">Figure 9</a>.</p> "> Figure 12
<p>Existence of supercritical Hopf bifurcation.</p> "> Figure 13
<p>Existence of subcritical Hopf bifurcation.</p> "> Figure 14
<p>Symmetric view of the temperature distribution profile.</p> "> Figure 15
<p>Wall heat effect on the stability boundary.</p> ">
Abstract
:1. Introduction
2. Modelling Methodology
- Homogenous flow is considered in both channels.
- The inlet temperature remains constant.
- The system pressure remains constant to use the thermodynamics property of the fluids.
- First ODE for the phase variable from the energy balance equation:
- Second ODE for the phase variable from the energy balance equation:
- Final ODEs for the inlet velocity of each channel:
3. Stability Analysis
3.1. Case I: Channels under Different Fluid (Supercritical and ) Flow Conditions
3.2. Case II: Channels under Equal and Unequal Heat Flux Conditions
3.3. Case III: Channels under Different Inclination Conditions
3.4. Case IV: Different Flow Rates in a Channel
3.5. Validation and Comparison
3.6. Non-Linear Analysis
3.7. Generalized Hopf Bifurcation
3.8. Subcritical and Supercritical Hopf Bifurcation
3.9. Bogdanov–Takens bifurcation
3.10. Wall Heat Effect
4. Conclusions
- Different working fluid conditions, namely, supercritical water and carbon dioxide, were considered. Thermodynamic properties show significant changes near the pseudo-critical temperature; therefore, the NIST fluid properties and appropriate approximation functions (density, enthalpy, and velocity) were used to capture the real fluid dynamics. For both types of fluids, the stability characteristics were marginally different, especially at the high pseudo subcooling number (). The Ledinegg instability region was observed in between the dynamic instability region. This interaction was observed only for supercritical water through Bogdanov–Takens bifurcation (BT point). In comparison, only Hopf bifurcation was observed for the supercritical carbon dioxide. Similar stability characteristics were previously reported for a single heated channel system.
- The heat flux distribution plays a crucial role, and in order to apply the unequal heat condition on each channel, a heat distribution coefficient ( ) was introduced. It was observed that as the heat flux ratio of both channels increased, the stability threshold shifted towards making the overall system more unstable.
- The channels used in several energy systems are often oriented at an angle. Henceforth, the effect of the inclination on the stability characteristics was analyzed. It was observed that at a high subcooling number (heavy fluid), the major component of the pressure drop is due to gravitational force, leading to the dominant inclination effect.
- The parallel channel system was also examined under different flow rate conditions in respective channels at a fixed total mass flow rate where channel 1 counterbalances the flow rate in channel 2. Here, out-of-phase oscillations were observed. Additionally, when the flow rate was fixed in channel 1 and variable in channel 2, oscillation characteristics were observed only in the latter channel.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
Cross-section area ( | |
Phase Variable | |
Specific heat at constant pressure | |
Hydraulic diameter () | |
Friction factor | |
Normalized distribution of heat flux | |
Froude number | |
Acceleration due to gravity (m/s) | |
Enthalpy (kJ/kg) | |
Localized pressure drop coefficient at the channel inlet | |
Localized pressure drop coefficient at the channel outlet | |
Channel length (m) | |
Frictional factor number | |
Sub-pseudo-critical number | |
Pseudo-critical number | |
Trans-pseudo-critical number | |
External pressure drop | |
Heat flux (W/) | |
Time (s) | |
T | Non-dimensional time |
Specific volume (/kg) | |
Velocity (m/s) | |
Distance along the axis of flow channel (m) | |
Thermal expansion number (K−1) | |
Dirac delta function ( | |
Friction dimensionless group (Euler number) | |
Heated perimeter (m) | |
Inclination angle | |
Density (kg/) | |
Subscripts | |
exit | Outlet of the channel |
In | Inlet of the channel |
i | Number of node |
Superscripts | |
Steady-state value | |
* | Dimensional quantity |
Abbreviations | |
Acceleration | |
DWOs | Density wave oscillations |
grav | Gravitational |
GH | Generalized Hopf |
fri | Frictional |
Odes | Ordinary differential equations |
PDEs | Partial differential equations |
SCFs | Super-critical fluids |
SC-CO2 | Super-critical carbon dioxide |
SCRs | Super-critical reactors |
SCW | Super-critical water |
Appendix A
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Singh, M.P.; Berrouk, A.S.; Singh, S. A Comparative Assessment on Different Aspects of the Non-Linear Instability Dynamics of Supercritical Fluid in Parallel Channel Systems. Energies 2022, 15, 3652. https://doi.org/10.3390/en15103652
Singh MP, Berrouk AS, Singh S. A Comparative Assessment on Different Aspects of the Non-Linear Instability Dynamics of Supercritical Fluid in Parallel Channel Systems. Energies. 2022; 15(10):3652. https://doi.org/10.3390/en15103652
Chicago/Turabian StyleSingh, Munendra Pal, Abdallah Sofiane Berrouk, and Suneet Singh. 2022. "A Comparative Assessment on Different Aspects of the Non-Linear Instability Dynamics of Supercritical Fluid in Parallel Channel Systems" Energies 15, no. 10: 3652. https://doi.org/10.3390/en15103652