Non-linear Stability Analysis for Supercritical Fluid Flow in Inclined Heated Channel

The nonlinear stability analysis of Supercritical fluids (SCFs) has been carried out using a reduced order nodalized model for the single inclined heated channel. The primary objective of the study is to portray the linear as well as nonlinear stability characteristics of SCFs flow channel along with the prospect of different types of bifurcation phenomena. The linear stability analysis is carried out with the help of the eigenvalues of the Jacobian at steady state conditions, and stability boundary is shown in the parameter plane of pseudo-sub-cooling (N_spc) and pseudo-phase-change numbers (N_tpc). The non-linear analysis includes detailed study of dynamic and static instabilities. Different types of bifurcation phenomena namely; sub-critical, super-critical and Generalized Hopf are observed; indicating various features of the dynamic instabilities. The first Lyapunov coefficient has been calculated to distinguish between sub-critical and super-critical Hopf bifurcations. Whereas in static instability; Ledinegg excursive phenomena, which is characterized as a saddle-node bifurcation, is observed. Additionally on saddle-node bifurcation curve, Bogdanov-Takens bifurcation points (as an interaction with Hopf bifurcation) appear. These bifurcations lead to complex dynamics in the system, therefore, various numerical simulations have been carried out around the stability threshold. This type of bifurcation analysis is rarely reported for SCFs in existing literature. To extend this analysis, the dependence of various system design parameters on the bifurcation curve has been investigated along with the shifting of Generalized Hopf (GH) bifurcation point. Furthermore, the effect of inclination channel on the stability threshold in parametric space is also investigated

Development of a novel nodalized reduced order model for stability analysis of supercritical fluid in a heated channel

A Novel Nodalized Reduced Order Model (NNROM) is developed in this paper to analyze the linear stability phenomena in a heated channel with supercritical water as a coolant. The existing models are based on finite volume approach, leading to a large number of non-linear time-dependent ODEs, making linear stability analysis (for infinitesimally perturbation) computationally expensive and tedious. Moreover, the non-linear stability analysis considers the effect of small but finite perturbations which becomes even more difficult. It is pointed out that the accuracy of the reduced order model developed here is not compromised, as the comparisons of the model results, with existing studies show good agreement. In ordered to develop the NNROM, the heated channel is divided into N number of nodes. The one-dimensional mass, energy and momentum conservation partial differential equations are converted into the corresponding time-dependent non-linear ordinary differential equations (ODEs) by applying the weighted residual method. The linear stability threshold of the system is determined by analyzing the eigenvalues of the Jacobian matrix at the steady states of the set of ODEs. Moreover, the linear stability boundary (Hopf bifurcation line) is represented in terms of trans-pseudo-critical phase change number, and pseudo-subcooling number. A parametric study is done to identify the change in linear stability behavior of the system with the design parameters. Furthermore, non-linear stability analysis is carried out to identify Generalized Hopf (GH) bifurcation points in the space. The GH points divide the stability boundary into sub-critical Hopf and super-critical Hopf parts, which is further varify by the numerical simulations. The identification of sub-critical region is quite important as it shows unstable limit cycles in the (linearly) stable region

A Comparative Assessment on Different Aspects of the Non-Linear Instability Dynamics of Supercritical Fluid in Parallel Channel Systems

The thermal-hydraulic behavior of supercritical water reactors with a parallel channel configuration was examined through a non-linear instability analysis. This analysis was performed under wide-ranging conditions and aspects, including different working supercritical fluids, varied heat-flux and flow-rate conditions, and channel inclinations. The supercritical fluid (SCFs) dynamics were captured using the density, enthalpy, and velocity analytical approximation functions. The major findings show that both SCFs (water and carbon dioxide) experienced density wave oscillations at a low pseudo-subcooling number. Static instability characteristics were observed for supercritical water, at a relatively high subcooling number. Further, it was found that at different heat flux, the hotter channel makes the overall system more unstable, whereas, at equal heat flux, parallel channels perform similar to a single-channel system. However, the effect of the inclination angle was found to be negligible owing to supercritical pressure conditions. Moreover, stable and unstable limit cycles along with out-of-phase oscillation characteristics were observed in dynamic stability regions. The present model was also compared with experimental and numerical data. Moreover, co-dimension and numerical simulations were performed to confirm the observed non-linear characteristics. This study helps to enhance the heat transfer characteristics during safe operation of heated channel systems, such as nuclear reactors and solar thermal systems

A Comparative Assessment on Different Aspects of the Non-Linear Instability Dynamics of Supercritical Fluid in Parallel Channel Systems

The thermal-hydraulic behavior of supercritical water reactors with a parallel channel configuration was examined through a non-linear instability analysis. This analysis was performed under wide-ranging conditions and aspects, including different working supercritical fluids, varied heat-flux and flow-rate conditions, and channel inclinations. The supercritical fluid (SCFs) dynamics were captured using the density, enthalpy, and velocity analytical approximation functions. The major findings show that both SCFs (water and carbon dioxide) experienced density wave oscillations at a low pseudo-subcooling number. Static instability characteristics were observed for supercritical water, at a relatively high subcooling number. Further, it was found that at different heat flux, the hotter channel makes the overall system more unstable, whereas, at equal heat flux, parallel channels perform similar to a single-channel system. However, the effect of the inclination angle was found to be negligible owing to supercritical pressure conditions. Moreover, stable and unstable limit cycles along with out-of-phase oscillation characteristics were observed in dynamic stability regions. The present model was also compared with experimental and numerical data. Moreover, co-dimension and numerical simulations were performed to confirm the observed non-linear characteristics. This study helps to enhance the heat transfer characteristics during safe operation of heated channel systems, such as nuclear reactors and solar thermal systems

The Effect of Nodalization Schemes on the Stability Characteristics of a Three Heated Channels under Supercritical Flow Condition

The present analysis is aimed at conducting node sensitivity analysis on the thermal–hydraulic performance of supercritical fluid in a three parallel channel configuration system. The heated channel was divided into different nodes and is examined under wide-ranging operating conditions. Firstly, the heated channel was divided into two nodes, like the two-phase flow system. In the second case, based on the correlation between the fluid properties, the heated channel was divided into three regions: heavy, mixture, and supercritical fluids. Finally, the channel was divided into N-nodes. Post the nodalization analysis, a non-linear analysis of three parallel channels was carried out under varied heat flux conditions. The analytical approximation functions were developed to capture the fluid flow dynamics. These functions were used to capture each node’s density, enthalpy, and velocity profiles under a wide range of operating conditions. The different flow instability (density wave oscillations and static) characteristics were observed at low pseudo- and relatively high subcooling numbers. In the density wave oscillations regime, out-of-phase oscillations and limit cycles are observed. A co-dimension parametric analysis with numerical simulations was carried out to confirm the obtained non-linear characteristics. Such analysis for parallel channel systems under supercritical working fluid flow conditions is missing in the literature which is limited to only linear stability analysis. This analysis can help to improve heat and mass transfer for designing efficient heated channel systems

Analysis of Supercritical CO2 Cycle Using Zigzag Channel Pre-Cooler: A Design Optimization Study Based on Deep Neural Network

The role of a pre-cooler is critical to the sCO2-BC as it not only acts as a sink but also controls the conditions at the main compressor’s inlet that are vital to the cycle’s overall performance. Despite their prime importance, studies on the pre-cooler’s design are hard to find in the literature. This is partly due to the unavailability of data around the complex thermohydraulic characteristics linked with their operation close to the critical point. Henceforth, the current work deals with designing and optimizing pre-cooler by utilizing machine learning (ML), an in-house recuperator and pre-cooler design, an analysis code (RPDAC), and a cycle design point code (CDPC). Initially, data computed using 3D Reynolds averaged Navier-Stokes (RANS) equation is used to train the machine learning (ML) model based on the deep neural network (DNN) to predict Nusselt number (Nu) and friction factor (f). The trained ML model is then used in the pre-cooler design and optimization code (RPDAC) to generate various designs of the pre-cooler. Later, RPDAC was linked with the cycle design point code (CDPC) to understand the impact of various designs of the pre-cooler on the cycle’s performance. Finally, a multi-objective genetic algorithm was used to optimize the pre-cooler geometry in the environment of the power cycle. Results suggest that the trained ML model can approximate 99% of the data with 90% certainty in the pre-cooler’s operating regime. Cycle simulation results suggest that the cycle’s performance calculation can be misleading without considering the pre-cooler’s pumping power. Moreover, the optimization study indicates that the compressor’s inlet temperature ranging from 307.5 to 308.5 and pre-cooler channel’s Reynolds number ranging from 28,000 to 30,000 would be a good compromise between the cycle’s efficiency and the pre-cooler’s size