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On the mechanism of final heat transfer restoration at the transition to gas-like fluid at supercritical pressure: A description by CFD analyses
... The use of CFD models was, at that time, highly unreliable, since the deficiencies in two-equation turbulence models were well-known, with k-ε models showing a tendency to overestimate heat transfer deterioration, while, in many instances, k-ω models showed some degree of insensitivity to the onset of heat transfer deterioration [14,15]. This situation could not be considered satisfactory, and subsequent works concentrated on the assessment and the development of more reliable turbulence models; this effort was rewarded by better predictions at least in some operating ranges [16][17][18][19][20] owing to the use of an algebraic heat flux model (AHFM) and to its implementation in a low-Re turbulence model adopted in a commercial code [21]. ...
... Acceleration and buoyancy phenomena do occur inside a vertical flow duct, owing to the huge density changes along and across the channel, caused by the variation of the temperature. Figure 3 describes one such case previously analysed by the authors in reference [20] making use of CFD, based on experimental data by Kline [28] obtained with carbon dioxide. In agreement with the experiment (comparisons are shown in reference [20]), the code predicts the occurrence of deteriorated heat transfer (DHT) soon after the entrance of the heated length in the test section (some 30 cm from the pipe inlet). ...
... Figure 3 describes one such case previously analysed by the authors in reference [20] making use of CFD, based on experimental data by Kline [28] obtained with carbon dioxide. In agreement with the experiment (comparisons are shown in reference [20]), the code predicts the occurrence of deteriorated heat transfer (DHT) soon after the entrance of the heated length in the test section (some 30 cm from the pipe inlet). In the downstream part of the pipe, the conditions are encountered for heat transfer restoration, with a subsequent increase of the turbulence. ...
The recent advancements achieved in the development of a fluid-to-fluid similarity theory for heat transfer with fluids at supercritical pressures are summarised. The prime mover for the development of the theory was the interest in the development of Supercritical Water nuclear Reactors (SCWRs) in the frame of research being developed worldwide; however, the theory is general and can be applied to any system involving fluids at a supercritical pressure. The steps involved in the development of the rationale at the basis of the theory are discussed and presented in a synthetic form, highlighting the relevance of the results achieved so far and separately published elsewhere, with the aim to provide a complete overview of the potential involved in the application of the theory. The adopted rationale, completely different from the ones in the previous literature on the subject, was based on a specific definition of similarity, aiming to achieve, as much as possible, similar distributions of enthalpies and fluid densities in a duct containing fluids at a supercritical pressure. This provides sufficient assurance that the complex phenomena governing heat transfer in the addressed conditions, which heavily depend on the changes in fluid density and in other thermophysical properties along and across the flow duct, are represented in sufficient similarity. The developed rationale can be used for planning possible counterpart experiments, with the aid of supporting computational fluid-dynamic (CFD) calculations, and it also clarifies the role of relevant dimensionless numbers in setting up semi-empirical correlations for heat transfer in these difficult conditions, experiencing normal, enhanced and deteriorated regimes. This paper is intended as a contribution to a common reflection on the results achieved so far in view of the assessment of a sufficient body of knowledge and understanding to base successful predictive capabilities for heat transfer with fluids at supercritical pressures.
... In the first applications (see e.g., [13] ), available two-equation turbulence models were adopted, which allowed for scarce success in predicting experimental data. Later on (see [12] ), a better rationale was allowed for by substantial refinements in the adopted turbulence model, making use of more than two turbulence transport equations (three to four ), which were achieved in steps [14][15][16][17][18][19] mainly using the STAR-CCM + code in its different versions (the latest, being presently used, is described in [20] ). These improvements were also accompanied by side analyses, aiming at establishing the role of wall roughness on heat transfer behaviour [21] , especially in laminarisation conditions, on heat transfer in bundles [ 22 , 23 ] and on the effect of conjugated heat transfer and its impact on predicted turbulence by resolved calculations near the heating wall [24] . ...
... The CFD code adopted in the calculations was STAR-CCM + [20] and the turbulence model was the one already adopted in recent works [ 10 , 11 , 19 ] based on the Lien k-ε low-Re turbulence model [30] equipped with an Algebraic Heat Flux Model (AHFM), as already discussed in several previous works devoted to turbulence modelling improvement (see [15][16][17][18][19] ). The modelling solutions here adopted are similar to the ones used in all the mentioned previous works, based on the use of segregated flow and energy equations, with second order treatment for most of the advection terms. ...
... Again the predictions by the turbulence model are coherent with experimental data, showing an occasional anticipation of deterioration (see e.g., Fig. 8 ) or its spurious prediction while experimental data do not show any (see the single case at T in = 16 °C in Fig. 7 ). While the obtained results are anyway to be considered relatively good in terms of predictive capabilities, considering the present state-of-the-art in the field, what is particularly relevant for the present discussion is that the heat transfer main phenom- In relation to the quality of the predicted data, as obtained by the use of an algebraic heat flux model (AHFM), it must be considered that previous work (see [14][15][16][17][18][19] ) showed a reasonable applicability of this approximation for a sufficiently wide range of addressed data. Though this model cannot be considered perfect, its capabilities and limitations were clearly shown in the previous mentioned papers and it is here used because of its sufficient quantitative and qualitative coherence with the experimental data to which it is applied, thus serving as a tool to support the validity of the similarity theory by numerical means. ...
Recent papers by the Authors proposed a successful rationale to establish a similarity theory for heat transfer at supercritical pressures, making use of original definitions that seemingly solved the problem of finding a closely similar phenomenological behaviour with very different fluids. As promised in the conclusions of the latest publication, this paper reports on the result of the work performed by further exploring the features of the theory, confirming its choices in front of additional evidence.
In particular, several further RANS calculations, made by the same improved turbulence model adopted in recent works, have been performed, assessing various additional features of the theory and finding excellent confirmations of its capabilities. While waiting for experimental confirmations, RANS analyses are here used as a suitable workbench to compare data obtained by four different fluids (water, carbon dioxide, ammonia and R23) for “similar” conditions. The predicted behaviour, in addition to be conditioned to the limited capabilities of the adopted turbulence model, can be attributed to the different thermodynamic and thermo-physical properties, i.e., to intrinsic features of supercritical fluids. The obtained results further clarify relevant issues, as the most appropriate selection of supercritical pressure for each fluid in front of a given reference condition, together with the level of accuracy achieved in establishing similarity and the role of the most important dimensionless numbers.
It is shown that the reported results depict a very clear picture of the role played by the different boundary conditions in determining the relevant heat transfer regimes, with specific reference to deterioration and recovery, opening the possibility for a thorough and physically based revision of the only partially successful engineering correlations for heat transfer proposed up to now.
... The two experimental cases reported in Fig. 16 from the carbon dioxide database by Kline [29] were selected both for the good prediction obtained by the adopted model (see [23] for a through discussion of model capabilities and limitations in relation to these data) and because they represent bounding phenomena. On the one side, the case with lower inlet temperature exhibits a severe deterioration, beyond the pseudocritical threshold, with wall temperature oscillations and the occurrence of a final recovery of heat transfer at the transition to the gas-like phase (see [23] for a discussion of the phenomenon). ...
... The two experimental cases reported in Fig. 16 from the carbon dioxide database by Kline [29] were selected both for the good prediction obtained by the adopted model (see [23] for a through discussion of model capabilities and limitations in relation to these data) and because they represent bounding phenomena. On the one side, the case with lower inlet temperature exhibits a severe deterioration, beyond the pseudocritical threshold, with wall temperature oscillations and the occurrence of a final recovery of heat transfer at the transition to the gas-like phase (see [23] for a discussion of the phenomenon). The second case is instead characterised by the lack of deterioration phenomena, since the inlet subcooling is very low and buoyancy forces have not enough strength to produce laminarisation and the subsequent deterioration. ...
The present paper introduces a successful and general fluid-to-fluid similarity theory for heat transfer to fluids at supercritical pressure, having a high degree of universality. This work shortly follows the recent publication of a “local” successful similarity theory developed for fluids at supercritical pressures in a range of conditions in which the values of their molecular Prandtl number were quantitatively similar, extending its conclusions to the case of different molecular Prandtl numbers. The reason why this further step requested a short time to be elaborated is due to recognising that previous work by the Authors had actually already solved the related problems, though in a slightly different way, now interpreted in a more significant frame owing to a better problem understanding.
The present similarity theory is based on first ideas developed more than one and a half decade ago by one of the authors, while addressing flow stability of supercritical fluids in heated channels, which encountered immediate problems to be applied in a straightforward way to heat transfer. These ideas were revised and considerably improved during the PhD thesis of the other author, also overcoming a sort of prejudicial assumption that finally resulted to limit their applicability. More recently, published DNS data triggered further reflections on the role of the Prandtl number, leading to the mentioned “local” form of the successful similarity theory. This led to the present step, by just recognising that the mentioned PhD thesis had already proposed a sufficient rationale to extend this local interpretation to a broader range of conditions.
The rather convincing results presented herein, obtained making use of RANS CFD analyses with four different fluids, demonstrate the interesting capabilities of this final form of the theory. The establishment of an effective set of dimensionless numbers for heat transfer problems is hoped to pave the way for the development of the still lacking successful engineering heat transfer correlations for supercritical pressure fluids. It further calls for dedicated experiments needed to confirm the suitability of the present theory beyond any reasonable doubt.
... Researchers have endeavored to enhance the accuracy of RANS modeling within SCF by introducing modified turbulence models. Noteworthy examples include the variable turbulent Prandtl number model [202,203], the variable damping function model [204], and the algebraic heat flux model [205]. These modifications have led to improvements in predicting turbulent heat transfer to SCFs in both vertical and horizontal flows, compared to conventional models like the k − ε model. ...
Supercritical fluids (SCFs) hold potential in the fields of energy and advanced propulsion, highlighting the significance of comprehensively investigating SCF flow and heat transfer characteristics. The intricate and nonlinear thermophysical property variations of SCFs coupled with the primitive variables in the conservation equations pose several challenges in effectively modeling and simulating SCF flows and heat transfer. This paper conducts a thorough assessment of commonly used equations of state and look-up tables for describing the thermophysical properties of SCFs. The data-driven methods based on machine learning for SCFs are also discussed. The challenges associated with direct numerical simulation, Reynolds-averaged simulation, and large-eddy simulation of SCFs are examined. Emphasis is placed on the evaluation and discussion of the issue of turbulence modeling strategies that stem from low-pressure or ideal-gas conditions directly applied to SCF flow and heat transfer. The primary objective is to provide guidance for future research, thereby advancing and promoting the modeling and simulations of SCF flows and heat transfer.
... In particular, Fig. 21 and Fig. 22 refer to one of Kline's CO 2 experiments that shows a quite interesting behaviour, consisting in an entrance region of nearly normal heat transfer, followed by the onset of deterioration and by huge wall temperature oscillations terminating with a higher level of wall temperature; this higher temperature persists up to the point in which heat transfer is restored when buoyancy effects become low enough as a consequence of the bulk fluid approaching the pseudo-critical conditions and causing a more uniform density distribution across the pipe. The latter phenomenon is clearly observed in data sets as the one by Kline that explore conditions in which the fluid enters as a liquid-like one and exits as a gas-like one (see also Kim et al., 2008 for additional examples) and was explained in previous work making use of CFD techniques (Buzzi et al., 2019). As it can be noted in the two figures, the adoption of either the Bishop et al. or the Mokry et al. formulation in "correlation mode", owing to the use of the experimental values of wall temperature, keeps in the predicted trend a weak reminder of the oscillating behaviour exhibited by the experiments; however, in "prediction mode", i.e. after repeated iterations on the wall temperature values, the correlations provide a completely smooth, mildly increasing trend of wall temperature, without any sign of heat transfer deterioration. ...
... In view of the heat transfer deterioration (HTD), many scholars have tried to analyze the intrinsic mechanism in detail. Buzzi et al. (Buzzi et al., 2019) tried to provide a description of the reasons for its occurrence with numerical simulation. The simulation results of wall temperature correspond well with the experimental results. ...
... In particular, Fig. 21 and Fig. 22 refer to one of Kline's CO 2 experiments that shows a quite interesting behaviour, consisting in an entrance region of nearly normal heat transfer, followed by the onset of deterioration and by huge wall temperature oscillations terminating with a higher level of wall temperature; this higher temperature persists up to the point in which heat transfer is restored when buoyancy effects become low enough as a consequence of the bulk fluid approaching the pseudo-critical conditions and causing a more uniform density distribution across the pipe. The latter phenomenon is clearly observed in data sets as the one by Kline that explore conditions in which the fluid enters as a liquid-like one and exits as a gas-like one (see also Kim et al., 2008 for additional examples) and was explained in previous work making use of CFD techniques (Buzzi et al., 2019). As it can be noted in the two figures, the adoption of either the Bishop et al. or the Mokry et al. formulation in "correlation mode", owing to the use of the experimental values of wall temperature, keeps in the predicted trend a weak reminder of the oscillating behaviour exhibited by the experiments; however, in "prediction mode", i.e. after repeated iterations on the wall temperature values, the correlations provide a completely smooth, mildly increasing trend of wall temperature, without any sign of heat transfer deterioration. ...
... The theoretical design of heat transfer structures is seriously affected by this phenomenon, and it is essential to figure out the physical mechanism of HTD. Regarding HTD studies of supercritical fluids, the steady heat transfer process is commonly focused on [11][12][13][14], particularly for various operating conditions [15]. Besides, the HTD characteristics can be affected by the instability of the flow and heat transfer [16], and the starting stage of heat transfer is also a critical step. ...
For the supercritical n-decane horizontally flowing in a rectangular channel of an active regenerative cooling system, a transient thermal-fluid-structure coupling method is employed to investigate the unsteady thermal-hydraulic characteristics and the wall deformation at a starting stage. The temperature distributions of the fluid domain and solid domain along the flow direction are investigated at fixed times as well as at a certain cross-section. Streamlines in cross-sections are employed to explain the temperature distribution. The velocity and pressure at a fixed point versus time are also given. Besides, the solid deformation is presented according to the uneven pressure distribution and temperature distribution. It is found that the response time is less than 30 seconds when the heat flux is less than 3.0 MW/m2. A larger heat flux contributes to promoting the steady-state. The high-temperature part of the solid domain is close to the heated wall, but the situation is reversed for the fluid domain. This is because a bunch of dead-zone vortices appears in the vicinity of the upper wall of the channel. The maximum deformation is 0.132 mm for the condition of heat flux 3.0 MW/m2 and it is exacerbated by the uneven temperature and pressure distributions on the solid domain.
... operating conditions (Pizzarelli et al., 2015;Sunden et al., 2016) such as heat flux (a crucial factor to the abnormal heat transfer), turbulence models (Pandey et al., 2017;Gustavo et al., 2018), predictive correlations (Zahlan et al., 2017;Shokri and Ebrahimi, 2019), the effect of buoyancy force and thermal acceleration (Jackson, 2017;Yu et al., 2013), flow instability (Wang et al., 2013a;Yan et al., 2018) and enhanced heat transfer Xu et al., 2015), also including a transient study of HTD (Dutta and Giridhar, 2017). In our opinion, the effect of the wall conduction is a matter of concern due to the sensibility to the heat flux (Buzzi et al., 2019;Ehsan et al., 2018). However, the impact of wall conduction on the fluid temperature distribution is rarely studied in the couple heat transfer process. ...
Purpose ¬- The purpose of this paper is to numerically study the influence of wall conduction on the heat transfer of supercritical n-decane in the active regenerative cooling channels.
Design/methodology/approach - A horizontally placed rectangular pipe with a solid zone and another one without a solid zone were employed. A drastic variation of thermo-physical properties was emphatically addressed. After the verification of mesh and turbulence models comparing with the experimental results, a mesh number of 4.5M and the low Reynolds number SST k-ω turbulence model were chosen, respectively. The solution of the governing equations and the acquisition of the numerical results were executed by the commercial software FLUENT 2020 R1.
Findings - The numerical results indicate that there is a heat transfer deterioration (HTD) potential for the upper wall, lower wall and sidewall with the decrease of mass flux. Due to wall conduction, the distribution of the fluid temperature at spanwise-normal planes becomes uniform and this feature also takes advantage of the relatively uniform transverse velocity. For the streamwise-normal planes, the low fluid temperature appears close to the upper wall at the region near the sidewall and vice versa for the region near the center. Undoubtedly, the secondary flow at the cross-section plays a crucial role in this process and the relatively cool mainstream is affected by the vortices.
Originality/value - This study warns that the wall conduction must be considered in the practical design and thermal optimization due to the sensibility of thermo-physical properties to the heat flux. The secondary flow caused by the buoyancy force (gravity) plays a significant role in the supercritical heat transfer and mixed convection heat transfer should be further studied.
Supercritical fluids are used as coolants in one of the Generation-IV reactors (i.e., supercritical water reactor) owing to their good diffusivity, low viscosity, and high specific heat. Additionally, these fluids exist at higher pressure and temperature which allows high thermal efficiency. Two heat transfer phenomena are related to supercritical fluids: heat transfer deterioration and enhancement. These phenomena made it difficult for Reynolds-averaged Navier–Stokes simulation (RANS)-based turbulence models to accurately predict the heat transfer. In this study, an assessment of RANS-based turbulence models is conducted for supercritical carbon dioxide under two different flow conditions (i.e., horizontal flow and natural circulation vertical flow). The two cases are simulated using current turbulence models (i.e., SST k-ω, k-ε, RNG k-ε) and a newly developed model based on the algebraic heat flux model (AHFM), hereafter called UniPi. It was found that for the horizontal flow case, the SST k-ω model captured the temperature difference induced by buoyancy between different regions of the wall, however, with poor accuracy in predicting wall temperatures. The RNG k-ε models captured the behavior of wall temperature across all regions with underestimated values. The enhanced wall treatment gives good predictions of wall temperatures compared to experimental data, but it underestimates the deterioration and recovery of heat transfer. In the natural circulation case, the recently developed model, which is based on AHFM, yielded better results compared to k-ε and the SST k-ω models. This is mainly because it explicitly considers the buoyancy production term and the turbulent heat flux.
The paper addresses the problem of predicting deteriorated heat transfer to supercritical pressure fluids, trying a novel approach whose main features are obtained by consideration of the experimental trends of a meaningful series of experimental data. This approach takes into consideration the modifications that the axial velocity profile undergoes at the entrance of a bare tube, when a supercritical pressure fluid is heated at the wall and decreases its density owing to thermal expansion, giving rise to buoyancy effects. Combining the observation of measured wall temperature trends with their successful predictions by CFD RANS models, the effects giving rise to flow laminarization are firstly interpreted as a very peculiar form of the entry length problem applicable to supercritical pressure fluids. Then, their quantitative prediction is addressed by the use of an iterative procedure based on shape functions tailored on the observed trends of the Nusselt number and of the consequent wall temperature trends. In this frame, the results of a recently developed fluid-to-fluid similarity theory for heat transfer at supercritical pressures provide indication about the relevant dimensionless numbers to be involved in determining the shape functions. Being based at the moment on a series of water experimental data collected at 25 MPa for vertical upward flow, the methodology may suggest a way for predicting heat transfer deterioration in the proposed supercritical water nuclear reactor conditions, providing an additional means to tackle this critical problem by a different approach that may result useful as a practical means to bound expected wall temperature trends.
The paper analyses experimental data exhibiting heat transfer deterioration phenomena at supercritical pressure resulting in quite different trends of wall temperature. The two modes of deteriorated heat transfer considered here are both due to the laminarisation of flow, owing to lighter fluid appearing at the wall as a consequence of heating. However, in one of the two modes, mainly occurring in the liquid-like region at sufficiently large bulk subcooling with respect to the pseudocritical threshold, after a local deterioration heat transfer is restored almost at normal levels, while in the other, at lower pseudo-subcooling and involving at a larger degree also the gas-like region, a jump of wall temperature to a stable deteriorated condition occurs. In the latter case, the wall temperature finally decreases only when the bulk fluid approaches the pseudocritical temperature, thus causing a smaller density difference between bulk and wall, whose increase was at the root of buoyancy effects leading to deterioration.
The two heat transfer deterioration modes are here investigated considering the measured trends of wall temperature in two relevant data sets exhibiting the two behaviours and by the use of a CFD model for predicting detailed data. It is shown that while the first mode of deterioration can be interpreted as an entry-length problem, related to strong buoyancy effects, the second mode is complicated by an excursion to a different manifold of what can be defined as a pseudo-boiling curve, occurring when the imposed heat flux cannot be sustained by normal or enhanced heat transfer. In the latter case, an interesting analogy can be considered with phenomena related to the boiling crisis occurring at subcritical pressures, though the causes of such similar behaviours are obviously quite different. The CFD model, validated by comparison with experimental data of wall temperature, is used as a tool to generalise the observed trends and to draw conclusions.
The work is carried out in the frame of studies for assessing and possibly improving heat transfer correlations for supercritical pressure fluids, for the use in the future licensing of supercritical water reactors (SCWRs), in the aim to gain better understanding and to suggest possible modelling strategies. Conclusions are proposed in this regard.
Under the background of carbon reduction, the development of efficient power conversion system has drawn more important attention. The advantages of high efficiency and compactness have triggered increasing industrial interests in the applications of supercritical carbon dioxide power system for broad heat sources. However, when the supercritical carbon dioxide power cycle is coupled with different heat sources, there exist both general problems mainly associated with the supercritical carbon dioxide cycles and unique problems due to the heating processes of supercritical carbon dioxide. Moreover, the multiscale features of the supercritical carbon dioxide power system lead to significant challenges to theoretical analysis and engineering design. This review work provides the recent advances of supercritical carbon dioxide power systems, especially for the applications in heat sources of nuclear, solar and fossil fuel. With a focus on the category of general problems and unique problems, the current studies concerning the coupling of supercritical carbon dioxide power cycle with a specific heat source are further divided into three scales of system, component and process. The designs of the reactor core in nuclear, the receiver in concentrated solar power plant and the boiler in coal-fired power plant are discussed in detail at the scales of system, component and process. Future research directions for supercritical carbon dioxide power system are identified at different scales and more accurate integrated modelling for the coupling of supercritical carbon dioxide power cycle with specific heat sources are encouraged. The present work will benefit the in-depth understanding and further promotion of supercritical carbon dioxide power systems in industrial applications.
Supercritical water (SCW) is one of the high-profile fluids between scholars, which has great potential in the thermal industry. Therefore, it may serve as a promising fluid utilized in nuclear and boiler systems. This paper presents a comprehensive review in terms of the published literature on supercritical water heat transfer in channels. Three types of heat transfer, involving normal, enhanced and deteriorated heat transfer, are classified in this article. Firstly, the thermo-physical properties of supercritical water are illustrated to explain the reasons for the different heat transfer behavior. Heat transfer enhancement and deterioration appear to have greater potential and draw some researchers’ interest. In short, transformation of geometry or operating conditions, such as, channel structure, mass flux and heat flux, would induce variations of thermo-physical properties. Consequently, heat transfer performance can be significantly affected. The heat transfer characterization and affecting factors were evaluated. Some unusual phenomena were explained. Then, the developed heat transfer coefficients (HTC) were compared with the experimental data available in the open literature. The results indicated that the reliable accuracy could be achieved through these correlations except the pseudo-critical region. Additionally, the mechanism and the onset of heat transfer deterioration were revealed and estimated. Finally, this review also identifies the current challenges regarding this research through analysis and comparison. These challenges deserve to be efficiently solved and optimized to enhance supercritical water heat transfer in the future.
This review discusses the recent application of artificial intelligence (AI) algorithms in five aspects of computational fluid dynamics: aerodynamic models, turbulence models, some specific flows, and mass and heat transfer. Currently, there are three main coupling models. The first is the data-driven model to obtain the input-output relationship without involving any physical mechanisms. The second is the physical model to optimize the existing models by AI algorithms. The third is the hybrid model involving both data and physical mechanisms. Among various AI algorithms, artificial neural network is usually applied to build data-driven models and has been successfully employed in the mentioned five fields. Other AI algorithms such as recursive neural network, support vector machine, and naive Bayes are mainly used for the physical models. Finally, the development tendency of coupling models and how to choose an appropriate model are given in the conclusions and prospects.
A similarity theory, proposed with a limited success some years ago and subsequently refined in a more complex form in further efforts, has been applied in a recent published work to perform Direct Numerical Simulations (DNS) of heat transfer to turbulent flow, with four different fluids at supercritical pressure. The obtained results showed an exceptionally good behaviour of the theory in the addressed cases, suggesting that the initial proposal, though it had only limited success in the cases considered at that time, possibly caught some of the basic features to be preserved in scaling.
The theory, based on dimensionless definitions that provided a reasonable degree of universality in the analysis of flow stability, found immediate difficulties to be applied with a comparable success to heat transfer problems. These difficulties mainly stemmed from the fact that, while it is relatively easy to scale fluid density, having a major role in stability analyses, it is definitely much harder to scale at a comparable level of accuracy the fluid thermo-physical properties, relevant in heat transfer. The very good results obtained in the recent work by DNS stimulated new reflections that shed light on the merits and limitations on the old theory.
The present paper, starting from these recent results and discussing them in front of RANS calculations, is aimed to highlight the promising features of this theory, envisaging the missing steps that should be completed to make it more general, in order to give to its consequences a higher level of universality.
There have been relatively few publications detailing heat transfer to supercritical water (SCW) flowing through a channel with a bundle or just with a single rod (annular channel) as compared to heat transfer to SCW in bare tubes. In the present paper, results of experimental heat transfer to SCW flowing upward in an annular channel with a heated rod equipped with four helical ribs and a 3-rod bundle (rods are also equipped with four helical ribs) are discussed. The experimental results include bulk-fluid-temperature, wall-temperature, and heat-transfer-coefficient (HTC) profiles along the heated length (485 mm) for these flow geometries. Data obtained from this study could be applicable as a reference estimation of heat transfer for future fuel-bundle designs.
The results on heat transfer to supercritical water heated above the pseudo-critical temperature or affected by mixed convection flowing upward and downward in vertical tubes of 6.28-mm and 9.50-mm inside diameter are presented. Supercritical water heat-transfer data were obtained at a pressure of 23.5 MPa, mass flux within the range from 250 to 2200 kg/(m{sup 2}s), inlet temperature from 100 to 415 deg. C and heat flux up to 3.2 MW/m{sup 2}. Temperature regimes of the tubes cooled with supercritical water in a gaseous state (i.e., supercritical water at temperatures beyond the pseudo-critical temperature) were stable and easily reproducible within a wide range of mass and heat fluxes. An analysis of the heat-transfer data for upward and downward flows enabled to determine a range of Gr/Re{sup 2} values corresponding to the maximum effect of free convection on the heat transfer. It was shown that: 1) the heat transfer coefficient at the downward flow of water can be higher by about 50% compared to that of the upward flow; and 2) the deteriorated heat-transfer regime is affected with the flow direction, i.e., at the same operating conditions, the deteriorated heat transfer may be delayed at the downward flow compared to that at the upward flow. These heat-transfer data are applicable as the reference dataset for future comparison with bundle data. (authors)
A numerical investigation of the heat transfer deterioration (HTD) phenomena is performed using the low-Re k-É turbulence model. Steady-state Reynolds-averaged Navier-Stokes equations are solved together with equations for the transport of enthalpy and turbulence. Equations are solved for the supercritical water flow at different pressures, using water properties from the standard IAPWS (International Association for the Properties of Water and Steam) tables. All cases are extensively validated against experimental data. The influence of buoyancy on the HTD is demonstrated for different mass flow rates in the heated pipes. Numerical results prove that the RANS low-Re turbulence modeling approach is fully capable of simulating the heat transfer in pipes with the water flow at supercritical pressures. A study of buoyancy influence shows that for the low-mass flow rates of coolant, the influence of buoyancy forces on the heat transfer in heated pipes is significant. For the high flow rates, buoyancy influence could be neglected and there are clearly other mechanisms causing the decrease in heat transfer at high coolant flow rates.
The present paper reports the results of analyses concerning heat transfer to supercritical pressure fluids, performed by adopting a k - epsilon turbulence model modified in association with the AHFM model, here used for calculating both buoyancy effects and the turbulent heat flux. The promising capabilities of this approach were already highlighted in past studies and the present paper represents a further step in this line of research.
Experimental data concerning supercritical carbon dioxide flowing in tubes are here considered, with operating conditions involving both high and low mass flux values and spanning from relatively low inlet
temperatures to values higher than the pseudocritical threshold. Some of the interesting features appearing in the experimental data are correctly reproduced by the model, which manages to predict reliable
wall temperature trends, both qualitatively and quantitatively.
The performed analyses, though reporting successes in a sufficiently wide range of operating conditions, suggest that some parameters of the proposed model should be varied in accordance with the boundary conditions, e.g. considering the mass flux, in order to improve predictions in the most challenging situations.
The paper discusses the latest improvements obtained in performing heat transfer calculations by RANS turbulence models when dealing with fluids at supercritical pressure.
An algebraic heat flux model (AHFM) is adopted as an advanced tool for calculating the turbulent Prandtl number distribution to be used in the energy equation. Though maintaining a simple gradient approach, the proposed model manages to obtain interesting results when dealing with temperatures spanning from low to supercritical values.
This is due to the introduction of a correlation for defining one of the relevant AHFM parameters. As stated in previous works, in fact, single fixed constant values could not be sometimes sufficient for dealing with very different operating conditions such as the ones occurring with supercritical fluids. In order to make the relation suitable for different fluids, a dependence on a non-dimensional quantity which proved to be relevant by parallel work is assumed.
Some sensitivity analyses are also performed, showing some interesting capabilities in reproducing a sort of threshold behaviour when working in transition regions.
Buoyancy induced phenomena are much better captured than in past attempts, though incomplete accuracy is observed for some boundary conditions.
The paper discusses capabilities and limitations of Algebraic Heat Flux Models in predicting heat transfer to supercritical fluids. The model was implemented in a commercial code and used as a basis for obtaining an advanced definition of the turbulent Prandtl number and an improved estimate of the buoyancy production of turbulence kinetic energy. A comparison between the obtained results and experimental data available in literature is performed highlighting promising features, in particular when dealing with trans-pseudo-critical conditions. Experimental conditions using different fluids where analysed showing improvements with respect to two-equation turbulence models; a reference DNS calculation is considered as well for comparison. Calculated wall temperature values are in general well reproduced by the methodology and sensitivity analyses show that improvements may be obtained in future works by selecting case-specific AHFM parameters in association with different turbulence models.
The High Performance Light Water Reactor is a nuclear reactor concept of the 4th generation which is cooled and moderated with supercritical water. The concept has been worked out by a consortium of European partners, co-funded by the European Commission. It features a once through steam cycle, a pressure vessel type reactor, and a compact containment with pressure suppression pool. The conceptual design enables to assess its feasibility, its safety features and its economic potential.
The paper summarises the results obtained in the assessment of different turbulence models including low-Reynolds k-epsilon and k(theta)-epsilon(theta) equations, in the attempt to improve the prediction by RANS techniques of heat transfer to fluids at supercritical pressure. The work has been mainly developed in two phases. Firstly, 4-equation models available in literature were applied to a broad range of experimental data making use of the relationships suggested in their formulations for evaluating turbulent thermal diffusivity. These models were herein used with an Algebraic Heat Flux Model (AHFM), aiming at evaluating the turbulent heat flux; in the present work it was used only in the formulation of turbulence production due to buoyancy while the SGDH was used in the energy equation. In a second phase, the same models were applied repeatedly to a subset of the addressed experimental information with different calculation options, including constant values of the turbulent Prandtl number, mixing models for k-epsilon and k(theta)-epsilon(theta) equations in order to identify possible improvements. The results show that recourse to these models, which are more complex than common 2-equation ones, provides limited improvements in the comparison with experimental data.
The feature of fluids at pressures just above the critical value which makes them of particular interest is that they change in a continuous manner from being liquid-like to gas-like with increase of temperature at constant pressure. As a consequence of the extreme dependence of fluid properties on temperature, non-uniformity of density can lead to important effects on the mean flow and turbulence fields and heat transfer effectiveness. When the author and his colleagues first commenced research on supercritical pressure fluids many years ago it was decided to begin with a novel experiment specifically designed to include effects of strong non-uniformity of fluid properties whilst avoiding other complications associated with the temperature dependence of density. This fundamental experiment on stably stratified turbulent flow of supercritical pressure carbon dioxide between two horizontal planes, with the upper one heated and the lower one cooled, in such a way that there was no net heat transfer to the fluid, yielded evidence of a special mechanism for enhancement of turbulent mixing. Later, experiments with uniformly heated vertical tubes using carbon dioxide at pressures very near to the critical value gave results, which exhibited further striking features. Severe localized non-uniformity of heat transfer developed in the case of upward flow, but was not found with downward flow. Gravitationally induced motion caused effects on heat transfer which could only be explained by postulating drastic modification of turbulence. Such results stimulated the development of physically based ideas concerning the mechanisms which might be involved and led to the development of a semi-empirical model of buoyancy-influenced turbulent flow and heat transfer. The main aim of this paper is to show how such early work is now providing a basis for correlating experimental data and enabling the complicated phenomena encountered in those early experiments to be properly accounted for in thermal design procedures.
Turbulent heat transfer is an extremely complex phenomenon, which has challenged turbulence modellers over various decades. The limitations of the commonly used eddy diffusivity approach have become more evident specially for innovative nuclear reactor applications with low-Prandtl fluids like liquid metals. One of the objectives of the THINS (Thermal Hydraulics of Innovative Nuclear Systems) project sponsored by the European Commission is to push forward the validation and adoption of more accurate closures for single-phase turbulence for innovative reactors in engineering codes. As a part of this THINS project, CD-adapco has implemented in its commercial code STAR-CCM+ an algebraic turbulent heat flux model. In the present work, this implemented model has been widely tested and further calibrated for the application to natural, mixed and forced convection flow regimes at low-Prandtl number. As an outcome, a modelling correlation is proposed in combination with a newly proposed set of model coefficients. This proposed correlation shows dependency of Reynolds and Prandtl numbers in a logarithmic manner to accommodate the wall-normal temperature gradient for the heat flux term. The use of this correlation brings significant improvements in the prediction of heat transfer in liquid metals in all flow regimes.
Performance of turbulence modelling for supercritical pressure heat transfer in a upward tube flow is investigated by modelling the case simulated by (Bae et al., 2005) with the direct numerical simulation (DNS). Three major assumptions, i.e. (∂p¯/∂xi)≅ρ¯gi, ρ′=−βρ¯t′ and ρu″ih″¯≅Cpρu″it″¯ are pointed out as the major limitation of existing turbulent models for their application in supercritical pressure heat transfer in an upward flow. For the sake of model evaluation three representative model combinations are selected, i.e. (i) the General Gradient Diffusion Hypothesis (GGDH) model (buoyancy production of turbulent kinetic energy) and the Simple Gradient Diffusion Hypothesis (SGDH) model (heat flux), (ii) the Algebraic Flux Model (AFM) (both buoyancy production of turbulent kinetic energy and heat flux), (iii) the Elliptic Blending-Algebraic Flux Model (EB-AFM) (both buoyancy production of turbulent kinetic energy and heat flux). In general, prediction of turbulent kinetic energy by the EB-AFM model agrees with the DNS data better than the other two. And the agreement is especially good in the near-wall region. Similar to the turbulence kinetic energy, prediction of the radial turbulent heat flux by the EB-AFM model is good in the near-wall region. Good performance of the EB-AFM model in the near-wall region implies that the model holds the correct asymptotic feature towards the wall. In the bulk performance of the EB-AFM model is unsatisfactory because of strong variation of fluid properties. The GGDH + SGDH and AFM model fail to predict the streamwise turbulent heat flux, while the EB-AFM model can qualitatively predict it. The direct simulation by Bae et al. showed that the heat transfer recovery in the downstream is due to negative ∂(ρu″xh″)/∂x∂(ρu″xh″)/∂x. The investigated models do not predict such a phenomenon. Recovery of heat transfer is predicted by the EB-AFM model mainly due to more efficient heat transfer in the radial direction in the downstream. Contrary to the other models, the EB-AFM appears promising aspects as a candidate for further optimization.
This paper presents an analysis of heat-transfer to supercritical water in bare vertical tubes. A large set of experimental data, obtained in Russia, was analyzed and a new heat-transfer correlation for supercritical water was developed. This experimental dataset was obtained within conditions similar to those in supercritical water-cooled nuclear reactor (SCWR) concepts.The experimental dataset was obtained in supercritical water flowing upward in a 4-m long vertical bare tube with 10-mm ID. The data were collected at pressures of about 24MPa, inlet temperatures from 320 to 350°C, values of mass flux ranged from 200 to 1500kg/m2s and heat fluxes up to 1250kW/m2 for several combinations of wall and bulk-fluid temperatures that were below, at, or above the pseudocritical temperature.A dimensional analysis was conducted using the Buckingham Π-theorem to derive the general form of an empirical supercritical water heat-transfer correlation for the Nusselt number, which was finalized based on the experimental data obtained at the normal and improved heat-transfer regimes. Also, experimental heat transfer coefficient (HTC) values at the normal and improved heat-transfer regimes were compared with those calculated according to several correlations from the open literature, with CFD code and with those of the proposed correlation.The comparison showed that the Dittus–Boelter correlation significantly overestimates experimental HTC values within the pseudocritical range. The Bishop et al. and Jackson correlations tended also to deviate substantially from the experimental data within the pseudocritical range. The Swenson et al. correlation provided a better fit for the experimental data than the previous three correlations at low mass flux (∼500kg/m2s), but tends to overpredict the experimental data within the entrance region and does not follow up closely the experimental data at higher mass fluxes. Also, HTC and wall temperature values calculated with the FLUENT CFD code might deviate significantly from the experimental data, for example, the k–ɛ model (wall function). However, the k–ɛ model (low Reynolds numbers) shows better fit within some flow conditions.Nevertheless, the proposed correlation showed the best fit for the experimental data within a wide range of flow conditions. This correlation has an uncertainty of about ±25% for calculated HTC values and about ±15% for calculated wall temperature. A final verification of the proposed correlation was conducted through a comparison with other datasets. It was determined that the proposed correlation closely represents the experimental data and follows trends closely, even within the pseudocritical range. Finally, a recent study determined that in the supercritical region, the proposed correlation showed the best prediction of the data for all three sub-regions investigated.Therefore, the proposed correlation can be used for HTC calculations in SCW heat exchangers, for preliminary HTC calculations in SCWR fuel bundles as a conservative approach, for future comparison with other datasets and for the verification of computer codes and scaling parameters between water and modelling fluids.
The paper presents results obtained on the prediction of three-dimensional turbulent heat transfer to CO2 at supercritical pressure, flowing upward through heated vertical passages of non-circular cross-section. The problem is relevant to certain technical applications, including the cooling of fuel bundles in supercritical water nuclear reactors. Several k–ε turbulence models, implemented in the FLUENT code, are used; they include low-Reynolds number formulations and an RNG k–ε model with a two-layer near wall treatment. The reference experimental data are from a recent published study in which CO2 at supercritical pressure was used to cool uniformly heated channels of triangular and square cross sections.In keeping with previous work on circular ducts, the results obtained using the low-Reynolds number models were able to reproduce the trend of heat transfer deterioration due to buoyancy influence, but with a relatively large overestimation of measured wall temperatures. On the other hand, the RNG model with the two-layer approach was unable to capture even the expected qualitative trends. The results obtained in the current study are useful in confirming the mechanisms of heat transfer deterioration and also highlight interesting three-dimensional features related to flow redistribution under strong buoyancy forces.
To calculate complex turbulent flows with separation and heat transfer, we have developed a new turbulence model for flow field, which is modified from the latest low-Reynolds-number kappa-epsilon model. The main improvement is achieved by the introduction of the Kolmogorov velocity scale, upsilon(sub c) equivalent to (nu epsilon)(exp 0.25), instead of the friction velocity upsilon(sub t), to account for the near-wall and low-Reynolds-number effects in both attached and detached flows. The present model predicts quite successfully the separating and reattaching flows downstream of a backward-facing step, which involve most of the essential physics of complex turbulent flows, under various flow conditions. We have also discussed in detail the structure of the separating and reattaching flow based on the computational results, and presented several important features closely related to the mechanism of turbulent heat transfer.
The book includes an appendix that describes advances made in two-phase flow as applicable to reactor safety code modeling. It is considered by many to be the definitive work in the field. Current theories on boiling and two-phase flow as well as supercritical heat transfer are presented. Also examined are the similarities and dissimilarities in the mechanism of the two modes of heat transfer. Research in two-phase has demonstrated that the behavior of boiling two-phase flow is highly affected by the stability and dynamics of the entire system. Interest in the study of dynamics of a boiling two-phase flow is increasing, possibly due to the increased interest in high pressure water nuclear reactors. Introductions and summaries are included for each chapter.
In order to study a turbulent flow having an externally imposed time
dependence, a turbulent channel flow apparatus was constructed. The
apparatus consists of a two dimensional channel having a flexible upper
wall that can be deformed by a series of cams which articulate the wall
as a sinusoidal travelling wave. Measurements of the periodic component
of the fluid pressure induced by this undulating wall were made for both
laminar and turbulent channel flows and these measurements compared with
results from theoretical models. The turbulence models include a
quasilaminar model, a mean eddy viscosity model, and a turbulent kinetic
energy closure model. For the long waves studied here, each of these
closure models produces satisfactory agreement with measured
perturbation pressures provided that the calculation is carried out in a
suitably transformed coordinate system.
The paper proposes dimensionless parameters for the analysis of stability in heated channels with supercritical fluids. The parameters are devised basing on the classical phase change and sub-cooling numbers adopted in the case of boiling channels, proposing a novel formulation making use of fluid properties at the pseudo-critical temperature as a function of pressure. The adopted formulation for dimensionless density of a given fluid provides a unique dependence with respect to dimensionless enthalpy, in a reasonably wide range of system pressures, thus giving generality to the predictions of unstable conditions obtained as a function of dimensionless parameters. It is shown that these parameters allow setting up quantitative stability maps for a single heated channel with imposed overall pressure drop, in analogy with the ones proposed in previous work concerning boiling channels. Similarities with the boiling channel stability phenomena are pointed out, also supporting the conclusions with system code predictions.
The reduction in turbulent, convective heat transfer parameters observed in some supercritical data and in experiments with common gases can be due to radial property variation, acceleration, buoyancy or combinations of these phenomena, depending on the conditions of the applications. To date criteria for the onsets of these effects have been developed for vertical circular tubes. This note presents extensions of these criteria to non-circular ducts with constant cross-sections as in the cooling channels of some advanced nuclear reactors.
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An experimental study on heat transfer deterioration at supercritical pressures
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Aerospace Engineering, University of Ottawa, Ottawa Canada.
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Super Light Water Reactors and Super Fast Reactors
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Analysis of heat transfer phenomena with supercritical fluids by four equation turbulence models
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Engineering, University of Pisa, Italy.
Analysis of fluiddynamic and heat transfer phenomena with supercritical fluids
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Numerical Analysis of HTD and PD in 3-rod bundle & Benchmark. 3rd RCM on ''Understanding and Prediction of Thermal-Hydraulics Phenomena Relevant to Supercritical Water-Cooled Reactors (SCWRs)
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Supercritical Water-Cooled Reactors (SCWRs)", 26-29 June 2017. University of
Wisconsin -Madison, Wisconsin, US.
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