•Constitutive equations for a vertical rod bundle are reviewed.•Dependence of distribution parameter on flow conditions is discussed.•1D momentum equation by considering void fraction distribution is ...discussed.•Effect of channel size on interfacial area concentration is discussed.
In view of the quality assurance of two-phase flow simulations, CSAU (Code Scalability, Applicability, and Uncertainty) methodology and code V & V (Verification and Validation) have been proposed. The estimation of simulation uncertainty is indispensable in using best-estimate computational codes. A key of successful two-phase flow simulations is to use the state-of-the-art constitutive equations to close the mathematical system used in two-phase flow analyses. The advanced constitutive equations should be developed based on “physics” behind phenomena and should consider scaling parameters which enable their application beyond test conditions used for a code validation. Two-phase flow simulations in a rod bundle is important in various industrial apparatuses such as heat exchangers and nuclear reactors. Constitutive equations for two-phase flows in a vertical rod bundles have been advanced in recent five years. In view of this, this paper provides a comprehensive review of most advanced constitutive equations for two-phase flow analyses in a vertical rod bundle. The constitutive equations of two-phase flow parameters reviewed in this paper are flow regime map, void fraction, void fraction covariance and relative velocity covariance, interfacial area concentration and wall friction. In addition, an exact formulation of one-dimensional momentum equation in two-fluid model considering void fraction distribution is discussed.
•Channel aspect ratio affected the distribution parameter in the drift-flux model.•Asymptotic distribution parameter was approximated at 1.2 for square channels.•Hydraulic diameter was used as ...characteristic length in a drift velocity correlation.•Drift-flux correlation predicted void fractions within ±13.9 % for adiabatic flow data.•Drift-flux correlation predicted void fractions within ±8.72 % for boiling flow data.
Numerous industrial components and production processes encounter two-phase flows in rectangular channels. The drift-flux model, mathematically expressed in terms of the drift-flux parameters, such as the drift velocity and distribution parameter, is often utilized to solve two-phase flow problems. The constitutive equations of the drift-flux parameters have been limitedly developed and assessed for dispersed two-phase flows in rectangular channels. This study discusses the effects of geometrical parameters, such as the characteristic length and aspect ratio of a rectangular channel, on dispersed two-phase flow parameters. For the flow conditions of collected data in rectangular channels, it was observed that the hydraulic diameter was appropriate as the characteristic length scale for the drift velocity calculation since the drift flux correlation using the hydraulic diameter correlated better with experimental data. The asymptotic distribution parameters in the constitutive equation of the distribution parameter were approximated to be 1.35 and 1.20 for rectangular channel aspect ratios less than 0.25 and greater than 0.33, respectively. Due to the lack of data, the asymptotic distribution parameters were tentatively determined based on a linear interpolation scheme between 1.35 and 1.20 for the aspect ratios between 0.25 and 0.33. The constitutive equations of the drift-flux model were assessed using the collected data with various fluids, pressures, and flow conditions. The constitutive equations of the drift-flux model could predict the void fractions with negligible bias and random uncertainty of ±0.0358 and ±0.0434 (absolute value measure) and ±13.9 % and ±8.72 % (relative value measure) for adiabatic and boiling two-phase flows in rectangular channels, respectively.
•Effects of inlet boundary conditions on interfacial area concentration changes are investigated.•Some non-uniform inlet boundaries significantly affect interfacial area concentration ...changes.•Predictive capability of the IATE calculation is evaluated under inlet boundary conditions.•IATE's predictive capability deteriorates for several non-uniform inlet flow boundary conditions.
Bubbly flows appear in many heat and mass transfer equipment. The interfacial area concentration is a critical parameter in determining the performance of the heat and mass transfer systems. The one-dimensional one-group interfacial area transport equation (IATE) was developed based on area-averaged bubbly flow data collected under uniform inlet flow boundary conditions. The applicability of the IATE to the flow systems with non-uniform inlet flow boundary conditions was not comprehensively examined. This study investigated the effect of inlet liquid and gas velocity profiles on measured interfacial area concentration changes along the flow direction. The data used in this study were collected for vertical upward air-water two-phase flow in a rectangular channel with a gap length of 10 mm and width of 200 mm. The test conditions were the superficial gas velocity ranging from 0.0870 to 0.288 m/s and the superficial liquid velocity ranging from 0.515 to 2.53 m/s. The interfacial area concentrations with uniform and non-uniform inlet flow boundary conditions measured in a large vertical rectangular channel were compared under the same area-averaged flow rate conditions. Then, the predictive capability of the IATE to the bubbly flow test data with uniform and non-uniform inlet flow boundary conditions was assessed. Some non-uniform inlet flow boundary conditions, including channel-center peaking of liquid flow (CPF), channel-center peaking of gas flow (CPG), single-side-wall peaking of liquid flow (SPF), and single-side-wall peaking of gas flow (SPG), significantly affected the interfacial area concentration changes along the axial location at relatively high superficial gas or liquid velocity conditions. The IATE's predictive capability deteriorated for several non-uniform inlet flow boundary conditions, including double-side-wall peaking of gas flow (DPG) and single-side-wall peaking of liquid flow (SPF), specifically at relatively high superficial liquid velocity conditions. The maximum percentage error of the interfacial area concentration prediction by the IATE for these non-uniform inlet flow boundary conditions reached 75%.
Evaluation of aerosol deposition in the containment vessel is an important step for the assessment of radioactive material release to the environment. ART Mod 2 is a calculation code that is used for ...evaluation of aerosol deposition in the containment vessel. The authors modified aerosol deposition models of ART Mod 2, namely, gravitational settling model, Brownian diffusion model, diffusiophoresis model, and thermophoresis model in order to increase potential of capturing the deposition phenomena. This study aims to compare the simulated results of modified ART Mod 2 with aerosol deposition of cesium compounds in the containment vessel of Phébus FPT3 experiment, in order to validate modified ART Mod 2 code. It is found that aerosol deposition using modified ART Mod 2 agrees with Phébus FPT3. Prediction of Brownian diffusion is significantly improved due to the consideration of turbulent damping process. Cesium mass flow rate and aerosol size are factors that can significantly influence the uncertainty of the results. When conditions of single volumes are carefully selected to match those of the Phébus FPT3 experiment, modified ART Mod 2 can predict aerosol deposition in Phébus FPT3 with relative accuracy.
Recent progress in nuclear thermal-hydraulics simulations has been largely focused on coupling with other computational packages, improved closure models for subcooled boiling and for bubbly flows, ...and the development of higher-fidelity simulation capabilities (Kulesza et al., 2016). While high-fidelity 3D simulation is important for model validation, scientific understanding, and some design calculations, it can be prohibitively expensive for system design applications or applications involving large geometries. Thus, there is also a need for practical, simplified approaches for those applications. The two-fluid model strikes a balance between detail and computational resources, but requires the accurate specification of several key constitutive models. These include (1) interfacial forces, (2) interfacial area concentration, (3) two-phase turbulence, and (4) wall and bulk boiling and condensation. In many modern CFD packages, uncertainties in the local interfacial area concentration can have strong effects on the ability to predict the other key parameters. This paper demonstrates that the drag force in 3D CFD can be formulated in much the same way as in 1D system analysis codes and that this approach can be used to formulate a model for interfacial area concentration. The method is also applied to two-group approaches to consider the difference in transport properties for different bubble size classes. This approach may open a method to calculate the interfacial forces without the need for interfacial area transport equations. This reduces the number of differential equations and avoids the modeling challenges associated with bubble breakup and coalescence kernels and the need to specify the inlet interfacial area concentration a priori. The new method is expected to decouple the effects of interfacial area uncertainty and calibrated coefficients, and should provide reasonable local bubble diameters for both group-1 and group-2 bubbles. The approaches proposed in this study are applicable to two-phase flow simulations in rather simple geometries such as upward two-phase flow in vertical channels. In view of many applications for upward two-phase flow in vertical channels, including nuclear reactor systems, the proposed methods are considered useful.
•Simplified two-group two-fluid model is proposed for CFD applications.•Drift velocity based drag force formulation is simpler and more robust for CFD calculations.•Local interfacial area concentration correlation is proposed based on new drag force formulation.•This approach is applicable to vertical upward two-phase flow.
•Existing models of void fraction and relative velocity covariance are reviewed.•Database of local void fraction distribution in horizontal bubbly flows is established.•Covariance is calculated by an ...interpolation scheme of void fraction distribution.•Void fraction distributions for intermittent horizontal flows are modeled.•Void fraction covariance for horizontal gas-dispersed two-phase flows is modeled.
The interfacial drag force governs the momentum exchange between gas and liquid phases and is a key parameter in predicting void fraction using a one-dimensional thermal-analysis code. The interfacial drag force is formulated by the product of the overall drag coefficient and the area-averaged relative velocity between two phases. The area-averaged relative velocity for gas-dispersed two-phase flows has been formulated with the aid of the distribution parameter, which expresses the covariance due to the distributions of void fraction and mixture volumetric flux. It has been pointed out that the term due to void fraction covariance is missing in the formulation of the area-averaged relative velocity adopted in one-dimensional thermal-analysis codes. This study has been conducted to develop void fraction and relative velocity covariance models for gas-dispersed two-phase flows in horizontal pipes. The database of local void fraction distributions for horizontal bubbly flows was established, and an artificial database for intermittent horizontal flows was modeled with assumed distributions of local void fractions. The developed void fraction covariance model for gas-dispersed horizontal two-phase flows could reproduce the data with the mean relative deviation (i.e., bias) of 8.33% and the mean absolute relative deviation (i.e., random uncertainty) of 21.4%. The developed relative velocity covariance model for gas-dispersed horizontal two-phase flows could reproduce the data with the mean relative deviation (i.e., bias) of -0.265% and the mean absolute relative deviation (i.e., random uncertainty) of 7.11%.
•A simple drift-flux correlation has developed for horizontal gas-liquid flows.•The drift-flux correlation is applicable for all void fraction range.•The correlation has been validated by air-water ...and air-kerosene data.•The correlation can predict void fraction with the mean absolute error of 0.00487.•The correlation can predict void fraction with the standard deviation of 0.0985.
A drift-flux correlation has been often used to predict void fraction of gas-liquid two-phase flow in a horizontal channel due to its simplicity and practicality. The drift-flux correlation includes two important drift-flux parameters, namely, the distribution parameter and void-fraction-weighted-mean drift velocity. In this study, an extensive literature survey for horizontal two-phase flow is conducted to establish void fraction database and to acquire existing drift-flux correlations. A total of 566 data is collected from 12 data sources and 4 flow-regime-dependent and 1 flow-regime-independent drift-flux correlations are identified. The predictive capability of the existing drift-flux correlations is assessed using the collected data. It is pointed out that the drift velocity determined by a regression analysis may include a significant error due to a compensation error between distribution parameter and drift velocity. In this study, a simple flow-regime-independent drift-flux correlation is developed. In the modeling approach, the void-fraction-weighted mean drift velocity is approximated to be 0 m/s, whereas the distribution parameter is given as a simple function of the ratio of non-dimensional superficial gas velocity to non-dimensional mixture volumetric flux. The newly developed correlation shows an excellent predictive capability of void fraction for horizontal two-phase flow. Mean absolute error (or bias), standard deviation (random error), mean relative deviation and mean absolute relative deviation of the correlation are 0.0487, 0.0985, 0.0758 and 0.206, respectively. The prediction accuracy of the correlation is similar to the correlation of Chexal et al. (1991), which was formulated based on the drift-flux parameters by means of many cascading constitutive relationships with numerous empirical parameters.
The two-fluid model is comprised of the mass, momentum, and energy equations for liquid and gas phases separately. The terms of interfacial transfer govern mass, momentum, and energy transfer across ...the gas-liquid interface. The interfacial area concentration is critical to formulating the interfacial transfer terms. The one-dimensional Interfacial Area Transport Equation (IATE) has excellent potential in dynamically predicting the interfacial area concentration. The sink and source terms modeling for the interfacial area concentration is vital in the successful IATE development. The current IATE in bubbly flows considers the sink terms caused by bubble coalescence owing to random bubble collision and wake entrainment. The source term is caused by bubble breakup owing to turbulent impact. The turbulent diffusion term has not been considered in the current practice of one-dimensional IATE. This study developed the equation to predict the turbulent diffusion terms in the one-dimensional IATE for bubbly flow in vertical round channels. The predictive capability of the developed equation was validated with the collected database. The non-dimensionalized turbulent diffusion terms predicted by the developed equation agreed with those directly calculated by the experimental data. The developed turbulent diffusion term was also implemented in the one-dimensional IATE. The interfacial area concentrations predicted by the one-dimensional IATE with the turbulent diffusion term had a satisfactory agreement with the experimental results by the statistical parameters of the mean relative deviation of 3.76% and the mean absolute relative deviation of 10.1%.
•The interfacial area transport equation (IATE) research was conducted.•Turbulent diffusion term for one-dimensional IATE in bubbly flows was developed.•The developed equation of turbulent diffusion term was validated with experimental data.•The IATE with turbulent diffusion term was validated with experimental data.
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