•Transferred charge signal appears primarily in mean value of the measured signal.•A critical frequency exists for decoupling induced and transferred charge signal.•The cross-correlation coefficient increases from 0.61 to 0.78.Comparing the frequency spectrums of the measured signals from both outer surface-mounted electrostatic sensor and inner flush-mounted electrostatic sensor, it is found that the transferred charge signal which used to be ignored appears primarily in the mean value of the measured signal. Furthermore the signal frequency spectrum of inner flush-mounted electrostatic sensor contains a critical frequency for decoupling induced charge signal and transferred charge signal. Then harmonic wavelet transform is proposed to decompose the measured signal. As a result, the induced charge signals from the upstream and downstream electrodes have higher similarity than the directly measured signals. Thus the mean cross-correlation coefficient increases from 0.61 to 0.78, which contributes to the cross-correlation velocity. In addition, the RMS value of the transferred charge signals increases linearly with the increasing of both the particle mass flow rate and the superficial air velocity, which is an effective prediction method for particle mass flow rate.

•Multi-fluid viscous solver using SST k-ω turbulent model for vapor condensation.•Validated by experimental data of both steady and unsteady condensations.•Results about the dynamics of self-excited oscillation, asymmetric bifurcation.•Effect of unsteady self-excited condensation on its mass flow-rate was clarified.The unsteady flow with vapor condensation in critical flow nozzle includes many different self-excited periodic oscillation modes and might affect its flow field and mass flow-rate. The instability of condensation in transonic nozzle flow was discussed firstly. To investigate the phenomenon of self-excited oscillation and bifurcation induced by non-equilibrium condensation of the pure superheated steam flow in critical flow nozzle, an Eulerian viscous flow solver using SST k-ω turbulent model for vapor condensation was established and validated by experimental data of both steady and unsteady condensations, and algebraic formula. Three self-excited symmetric oscillation modes were classified by the oscillation amplitude and frequency of induced shock. The phenomenon of symmetric oscillation modes and asymmetric bifurcation were discussed in detail. Moreover, the effect of self-excited oscillation on its mass flow-rate was clarified. The maximum deviation of mass flow-rate is 1.2%, which is worthy of attention.

•Transient conjugate heat transfer with shock separation in critical flow nozzle.•SST k–ω model coupling with solid-phase heat conduction equation.•Separation criteria and the asymmetry of body temperature were investigated.•Steady conjugate heat transfer was analyzed by Mach number, isotherm and heatline.•The dynamic response characteristics of body temperature were discussed.The body temperature of critical flow nozzle is cooled by expanding cold gas. Subsequently, the thermal boundary layer and throat area are influenced by the conjugate heat transfer at the solid–fluid interface which is called thermal effect. This conjugate heat transfer process in nozzle flow with shock-induced separation were investigated experimentally and numerically, involving three-dimensional wall conduction and fluid convection. Numerical computations solved Reynolds-averaged equations based on SST k–ω model coupling with solid-phase heat conduction equation and were validated by some experiments. Three-dimensional separation criteria and the asymmetry of body temperature were investigated. The maximum asymmetry of body temperature appears at d = 5.25 mm and p0 = 4.5 bar. For this asymmetric flow, the experimental isotherm upstream of separation point obtained by twelve temperature points in different sides is accuracy which is enough to study the thermal effect and downstream isotherm with a maximum error of 0.5 °C is mapped merely for reference. At steady-state, minimum temperature point is close to separation point rather than nozzle exit. The body temperature gradually drops with the increase of throat diameter and inlet pressure. The maximum body temperature drop can reach to 15.0 °C. The detailed process of conjugate heat transfer was analyzed by Mach contour in fluid region, and isotherm, heatline in solid region. Finally, the dynamic response characteristics, especially thermal time constant of the body temperature were discussed.

•The effect of real gas state equation on discharge coefficient of hydrogen gas flow.•An analytic solution of real gas critical flow factor calculated by state equations.•Accurate empirical equation for real gas critical flow factor was obtained.•This equation was in good agreement with experimental data and CFD solutions.•Lots of detailed results about the effect of real gas state equation were obtained.Critical nozzles are widely used in the flow measurement and can be used for mass flow-rate measurement of hydrogen gas. The effect of real gas state equation on discharge coefficient of hydrogen gas flow through a critical nozzle was investigated. The real gas critical flow factor was introduced which considers the effect of the real gas on discharge coefficient. An analytic solution of real gas critical flow factor of hydrogen gas calculated from the modern equations of state based on Helmholtz energy, over a wider range of temperature 150–600 K and pressure up to 100 MPa was presented. An accurate empirical equation for real gas critical flow factor was determined by the nonlinear regression analysis. The equation was in good agreement with the high-pressure hydrogen gas experimental data by Morioka and CFD solutions by Nagao and Kim. Using this equation, the discharge coefficient can be directly and accurately calculated. It indicates that the discharge coefficient of hydrogen gas should be comprehensively taken into consideration with stagnation temperature, stagnation pressure and nozzle throat diameter. A lot of detailed results about the effect of real gas state equation were obtained.

•MBWR EOS, Helmholtz energy EOS and ECS real gas models were used.•Numerical model is verified by grid independence studies and experimental data.•Roughness effect related with Rent, Rc and γ were discussed.•Roughness effects on Cd of real gas through a critical nozzle were obtained.Critical nozzles are widely used in the flow measurement. The discharge coefficient is the most important performance parameter of critical nozzle. At present, the critical nozzle has increasingly been applied in high-pressure and high Reynolds number flow. Thus, the effect of surface roughness on discharge coefficient should be noticed. The effects of relative equivalent sand roughness on discharge coefficient of real gas in a wide range of temperature and pressure were investigated. The thermodynamic properties and transport properties of real gas can be calculated by MBWR EoS, Helmholtz energy EoS and ECS real gas models. Standard k–ε model with the wall function was used to predict the discharge coefficient with good accuracy. Numerical results were validated by existing experimental data. The effect of roughness related with parametric Reynolds number, nozzle curvature radius and specific heat ratio were discussed in details. The roughness effect will become larger with the increase of all these parameters. At last, the roughness effects on discharge coefficient of real gas, such as argon, nitrogen and methane with a wide range of pressure while temperature ranges from 300 K to 500 K and Ks/d ranges from 10−2 to 10−5 were studied. All of results coincided well with previous parametric studies.

•Approximate solution for discharge coefficient of sonic nozzle with roughness.•Universal logarithmic law and principle of equivalent velocity profiles were used.•Experiments conducted by others showed agreement with present algebraic method.•The agreement between this method and exact numerical calculation is also good.Sonic nozzle is widely used in the flow measurement and control. Nowadays, it has been applied to higher Reynolds number flow increasingly. The effect of surface roughness on discharge coefficient of the sonic nozzle should be discussed. An approximate analytic solution for discharge coefficient of the sonic nozzle with surface roughness was proposed in detail. The determination of this coefficient was based on universal logarithmic velocity-distribution law and the principle of equivalent velocity profile. Although there are some apparently approximations, this algebraic method accurately predicts the discharge coefficient of the sonic nozzle with surface roughness in Reynolds number range from 104 to 109, a relative equivalent roughness range of 10−6 to 10−2. Some experiments of sonic nozzle conducted by others showed agreement with present algebraic method. Besides, the agreement between this method and the corresponding exact numerical calculation is also good. The present method provides an excellent tool to deeply investigate the roughness effect and promote further improvement of the standard.