Calculation of terminal velocity in transitional flow for spherical particle

The terminal velocity has been widely used in extensive fields, but the complexity of drag coefficient expression leads to the calculation of terminal velocity in transitional flow （1 〈 Re ≤ 1000） with much more difficulty than those in laminar flow （Re ≤ 1） and turbulent flow （Re ≥ 1000）. This paper summarized and compared 24 drag coefficient correlations, and developed an expression for calculating the terminal velocity in transitional flow, and also analyzed the effects of particle density and size, fluid density and viscosity on terminal velocity. The results show that 19 of 24 previously published correlations for drag coefficient have good prediction performance and can be used for calculating the terminal velocity in the entire transitional flow with higher accuracy. Adapting two dimensionless parameters （w＊, d＊）, a proposed explicit correlation, w＊=-25.68654 × exp （-d＊/77.02069）＋ 24.89826, is attained in transitional flow with good performance, which is helpful in calculating the terminal velocity.

De Broglie’s Velocity of Transition between Quantum Levels and the Quantum of the Magnetic Spin Moment Obtained from the Uncertainty Principle for Energy and Time

The De Broglie’s approach to the quantum theory, when combined with the conservation rule of momentum, allows one to calculate the velocity of the electron transition from a quantum state n to its neighbouring state as a function of n. The paper shows, for the case of the harmonic oscillator taken as an example, that the De Broglie’s dependence of the transition velocity on n is equal to the n-dependence of that velocity calculated with the aid of the uncertainty principle for the energy and time. In the next step the minimal distance parameter provided by the uncertainty principle is applied in calculating the magnetic moment of the electron which effectuates its orbital motion in the magnetic field. This application gives readily the electron spin magnetic moment as well as the quantum of the magnetic flux known in superconductors as its result.

The transition to turbulent fluidization in a gas-solid fluidized bed operating from ambient temperature to 1600℃

Turbulent fluidized bed proves effective in industrial processes due to superior heat and mass transfer and chemical reaction performance. However, understanding the transition to turbulent fluidization remains limited, especially at temperatures exceeding 1000 ℃, making it challenging to develop high-temperature fluidized bed applications. This paper presents an experimental investigation on the turbulent fluidization onset velocity (U_(c)), measured in a 30 mm diameter bed using corundum particles with average diameters from 0.68 mm to 1.58 mm in temperatures from ambient to 1600 ℃. Experimental results reveal that U_(c) increases with temperature up to 600 ℃, stabilizes within the 600–1200 ℃ range, and then decreases above 1200 ℃, demonstrating the varying relative significance of hydrodynamic and interparticle forces at different temperatures. To help design and operate high-temperature applications of turbulent fluidization, we developed U_(c) correlations based on experimental data from both literature sources and this study, covering temperatures of up to 1600 ℃ and particles of Groups A to D.

Solids behavior in dilute zone of a CFB riser under turbulent conditions

The behavior of the solid phase in the upper zone of a circulating fluidized bed riser was studied using a phase Doppler anemometer. Glass particles of mean diameter 107μm and superficial gas velocities UE covering the turbulent and the beginning of the fast fluidization regime were investigated. Three static bed heights were tested. Ascending and descending particles were found co-existing under all oper ating conditions tested, and at all measurement locations. Superficial gas velocity proved/happened to have a larger effect on descending particles at the wall and on ascending particles in the central region. Transversal particle velocities in both directions （toward the center and toward the wall） behaved rela- tively equivalently, with only slight difference observed at the wall. However, observation of the number of particles moving in either transversal direction showed a change in bed structure when increasing Ug. Furthermore, a balance was constantly observed between the core zone and the annulus zone where the mutual mass transfer between these two zones occurred continuously. Transition from a slow to a fast particle motion was accompanied by a transition to high levels of velocity fluctuations, and was found corresponding to the appearance of significant solid particle flow rate.

Statistical and frequency analysis of the pressure fluctuation in a fluidized bed of non-spherical particles

In this paper, the pressure fluctuation in a fluidized bed was measured and processed via standard devia- tion and power spectrum analysis to investigate the dynamic behavior of the transition from the bubbling to turbulent regime. Two types （Geldart B and D） of non-spherical particles, screened from real bed materials, and their mixture were used as the bed materials. The experiments were conducted in a semi- industrial testing apparatus. The experimental results indicated that the fluidization characteristics of the non-spherical Geldart D particles differed from that of the spherical particles at gas velocities beyond the transition velocity Uo The standard deviation of the pressure fluctuation measured in the bed increased with the gas velocity, while that measured in the plenum remained constant. Compared to the coarse particles, the fine particles exerted a stronger influence on the dynamic behavior of the fluidized bed and promoted the fluidization regime transition from bubbling toward turbulent. The power spectrum of the pressure fluctuation was calculated using the auto-regressive （AR） model; the hydrodynamics of the flu- idized bed were characterized by the major frequency of the power spectrum of the pressure fluctuation. By combining the standard deviation analysis, a new method was proposed to determine the transition velocity Uk via the analysis of the change in the major frequency. The first major frequency was observed to vary within the range of 1.5 to 3 Hz.

Effect of hydration on the elasticity of mantle minerals and its geophysical implications

Recent studies have shown that major nominally anhydrous minerals in the Earth's mantle, such as olivine, pyroxene and garnet, can incorporate considerable amounts of water as structurally bound hydroxyl. Even a small amount of water is present in mantle minerals, it can strongly affect a number of physical properties, including density, sound velocity, melting temperature, and electrical conductivities. The presence of water can also influence the dynamic behavior, lead to lateral velocity heterogeneities, and affect the material circulation of the Earth's deep interior. In particular, seismic studies have reported the existence of low-velocity zones in various locations of the Earth's upper mantle and transition zone, which has been expected to be associated with the presence of water in the region. In the past two decades, the effect of water on the elasticity and sound velocities of minerals at relevant pressure-temperature(P-T) conditions of the Earth's mantle attracted extensive interests. Combining the high P-T experimental and theoretical mineralogical results with seismic observations provides crucial constraints on the distribution of water in the Earth's mantle. In this study, we summarize recent experimental and theoretical mineral physics results on how water affects the elasticity and sound velocity of nominally anhydrous minerals in the Earth's mantle, which aims to provide new insights into the effect of hydration on the density and velocity profile of the Earth's mantle, which are of particular importance in understanding of water distribution in the region.

Dynamic characteristics of bubbling fluidization through recurrence rate analysis of pressure fluctuations

Pressure fluctuations signals of a lab-scale fiuidized bed （15 cm inner diameter and 2 m height） at different superficial gas velocities were measured. Recurrence plot （RP） and recurrence rate （RR）, and the simplest variable of recurrence quantification analysis （RQA） were used to analyze the pressure signals. Different patterns observed in RP reflect different dynamic behavior of the system under study. It was also found that the variance of RR （a2R） Could reveal the peak dominant frequencies （PDF） of different dynamic systems： completely periodic, completely stochastic, Lorenz system, and fluidized bed. The results were compared with power spectral density. Additionally, the diagram of σ^2RR provides a new technique for prediction of transition velocity from bubbling to turbulent fluidization regime.

Characterization of gas-liquid-solid fluidized beds by S statistics

The hydrodynamics of a gas-liquid-solid fluidized bed was investigated by applying the S statistics method to pressure fluctuations measured under various operating conditions in a laboratory-scale bed. S statistics tests reveal the existence of three transition velocities, especially at low gas velocities. Four distinct fluidization regimes, namely, the compacted bed, agitated bed and coalesced and discrete bubble regimes were detected. A comparison of reconstructed attractors of pressure fluctuations measured at different axial positions along the riser and with various solid loadings showed significant differences in the signals compared before fluidization, especially at minimum liquid agitation velocity. Close to the minimum liquid fluidization velocity and high liquid velocities, the variation in particle size has an insignificant effect on the bed hydrodynamics. Therefore, S statistics is a reliable method to demar- cate different fluidization regimes and to characterize the influence of various operating conditions on the hydrodynamics of gas-liquid-solid fluidized beds. The method is applicable in large-scale industrial installations to detect dynamic changes within a bed, such as regime transitions or agglomeration.