Effective drag coefficient investigation in the acceleration zone of an upward gas–solid flow
Introduction
With the current rapid progress in computer science, hydrodynamic modeling of biphasic gas–solid flow is no longer a dream. Models developed and used in computational fluid dynamics codes (CFD) and/or simple unidirectional models provide a certain degree of understanding of phenomena occurring in the system.
These models are usually a set of mathematical equations—continuity, energy and momentum equations—containing parameters developed from experimental data. The drag coefficient encountered in the drag force is a concrete example of such a parameter.
In spite of the efforts made by scientists to better understand and model the momentum transfer between gas and solid phases through the force called drag force, the results still do not meet the expectations of the engineering and scientific communities. Reviewing previous work done on the characterization of drag forces in fluidized systems, three categories of expressions were determined:
- •
Semi-empirical correlation expressions developed based on data from pressure-drop measurements. Typical and well-known examples of this are the expressions developed by Ergun (1952) and Wen and Yu (1966).
- •
Expressions developed based on bed expansion. Correlations link the superficial velocity of the fluid to the bed expansion (e.g. Richardson and Zaki, 1954).
- •
Expressions developed based on numerical simulation. This type of expression is based on simulation of the flow around particles, taking particle interactions into consideration (e.g. lattice Boltzmann simulation).
In expression (1) the drag coefficient is explicit while in expression (2) it is implicit. Table 1 summarizes some of the expressions found in the literature representing the gas–solid momentum transfer coefficient.
The correlations developed to express the drag coefficient are many and various. Decades ago, Torobin and Gauvin (1961), Ko and Graf (1972) and others concluded that the drag coefficient in fluidization modeling could not be calculated or deduced from the standard curve. Many expressions developed before the 1970s were reported by Clift et al. (1978). Table 2 summarizes some of the most commonly used and recent expressions.
Ding and Gidaspow (1990) and many other scientists and researchers working on CFD simulation in general suggested the correlation proposed by Ergun for the condition of gas holdup and the correlation proposed by Wen and Yu (1966) for gas holdup . The correlations developed for the drag coefficient can be classified into two categories, a composed and a non-composed . The first class is usually formed by , the drag coefficient around a spherical particle and a correction function f to consider the effect and interactions of surrounding particles (Mostoufi and Chaouki, 1999; Di Felice, 1994, among others). In the second class is expressed in one formula, which includes particle–particle interactions through solid holdup (e.g. Haider and Levenspiel, 1989). The inconvenience involved in using the expressions developed from the first class lies in the fact that most of the expressions and correlations found in the literature were developed for free-falling particle conditions in liquid–solid media and extrapolated to upward gas–solid flow. At this point a question arises: are the phenomena occurring in gas–solid fluidization well represented?
In this study, a new correlation is developed for the purpose of evaluating the effective drag coefficient of gas–solid systems of ascending flow. In the expression proposed, the effects of particle acceleration and the Basset force are considered.
Section snippets
Particle motion through fluids
Most contributions to the analysis of dispersed-phase behavior in continuous flow fields are based on force balance. Let us consider Fig. 1, where the most relevant forces, as listed below, are exerted on spherical solid:
- •
- •
- •
- •
In turbulent regime flows, other forces should be considered such as the added mass and particle history force. The added mass is defined as the acceleration of the
Experimental setup
The experimental apparatus is illustrated in Fig. 2. It is a cold model representing an internal circulating fluidized bed (ICFB). Milne et al. (1992) were the first to pioneer this technology. Briefly, the ICFB consists of two coaxial tubes of different heights, which delimit two zones, a riser zone and an annulus zone. The two zones are characterized by two different regimes: a moving bed in the annulus and fast fluidization in the riser. The solid is fed to the annulus zone where it is
Measurement technique and procedure
The radioactive particle tracking (RPT) technique was used during this work. A radioactive tracer is used to track the behavior of the solid in the ICFB. The photons emitted from the tracer are counted by the discriminators, which are linked to the detectors surrounding the column. The RPT data consist of locating the particle positions (, and ) at each time point, with absolute errors of 2 mm for and coordinates and 3 mm for the coordinate. A sampling time of 10 ms was used for the
Results and discussion
Drag coefficients were calculated from the experimental data, using expression (17) presented above. Three cases were distinguished:
- 1.
both particle acceleration and Basset force are neglected, ;
- 2.
particle acceleration is accounted for but Basset force is not, ;
- 3.
both particle acceleration and Basset force are accounted for, .
Fig. 4 shows the variations of different drag coefficients (, ,
Comparison of model results and experimental data
The results of numerical solutions for Eqs. (19)–(22) are shown in Fig. 8, Fig. 9, Fig. 10 and compared with the experimental data. Experimental data were taken from two sources: the modeling exercise proposed at Fluidization VIII for evaluating the best model able to predict hydrodynamic properties along the riser (axial pressure-drop profile, axial and radial gas hold-up) and the experimental data from the work done by Pugsley and Berutti (1996).
Theoretical predictions of the pressure-drop
Conclusion
A new correlation for the drag coefficient is proposed in which the acceleration and Basset forces are considered: The correlation is developed from experimental data obtained in ICFB lab-scale using sand and alumina particles for several gas superficial velocities.
The importance of particle acceleration and Basset force terms is proven in a comparison with different experimental data from the literature. Numerical analysis of the one-dimensional two-phase flow model demonstrates
Notation
surface area of a particle, sphericity, Drag coefficient particle mean diameter , m particle mean diameter , m riser diameter, m DP/DZ pressure-drop, Pa/m particle wall friction factor gravity acceleration, solid flow rate, Reynolds number, Particle Reynolds number, Reynolds number, Reynolds number, time, s gasvelocity, m/s particles velocity, m/s relative velocity (, m/s relative
References (42)
The voidage function for fluid particle interaction systems
International Journal of Multiphase Flow
(1994)- et al.
Generalized friction factor and drag coefficient correlations for fluid particle interactions
Chemical Engineering Science
(1985) - et al.
Hydrodynamic modeling of vertical accelerating gas–solids flow
Powder Technology
(1997) - et al.
Drag coefficient and terminal velocity of spherical and nonspherical particles
Powder Technology
(1989) - et al.
A -ray detection system for 3-D particle tracking in multiphase reactors
Nuclear Instruments and Methods A
(1994) - et al.
Modeling and measurement of the effective drag coefficient in decelerating and non-accelerating turbulent gas–solids dilute phase flow of large particles in a vertical transport pipe
Powder Technology
(1993) - et al.
The voidage function and effective drag force for fluidized beds
Chemical Engineering Science
(2003) - et al.
Prediction of effective drag coefficient in fluidized beds
Chemical Engineering Science
(1999) - et al.
Apredictive hydrodynamic model for circulating fluidized bed risers
Powder Technology
(1996) - et al.
Solids friction factors in upward, lean gas–solids flows
Powder Technology
(1998)
A convenient empirical equation for estimation of Richard–Zaki exponent
Chemical Engineering Science
Drag coefficients of irregularly shaped particles
Powder Technology
Effect of the history term on the motion of rigid spheres in viscous fluid
International Journal of Multiphase Flow
The importance of the forces acting on particles in turbulent flows
Physics of Fluids
Treatise on Hydrodynamics
Sur la résistance qu’oppose un liquide indéfini en repos’
C.R. Academie des Sciences
Vertical pneumatic conveying: an experimental study with particles in the intermediate and turbulent flow regimes
The Canadian Journal of Chemical Engineering
Motion of entrained particles in gas streams
Canadian Journal of Chemical Engineering
Bubbles, Drops, and Particles
A bubbling fluidization model using kinetic theory of granular flow
A.I.C.H.E Journal
Fluid flow through packed columns
Chemical Engineering and Processing
Cited by (31)
Free falling of nonspherical particles in Newtonian fluids, B: Acceleration
2024, Powder TechnologyAcceleration length and time of falling spherical particles
2023, Powder TechnologyNew model to predict the velocity and acceleration of accelerating spherical particles
2023, Powder TechnologyCitation Excerpt :Most other researchers found that the effect of the Basset-history force decreases as Re increases. Accordingly, Mabrouk et al. [8] claimed that the effect of Basset-history force cannot be neglected for Re < 80. This was confirmed by Keshav [14], who found that the effect of the Basset-history force was significant for 0.01 < Re < 100.
GIPPE-RPT: Geant4 interface for particle physics experiments applied to Radioactive Particle Tracking
2022, Applied Radiation and IsotopesExperimental pressure drop analysis for horizontal dilute phase particle-fluid flows
2017, Powder TechnologyCitation Excerpt :Indeed, added mass effect might be negligible in pneumatic conveying, as the densities ratio of gas and solids is mostly very small. However, Basset history force is significant: Mabrouk et al. [62] accelerated particles in a highly dilute (ε < 0.01) riser, observed deviations from the standard curve, and accounted some of it to Basset force. Considering horizontal conveying, when a particle regains speed after impact, the relative velocity decreases as the particle approaches fluid velocity, hence the acceleration, with regard to the drag force, is negative.