Effective drag coefficient investigation in the acceleration zone of an upward gas–solid flow

doi:10.1016/j.ces.2006.08.055

Chemical Engineering Science

Volume 62, Issues 1–2, January 2007, Pages 318-327

https://doi.org/10.1016/j.ces.2006.08.055 Get rights and content

Abstract

A new correlation has been developed for the purpose of evaluating the effective drag coefficient of gas–solid systems of ascending flow. In the proposed expression, the effects of particle acceleration in the acceleration zone and Basset force have been considered.

The radioactive particle tracking measurement technique (RPT) was used to obtain a dynamic picture of particle trajectory in the system. The Cartesian coordinates ( $x$ , $y$ and $z$ ) of the radiotracer were registered every 10 ms and other useful variables, such as particle velocity, particle velocity fluctuation and acceleration, were calculated. The effects of particle velocity, acceleration and Basset force on the measurement of the drag coefficient were investigated in the internal circulating fluidized bed (ICFB). The experiments were carried out using sand and alumina particles in an ICFB unit with a 1 m riser length and a 0.052 m riser diameter. The gas velocity range studied was between 2 and 12 m/s. The most common correlations for calculating the drag coefficient were reviewed and compared to the one developed in this work. Numerical analysis of the one-dimensional two-phase flow model demonstrates that the drag coefficient proposed here is in good agreement with experimental data and covers a variety of operating conditions $(G_{s} = 29$ , 50 and $240 kg / m^{2} s$ ; $U = 4.2$ , 5.8 and 8.5 m/s; $D = 0.05$ , 0.2 and 0.4 m ID).

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).

The drag force is often given by one of the following expressions:

F_{D} = \frac{1}{2} C_{D} . A_{p} . ρ_{g} . (u_{g} - u_{p}) . | u_{g} - u_{p} |,

F_{D} = β (u_{g} - u_{p}),

where

β

represents the gas–solid momentum transfer coefficient.

In expression (1) the drag coefficient $C_{D}$ 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 $ɛ_{g} ⩽ 0.8$ and the correlation proposed by Wen and Yu (1966) for gas holdup $ɛ_{g} ⩾ 0.8$ . The correlations developed for the drag coefficient $C_{D}$ can be classified into two categories, a composed and a non-composed $C_{D}$ . The first class is usually formed by $C_{D 0}$ , 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 $C_{D}$ 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:

•
$F_{i} = - ρ_{s} V_{p} \frac{d u_{p}}{d t} inertial force,$
•
$F_{g} = ρ_{s} V_{p} g gravitational force,$
•
$F_{b} = - ρ_{g} V_{p} g buoyancy force,$
•
$F_{D} = \frac{1}{2} ρ_{g} A_{p} C_{D} (u_{g} - u_{p}) | u_{g} - u_{p} | drag force .$

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 ( $x$ , $y$ and $z$ ) at each time point, with absolute errors of 2 mm for $x$ and $y$ coordinates and 3 mm for the $z$ 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, $C_{D}$ ;
2.
particle acceleration is accounted for but Basset force is not, $C_{D}^{'}$ ;
3.
both particle acceleration and Basset force are accounted for, $C_{D}^{″}$ .

The experimental results obtained for sand and alumina particles are presented in the figures below.

Fig. 4 shows the variations of different drag coefficients ( $C_{D}$ , $C_{D}^{'}$ ,

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: $C_{D} = 4330 * {Re}_{p}^{- 2} .$ 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

$A_{p}$	surface area of a particle, $m^{2}$
$c$	sphericity, $π d_{A} / P_{p}$
$C_{D}$	Drag coefficient
$d_{A}$	particle mean diameter $\sqrt{\frac{4 A_{p}}{π}}$ , m
$d_{p}$	particle mean diameter $\sqrt[3]{\frac{6 V_{p}}{π}}$ , m
$D$	riser diameter, m
DP/DZ	pressure-drop, Pa/m
$f$	particle wall friction factor
$g$	gravity acceleration, $m / s^{2}$
$G_{s}$	solid flow rate, $kg / m^{2} s$
$Re$	Reynolds number, $Re = \frac{ρ_{g} UD}{μ}$
${Re}_{p}$	Particle Reynolds number, ${Re}_{p} = \frac{ρ_{g} d_{p} ɛ (u_{g} - u_{p})}{μ}$
${Re}_{s}$	Reynolds number, ${Re}_{s} = \frac{ρ_{g} d_{p} u_{p}}{μ}$
${Re}_{s, t}$	Reynolds number, ${Re}_{s, t} = \frac{ρ_{g} d_{p} u_{t}}{μ}$
$t$	time, s
$u_{g}$	gasvelocity, m/s
$u_{p}$	particles velocity, m/s
$u_{r}$	relative velocity ( $u_{g} - u_{p})$ , m/s
$u_{r 0}$	relative

References (42)

R. Di Felice
The voidage function for fluid particle interaction systems
International Journal of Multiphase Flow
(1994)
L.G. Gibilaro et al.
Generalized friction factor and drag coefficient correlations for fluid particle interactions
Chemical Engineering Science
(1985)
Z.B. Grbavcic et al.
Hydrodynamic modeling of vertical accelerating gas–solids flow
Powder Technology
(1997)
A. Haider et al.
Drag coefficient and terminal velocity of spherical and nonspherical particles
Powder Technology
(1989)
F. Larachi et al.
A $γ$ -ray detection system for 3-D particle tracking in multiphase reactors
Nuclear Instruments and Methods A
(1994)
H. Littman 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)
Y.T. Makkawi et al.
The voidage function and effective drag force for fluidized beds
Chemical Engineering Science
(2003)
N. Mostoufi et al.
Prediction of effective drag coefficient in fluidized beds
Chemical Engineering Science
(1999)
T.S. Pugsley et al.
Apredictive hydrodynamic model for circulating fluidized bed risers
Powder Technology
(1996)
A. Rautiainen et al.
Solids friction factors in upward, lean gas–solids flows
Powder Technology
(1998)

P.N. Rowe

A convenient empirical equation for estimation of Richard–Zaki exponent

Chemical Engineering Science

(1987)

S. Tran-cong et al.

Drag coefficients of irregularly shaped particles

Powder Technology

(2004)

D.J. Vojir et al.

Effect of the history term on the motion of rigid spheres in viscous fluid

International Journal of Multiphase Flow

(1994)

V. Armenio et al.

The importance of the forces acting on particles in turbulent flows

Physics of Fluids

(2001)

A.B. Basset

Treatise on Hydrodynamics

(1888)

V.J. Boussinesq

Sur la résistance qu’oppose un liquide indéfini en repos’

C.R. Academie des Sciences

(1885)

C.E. Capes et al.

Vertical pneumatic conveying: an experimental study with particles in the intermediate and turbulent flow regimes

The Canadian Journal of Chemical Engineering

(1973)

R. Clift et al.

Motion of entrained particles in gas streams

Canadian Journal of Chemical Engineering

(1971)

R. Clift et al.

Bubbles, Drops, and Particles

(1978)

J. Ding et al.

A bubbling fluidization model using kinetic theory of granular flow

A.I.C.H.E Journal

(1990)

S. Ergun

Fluid flow through packed columns

Chemical Engineering and Processing

(1952)

Cited by (31)

Free falling of nonspherical particles in Newtonian fluids, B: Acceleration
2024, Powder Technology
The acceleration length of nonspherical particles is required for various aspects of system design. However, no simple or usable equations can be found in literature. Therefore, this study is designed to show that the force balance developed earlier for spherical particles, including the new exponential function for the history force, is also valid for nonspherical particles. To achieve this, a new history force parameter definition considering sphericity has been developed based on numerous experiments. The experimental results confirm the definitions of particle acceleration and velocity of nonspherical particles. The acceleration length can be described as a function of the Archimedes number, density ratio, and particle diameter by analogy to spherical particles, considering the terminal Reynolds number, drag coefficient, and history force parameter as functions of the sphericity. In most cases, the acceleration length describes the experiments well, but it is less accurate for cases of orientation change during acceleration.
Free falling of non-spherical particles in Newtonian fluids, A: Terminal velocity and drag coefficient
2024, Powder Technology
The terminal velocity and drag coefficient are vital parameters for designing industrial systems involving free-falling particles. While the literature on spherical particles is abundant, that on non-spherical particles is scarce. Therefore, a comprehensive experimental study was conducted using discs, cylinders, and irregular particles of various sphericities in various fluids. First, the flow mode was analyzed, and seven flow modes were identified and described in a flow regime map. Subsequently, the terminal velocity and drag coefficient were correlated depending on the sphericity of three trends. The first is for nearly spherical particles with sphericities over 0.87, and the other two trend lines are for sphericities lower than 0.87, one for discs and the other for cylinders. Additional experiments enabled the extension of the models to nearly spherical, flat, and long particles within an error of ±30%.
Acceleration length and time of falling spherical particles
2023, Powder Technology
Acceleration length and time are required for various aspects of system design. However, simple and accurate equations that can be used by designers are lacking. Therefore, in this study, we used previously conducted tests and the simple exponential model to describe particle velocity and acceleration and thereby calculate the acceleration length, which is the length required for a particle to achieve 99% of the terminal velocity. Results indicated that the acceleration length can be well described by a function of the Archimedes number, density ratio, and particle diameter. Several new tests were conducted to measure the particle velocity as a function of time. These tests confirmed the newly developed model that describes the particle location, velocity, and acceleration as functions of time. Based on the new model, the acceleration time was calculated and found to be well described by the acceleration length and terminal velocity.
New model to predict the velocity and acceleration of accelerating spherical particles
2023, Powder Technology
Citation 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.
The equation of motion for an accelerating particle is complicated to solve, primarily because of the Basset-history term. Therefore, most solutions found in the literature are numerical. Various experiments conducted with different particles and fluids have been presented and analyzed in this study. A simple exponential function for the particle velocity was found to fit all the experiments, describing the acceleration as a function of distance. Consequently, a simple exponential equation was developed for velocity. These functions depend on the initial acceleration and terminal velocity. While the terminal velocity can be predicted by several known equations, the initial acceleration was defined without any empirical parameter for cases of negligible history forces. In addition, the effect of history forces was correlated. Consequently, the velocity during the acceleration of 51 experiments was predicted to be ±10%.
GIPPE-RPT: Geant4 interface for particle physics experiments applied to Radioactive Particle Tracking
2022, Applied Radiation and Isotopes
Radioactive Particle Tracking (RPT) is a non-invasive experimental technique that tracks the motion of a gamma-emitting radionuclide. Despite the RPT’s high versatility, the lack of dedicated software represents a significant barrier to its wider adoption. This article introduces a new software, GIPPE-RPT, designed to bring the technique closer to a wider group of users. GIPPE-RPT is a user-friendly software that enables the creation, execution, and post-processing of high-energy physics simulations in Geant4 employing a graphical user interface. Under the platform, the user can specify all critical RPT parameters, such as reactor geometry, materials, detector number and type, and tracer type and activity. Multiple configurations can be designed and compared for optimization. GIPPE-RPT also integrates the OpenFOAM solver, which enables the setup and execution of Computational Fluid Dynamics simulations. The simulation results can be imported in Geant4 allowing an accurate description of density profiles inside the reactor. The main software modules, functions, and workflow are demonstrated using a virtual NETL SSCP-I fluidized bed reactor as a test case. The main steps in setting up an RPT experiment in GIPPE-RPT, including domain design, tracer selection, detector placement, and calibration strategy are presented in detail. The performance of six position reconstruction methods implemented in GIPPE-RPT is compared for several RPT scenarios highlighting the strengths and weaknesses of each method. Finally, the benefits of capturing heterogeneous density profiles using simulations are demonstrated by comparing reconstruction errors for test cases with heterogeneous and homogeneous media.
Experimental pressure drop analysis for horizontal dilute phase particle-fluid flows
2017, Powder Technology
Citation 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.
Prediction of horizontal pressure drop in dilute phase pneumatic conveying or slurry transport is among the fundamental issues of designing these systems. In this work a thorough literature survey demonstrates that there is no consensus on an integrative solution for conveying pressure drop, therefore a large data base of conveying systems' operating points at dilute phase is established and analyzed. It includes measurements taken at our 52 mm bore laboratory pneumatic conveying line, as well as thousands of data points collected from the literature. Two modelling approaches are attempted – the first is frictional representation based on dimensional analysis, which presents linear dependency between the pressure drop and the superficial fluid velocity normalized by the minimum pressure velocity of the transport curve. It had also been found that the pressure drop increase rate has a strong dependency on the Archimedes number. The second approach is an energy balance between the power of the drag force applied to the particles and the additional power required to drive the fluid due to particle presence. Unlike the common expressions, a modification to the drag coefficient is suggested, which takes into account pipe turbulence and particle acceleration. The two presented models do not require pilot plant calibration, and an applicability of one of them to slurry transport is demonstrated. Some future research routes are offered.

View all citing articles on Scopus

View full text

Effective drag coefficient investigation in the acceleration zone of an upward gas–solid flow

Abstract

Introduction

Section snippets

Particle motion through fluids

Experimental setup

Measurement technique and procedure

Results and discussion

Comparison of model results and experimental data

Conclusion

Notation

International Journal of Multiphase Flow

Chemical Engineering Science

Powder Technology

Powder Technology

Nuclear Instruments and Methods A

Powder Technology

Chemical Engineering Science

Chemical Engineering Science

Powder Technology

Powder Technology

Chemical Engineering Science

Powder Technology

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