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A Baseline Drag Force Correlation for CFD Simulations of Gas-Solid Systems

James M. Parker

CPFD Software, LLC

2016 NETL Workshop on Multiphase Flow Science August 9 – 10, 2016Morgantown, WV

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Momentum exchange between fluid and collection of particles is affected by hydrodynamics:

• Reynolds number• Voidage

As well as particle-level phenomena

• Particle shape• Morphology• Static electricity• Van der Waals forces• Liquid films

Appears as a deviation from ideal fluidization (particulate)

Particle Drag Force in Gas-Solid Systems

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Effect on CFD Modeling of Gas-Solid Systems

• Agglomerative phenomena may not be calculated or characterized but is evident in operational data

• Available operational or experimental data used to improve CFD model

• Particle drag force models are needed that can be calibrated to account for the effect that agglomerative effects have on particle drag as a deviation from an ideal baseline drag

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ideal( (,Re, ) ,Re, ) ,Re, )(p FF f

Ideal drag Deviation

Particle Drag Force Map

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FLUIDIZATION RANGEF(Re,ε)

SIN

GLE

PA

RTI

CLE

Voidage (ε)mf 1

0

CLO

SE P

AC

K

Re

Less known KnownKnown

Single particle (Schiller-Naumann correlation)

Close pack or minimum fluidization (Ergun’s)

Dimensionless Particle Drag Force

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0.6871 0.15Re Re 1000

,Re) (Re) 0.44Re Re 1000

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(1 spp FF

2 2 2

1,Re, ) ,Re, )

1

R

8

e(

18( mf

p mf mf mf

mf mf

a bF F

,Re, 3 Redrag Stokes Stokes p f p fF F dF F U d U

Dimensionless

Correlation for Baseline Particle Drag Model

• Requirements:

• Satisfy both close pack and single particle extremes

• Continuous function of voidage

• Popular models do not satisfy these requirements

• Wen and Yu does not satisfy close pack

• Ergun does not satisfy single particle drag

• Wen-Yu/Ergun Blend (Gidaspow) is discontinuous at ε = 0.8

• Two different forms (A & B) which meet these requirements are proposed and tested against experimental data

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Proposed models vs Existing models

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New drag forms for particulate fluidization

Form A: A Wen-Yu form (power of voidage) is used with an exponent that is adjusted to match Ergun’s drag for all Re at minimum fluidization

The expression can be further simplified by rearrangement

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ln,Re)

l

(Re) / Re( Re

n

mf sp

p sp

mf

F FF F

Form A 1( Rel

ln,Re) 1

np sp mf

mf

F FF

New drag forms for particulate fluidization

Form B: It was observed by Leva (1959) that the Ergun’s model is applicable up to voidages of 0.8 and this same cutoff is commonly used in the Gidaspow Wen-Yu / Ergun blend. Logarithmic interpolation between Ergun’s drag and the Schiller-Naumann drag reproduces this observation well

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1,Re) ( , )( Re1

mfp sp mf

mf

F FF Re

Form B

Validation against experimental data

Particulate fluidization data used for validation (177 points)

• Wen and Yu (1966): Bed expansion data for 191 & 500 micron glass balls in water

• Liu, Kwauk, and Li (1996): Bed expansion data for 54 micron FCC catalyst in supercritical CO2 (8 and 9.4 MPa)

• Jottrand (1952): Bed expansion data for 20, 29, 43, 61, 86, and 113 micron sand in water

• Wilhelm and Kwauk (1948): Bed expansion data for 373, 556, and 1000 micron sea sand in water

• Lewis, Gilliland, and Bauer (1949): Settling of 100 and 150 micron glass in water

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Analysis Approach

• Bed expansion is generally comprised of data showing• Superficial velocity• Bed height or voidage• Pressure drop

• Non-dimensional drag is related to the Archimedes number in a fluidized bed (di Felice, 1994)

• Particle sphericity is estimated from close pack pressure drop data where possible using Ergun coefficients recommended by Macdonald et al (1979) of a = 180 and b = 1.8.

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3

meas 2( ,Re) Ar Ar

18Re

p f s f

f

d gF

Drag Model Error Ergun Coeffs

Form A 11.9% a = 180, b = 1.8

Ergun* 20.5% a = 180, b = 1.8

Gidaspow* 23.1% a = 150, b = 1.75

Wen and Yu 26.0%

Ergun* 26.1% a = 150, b = 1.75

Form B 28.0% a = 180, b = 1.8

Ergun 64.2% a = 180, b = 1.8

Error Analysis of different drag models

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1

1Average error

Nmeas corr

i meas i

F F

FN

* Sphericity is assumed = 1, as is common practice for these models

Correlations vs Measured Data

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Correlations vs Measured Data

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Particulate Drag Error dependence on Re and Voidage

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ClosePack

SingleParticle

Comparison of Form A with expression of di Felice

• The analysis of di Felice (1994) found a Reynolds number dependence for the exponent and proposed the following expression:

• The di Felice expression does not explicitly guarantee close-pack drag but the shape is similar

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21.5 log

e3.7 0.6 x5 p2

Re

Conclusions

• CFD modeling of gas-solid systems can benefit from calibration of drag models to account for particle-level phenomena that affect particle drag

• For this effort, it is convenient to look at deviations from an ideal baseline drag model

• Two models for a baseline drag model were proposed and validated against available experimental data for particulate fluidization from literature

• Form “A” was found to match experimental data with an average error that was nearly half (11.9%) of the next best model tested

• This form is recommended as a baseline model for future work studying deviations in drag force due to agglomerative effects

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References

• Di Felice, R (1994). The Voidage Function for Fluid-Particle Interaction Systems. International Journal of Multiphase Flow, 20(1):153.

• Leva, M (1959). Fluidization. McGraw-Hill Book Company, Inc., New York.

• Lewis, W., Gilliland, E., and Bauer, W. (1949). Characteristics of Fluidized Particles. Industrial and Engineering Chemistry, 41(6): 1104.

• Liu, D., Kwauk, M., and Li, H. (1996). Aggregative and Particulate Fluidization – The Two Extremes of a Continuous Spectrum. Chemical Engineering Science, 51(17): 4045.

• Jottrand, R. (1952). An Experimental Study of the Mechanism of Fluidization. Journal of Applied Chemistry, Supplementary Issue 1: S17.

• Macdonald, I., El-Sayed, M., Mow, K., and Dullien, F. (1979) Flow through Porous Media-the Ergun Equation Revisited. Ind. Eng. Chem. Fundam., 18 (3): 199

• Wen, C. and Yu, Y. (1966). Mechanics of Fluidization. Chemical Engineering Progress Symposium Series, 62(62): 100

• Wilhelm, R. and Kwauk, M. (1948). Fluidization of Solid Particles. Chemical Engineering Progress, 44(3): 201.

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