^{1,a)}, Amsini Sadiki

^{2}and Amirfarhang Mehdizadeh

^{3}

### Abstract

Based on the notion of a construction process consisting of the stepwise addition of particles to the pure fluid, a discrete model for the apparent viscosity as well as for the maximum packing fraction of polydisperse suspensions of spherical, noncolloidal particles is derived. The model connects the approaches by Bruggeman and Farris and is valid for large size ratios of consecutive particle classes during the construction process, appearing to be the first model consistently describing polydisperse volume fractions and maximum packing fraction within a single approach. In that context, the consistent inclusion of the maximum packing fraction into effective medium models is discussed. Furthermore, new generalized forms of the well-known Quemada and Krieger–Dougherty equations allowing for the choice of a second-order Taylor coefficient for the volume fraction ( -coefficient), found by asymptotic matching, are proposed. The model for the maximum packing fraction as well as the complete viscosity model is compared to experimental data from the literature showing good agreement. As a result, the new model is shown to replace the empirical Sudduth model for large diameter ratios. The extension of the model to the case of small size ratios is left for future work.

I. INTRODUCTION

II. BASICS OF APPARENT VISCOSITY AND REVIEW OF RECENT WORKS

III. GENERALIZATION OF THE VISCOSITY CORRELATION FOR MONODISPERSE SUSPENSIONS

IV. DEVELOPMENT OF A VISCOSITY CORRELATION FOR POLYDISPERSE SUSPENSIONS

A. Starting point: The differential Bruggeman model

B. Assumptions

C. Construction process

D. A discrete model for the relative viscosity

E. Determination of the maximum packing fraction

1. The Furnas model

2. The Sudduth model

3. A new model for the maximum packing fraction

F. Comparison with experiments and models from the literature

V. CONCLUSIONS

### Key Topics

- Viscosity
- 81.0
- Suspensions
- 62.0
- Shear rate dependent viscosity
- 5.0
- Viscosity measurements
- 5.0
- Numerical modeling
- 4.0

## Figures

Schematic representation of the two possible situations for the bidisperse maximum packing fraction: (a) Situation ①: The fraction of large particles lies below while the small particles fill the interstices with a fraction of . (b) Situation ②: The large particles occupy a fraction of while the small particles are present in the interstices according to the volume fraction ratio.

Schematic representation of the two possible situations for the bidisperse maximum packing fraction: (a) Situation ①: The fraction of large particles lies below while the small particles fill the interstices with a fraction of . (b) Situation ②: The large particles occupy a fraction of while the small particles are present in the interstices according to the volume fraction ratio.

Comparison of Eqs. (56) and (59) for the maximum packing fraction of bidisperse suspension with the data from McGeary (1961) using , accordingly; the Furnas model (48) gives the maximum value of is the diameter ratio (18) .

Comparison of Eqs. (56) and (59) for the maximum packing fraction of bidisperse suspension with the data from McGeary (1961) using , accordingly; the Furnas model (48) gives the maximum value of is the diameter ratio (18) .

Comparison of the model consisting of Eq. (46) for n = 2, Eqs. (56) and (59) with experimental data from Poslinski et al. (1988), , Chong et al. (1971) , and Sweeny (1959) for various total volume fractions ϕ ( ); missing total volume fractions are [ Chong et al. (1971) ] and [ Sweeny (1959) ]; diameter ratios are [ Poslinski et al. (1988) ], [ Chong et al. (1971) ], and [ Sweeny (1959) ], respectively.

Comparison of the model consisting of Eq. (46) for n = 2, Eqs. (56) and (59) with experimental data from Poslinski et al. (1988), , Chong et al. (1971) , and Sweeny (1959) for various total volume fractions ϕ ( ); missing total volume fractions are [ Chong et al. (1971) ] and [ Sweeny (1959) ]; diameter ratios are [ Poslinski et al. (1988) ], [ Chong et al. (1971) ], and [ Sweeny (1959) ], respectively.

Scheme for the calculation of maximum packing fraction and relative viscosity after the ith construction step for n particle size classes with references to the respective equations.

Scheme for the calculation of maximum packing fraction and relative viscosity after the ith construction step for n particle size classes with references to the respective equations.

## Tables

Correlations between relative viscosity and volume fraction ϕ for .

Correlations between relative viscosity and volume fraction ϕ for .

Values of the second-order intrinsic viscosity taken from the literature.

Values of the second-order intrinsic viscosity taken from the literature.

Scheme for the volume fractions during the construction process (SC = size class).

Scheme for the volume fractions during the construction process (SC = size class).

Notation for the maximum packing fraction during the construction process (SC = size class); the arrows indicate that the far right values are calculated recursively from the previous values (for details of the calculation, see Sec. IV E ).

Notation for the maximum packing fraction during the construction process (SC = size class); the arrows indicate that the far right values are calculated recursively from the previous values (for details of the calculation, see Sec. IV E ).

Maximum packing fraction from McGeary (1961) compared against three different models: This work [Eqs. (65) and (67) ], the model by Sudduth (1993b), and the Furnas model [Eq. (49) ]; i denotes the index of a size class, λ is the diameter ratio between two consecutive classes.

Maximum packing fraction from McGeary (1961) compared against three different models: This work [Eqs. (65) and (67) ], the model by Sudduth (1993b), and the Furnas model [Eq. (49) ]; i denotes the index of a size class, λ is the diameter ratio between two consecutive classes.

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