Skip to main content
Erschienen in: Rheologica Acta 5/2017

17.04.2017 | Original Contribution

A simple semiempirical model for the effective viscosity of multicomponent suspensions

verfasst von: Carlos I. Mendoza

Erschienen in: Rheologica Acta | Ausgabe 5/2017

Einloggen

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

We propose a methodology to approximate the viscosity of multicomponent suspensions. The procedure consists of successive applications of expressions for the viscosity of binary mixtures, originally written as the product of monomodal stiffening functions. First, the viscosity of a binary mixture made of the two smallest components is calculated. This allows to extract a volume fraction that will be used, together with the volume fraction of the third component, to feed the next iteration of the procedure to calculate the viscosity of a trimodal mixture and so on. The application of this approach to arbitrary mixtures requires the detailed knowledge of the geometry of the system in the form of size ratios and compositions. When this information is unknown, an approximation of the model can still be used as a fitting tool. With that purpose, the final expression for the viscosity is written in terms of an effective volume fraction that is further approximated by the use of a (1,2) Padé approximant. This approximation allows to incorporate the crowding effects due to different species in a volume fraction-dependent crowding factor that can be used as a fitting parameter to match experimental or simulation data. We have applied the model to mixtures of particles with different sizes and tested its accuracy comparing with experimental results obtaining very good agreement.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Anhänge
Nur mit Berechtigung zugänglich
Fußnoten
1
1 A Padé approximant of order (m, n) to an analytic function f(x) at x = 0 is defined by the rational function \(R(x)=\frac {{\sum }_{j=0}^{m}a_{j}x^{j}}{1+{\sum }_{k=1}^{n}b_{k}x^{k}}\) and gives the “best” approximation of a function by a rational function of given order.
 
Literatur
Zurück zum Zitat Brinkman HC (1952) The viscosity of concentrated suspensions and solutions. J Chem Phys 20:571CrossRef Brinkman HC (1952) The viscosity of concentrated suspensions and solutions. J Chem Phys 20:571CrossRef
Zurück zum Zitat Brodnyan JG (1959) The concentration dependence of the newtonian viscosity of prolate ellipsoids. Trans Soc Rheol 3:61–68CrossRef Brodnyan JG (1959) The concentration dependence of the newtonian viscosity of prolate ellipsoids. Trans Soc Rheol 3:61–68CrossRef
Zurück zum Zitat Brouwers HJH (2011) Packing fraction of geometric random packings of discretely sized particles. Phys Rev E 84:042301CrossRef Brouwers HJH (2011) Packing fraction of geometric random packings of discretely sized particles. Phys Rev E 84:042301CrossRef
Zurück zum Zitat Brouwers HJH (2013) Packing fraction of trimodal spheres with small size ratio: an analytical expression. Phys Rev E 88:032204CrossRef Brouwers HJH (2013) Packing fraction of trimodal spheres with small size ratio: an analytical expression. Phys Rev E 88:032204CrossRef
Zurück zum Zitat Bullard JW, Pauli AT, Garboczi EJ, Martys NS (2009) A comparison of viscosity–concentration relationships for emulsions. J Colloid Interface Sci 330:186–193CrossRef Bullard JW, Pauli AT, Garboczi EJ, Martys NS (2009) A comparison of viscosity–concentration relationships for emulsions. J Colloid Interface Sci 330:186–193CrossRef
Zurück zum Zitat Chang C, Powell RL (1994) Effect of particle size distributions on the rheology of concentrated bimodal suspensions. J Rheol 38:85–98CrossRef Chang C, Powell RL (1994) Effect of particle size distributions on the rheology of concentrated bimodal suspensions. J Rheol 38:85–98CrossRef
Zurück zum Zitat Chong JS, Christiansen EB, Daer AD (1971) Rheology of concentrated suspensions. J Appl Polymer Sci 15:2007–2021CrossRef Chong JS, Christiansen EB, Daer AD (1971) Rheology of concentrated suspensions. J Appl Polymer Sci 15:2007–2021CrossRef
Zurück zum Zitat Dames B, Morrison BR, Willenbacher N (2001) An empirical model predicting the viscosity of highly concentrated, bimodal dispersions with colloidal interactions. Rheol Acta 40:434–440CrossRef Dames B, Morrison BR, Willenbacher N (2001) An empirical model predicting the viscosity of highly concentrated, bimodal dispersions with colloidal interactions. Rheol Acta 40:434–440CrossRef
Zurück zum Zitat de Kruif CG, van Iersel EMF, Vrij A, Russel WB (1985) Hard sphere colloidal dispersions: viscosity as a function of shear rate and volume fraction. J Chem Phys 83:4717–4725CrossRef de Kruif CG, van Iersel EMF, Vrij A, Russel WB (1985) Hard sphere colloidal dispersions: viscosity as a function of shear rate and volume fraction. J Chem Phys 83:4717–4725CrossRef
Zurück zum Zitat D’Haene P, Mewis J (1994) Rheological characterization of bimodal colloidal dispersions. Rheol Acta 33:165–174CrossRef D’Haene P, Mewis J (1994) Rheological characterization of bimodal colloidal dispersions. Rheol Acta 33:165–174CrossRef
Zurück zum Zitat Donev A, Cisse I, Sachs D, Variano E, Stillinger FH, Connelly R, Torquato S, Chaikin PM (2004) Improving the density of jammed disordered packings using ellipsoids. Science 303:990– 992CrossRef Donev A, Cisse I, Sachs D, Variano E, Stillinger FH, Connelly R, Torquato S, Chaikin PM (2004) Improving the density of jammed disordered packings using ellipsoids. Science 303:990– 992CrossRef
Zurück zum Zitat Dörr A, Sadiki A, Mehdizadeh A (2013) A discrete model for the apparent viscosity of polydisperse suspensions including maximum packing fraction. J Rheol 57:743–765CrossRef Dörr A, Sadiki A, Mehdizadeh A (2013) A discrete model for the apparent viscosity of polydisperse suspensions including maximum packing fraction. J Rheol 57:743–765CrossRef
Zurück zum Zitat Einstein A (1906) Eine neue Bestimmung der moleküldimensionen. Ann Phys 19:289–306CrossRef Einstein A (1906) Eine neue Bestimmung der moleküldimensionen. Ann Phys 19:289–306CrossRef
Zurück zum Zitat Einstein A (1911) Berichtigung zu meiner Arbeit: Eine neue Bestimmung der moleküldimensionen. Ann Phys 34:591–592CrossRef Einstein A (1911) Berichtigung zu meiner Arbeit: Eine neue Bestimmung der moleküldimensionen. Ann Phys 34:591–592CrossRef
Zurück zum Zitat Eveson GF (1959) The viscosity of stable suspensions of spheres at low rates of shear. In: Mill CC (ed) Rheology of disperse systems. Pergamon Press, London, pp 61–83 Eveson GF (1959) The viscosity of stable suspensions of spheres at low rates of shear. In: Mill CC (ed) Rheology of disperse systems. Pergamon Press, London, pp 61–83
Zurück zum Zitat Faroughi SA, Huber C (2014) Crowding-based rheological model for suspensions of rigid bimodal-sized particles with interfering size ratios. Phys Rev E 90:052303CrossRef Faroughi SA, Huber C (2014) Crowding-based rheological model for suspensions of rigid bimodal-sized particles with interfering size ratios. Phys Rev E 90:052303CrossRef
Zurück zum Zitat Farr RS (2014) Simple heuristic for the viscosity of polydisperse hard spheres. J Chem Phys 141:214503CrossRef Farr RS (2014) Simple heuristic for the viscosity of polydisperse hard spheres. J Chem Phys 141:214503CrossRef
Zurück zum Zitat Farris RJ (1968) Prediction of the viscosity of multimodal suspensions from unimodal viscosity data. Trans Soc Rheol 12:281–301CrossRef Farris RJ (1968) Prediction of the viscosity of multimodal suspensions from unimodal viscosity data. Trans Soc Rheol 12:281–301CrossRef
Zurück zum Zitat Furnas CC (1931) Grading aggregates - I. - mathematical relations for beds of broken solids of maximum density. Ind Eng Chem 23:1052–1058CrossRef Furnas CC (1931) Grading aggregates - I. - mathematical relations for beds of broken solids of maximum density. Ind Eng Chem 23:1052–1058CrossRef
Zurück zum Zitat Genovese DB (2012) Shear rheology of hard-sphere, dispersed, and aggregated suspensions, and filler-matrix composites. Adv Colloid Interf Sci 171–172:1–16CrossRef Genovese DB (2012) Shear rheology of hard-sphere, dispersed, and aggregated suspensions, and filler-matrix composites. Adv Colloid Interf Sci 171–172:1–16CrossRef
Zurück zum Zitat Gondret P, Petit L (1997) Dynamic viscosity of macroscopic suspensions of bimodal sized solid spheres. J Rheol 41:1261–1274CrossRef Gondret P, Petit L (1997) Dynamic viscosity of macroscopic suspensions of bimodal sized solid spheres. J Rheol 41:1261–1274CrossRef
Zurück zum Zitat Greenwood R, Luckham PF, Gregory T (1997) The effect of diameter ratio and volume ratio on the viscosity of bimodal suspensions of polymer latices. J Colloid Interface Sci 191:11–21CrossRef Greenwood R, Luckham PF, Gregory T (1997) The effect of diameter ratio and volume ratio on the viscosity of bimodal suspensions of polymer latices. J Colloid Interface Sci 191:11–21CrossRef
Zurück zum Zitat He D, Ekere NN (2001) Viscosity of concentrated noncolloidal bidisperse suspensions. Rheol Acta 40:591–598CrossRef He D, Ekere NN (2001) Viscosity of concentrated noncolloidal bidisperse suspensions. Rheol Acta 40:591–598CrossRef
Zurück zum Zitat Hoffman RL (1992) Factors affecting the viscosity of unimodal and multimodal colloidal dispersions. J Rheol 36:947–965CrossRef Hoffman RL (1992) Factors affecting the viscosity of unimodal and multimodal colloidal dispersions. J Rheol 36:947–965CrossRef
Zurück zum Zitat Krieger IM, Dougherty TJ (1959) A mechanism for Non-Newtonian flow in suspensions of rigid spheres. Trans Soc Rheol 3:137–152CrossRef Krieger IM, Dougherty TJ (1959) A mechanism for Non-Newtonian flow in suspensions of rigid spheres. Trans Soc Rheol 3:137–152CrossRef
Zurück zum Zitat Landau LD, Lifshitz EM, Pitaevskii LP (1984) Electrodynamics of continuous media. Pergamon Press, Oxford Landau LD, Lifshitz EM, Pitaevskii LP (1984) Electrodynamics of continuous media. Pergamon Press, Oxford
Zurück zum Zitat Lionberger RA (2002) Viscosity of bimodal and polydisperse colloidal suspensions. Phys Rev E 65:061408CrossRef Lionberger RA (2002) Viscosity of bimodal and polydisperse colloidal suspensions. Phys Rev E 65:061408CrossRef
Zurück zum Zitat Luckham PF, Ukeje MA (1999) Effect of particle size distribution on the rheology of dispersed systems. J Colloid Interface Sci 220:347–356CrossRef Luckham PF, Ukeje MA (1999) Effect of particle size distribution on the rheology of dispersed systems. J Colloid Interface Sci 220:347–356CrossRef
Zurück zum Zitat Maron SH, Pierce PE (1956) Application of Ree-Eyring generalized flow theory to suspensions of spherical particles. J Colloid Sci 11:80–95CrossRef Maron SH, Pierce PE (1956) Application of Ree-Eyring generalized flow theory to suspensions of spherical particles. J Colloid Sci 11:80–95CrossRef
Zurück zum Zitat Mendoza CI, Santamaría-Holek I (2009) The rheology of hard sphere suspensions at arbitrary volume fractions: An improved differential viscosity model. J Chem Phys 130:044904CrossRef Mendoza CI, Santamaría-Holek I (2009) The rheology of hard sphere suspensions at arbitrary volume fractions: An improved differential viscosity model. J Chem Phys 130:044904CrossRef
Zurück zum Zitat Mendoza CI, Santamaría-Holek I (2010) Rheology of concentrated emulsions of spherical droplets. Appl Rheol 20:23493 Mendoza CI, Santamaría-Holek I (2010) Rheology of concentrated emulsions of spherical droplets. Appl Rheol 20:23493
Zurück zum Zitat Mendoza CI (2011) Effective static and high-frequency viscosities of concentrated suspensions of soft particles. J Chem Phys 135:054904CrossRef Mendoza CI (2011) Effective static and high-frequency viscosities of concentrated suspensions of soft particles. J Chem Phys 135:054904CrossRef
Zurück zum Zitat Mendoza CI (2013) Model for the shear viscosity of suspensions of star polymers and other soft particles. Macromol Chem Phys 214:599–604CrossRef Mendoza CI (2013) Model for the shear viscosity of suspensions of star polymers and other soft particles. Macromol Chem Phys 214:599–604CrossRef
Zurück zum Zitat Mooney M (1951) The viscosity of a concentrated suspension of spherical particles. J Colloid Sci 6:162–170CrossRef Mooney M (1951) The viscosity of a concentrated suspension of spherical particles. J Colloid Sci 6:162–170CrossRef
Zurück zum Zitat Mwasame PM, Wagner NJ, Beris AN (2016a) Modeling the effects of polydispersity on the viscosity of noncolloidal hard sphere suspensions. J Rheol 60:225–240CrossRef Mwasame PM, Wagner NJ, Beris AN (2016a) Modeling the effects of polydispersity on the viscosity of noncolloidal hard sphere suspensions. J Rheol 60:225–240CrossRef
Zurück zum Zitat Mwasame PM, Wagner NJ, Beris AN (2016b) Modeling the viscosity of polydisperse suspensions: improvements in prediction of limiting behavior. Phys Fluids 28:061701CrossRef Mwasame PM, Wagner NJ, Beris AN (2016b) Modeling the viscosity of polydisperse suspensions: improvements in prediction of limiting behavior. Phys Fluids 28:061701CrossRef
Zurück zum Zitat Núñez A et al (2002) Viscosity minimum in bimodal concentrated suspensions under shear. Eur Phys J E 9:327–334CrossRef Núñez A et al (2002) Viscosity minimum in bimodal concentrated suspensions under shear. Eur Phys J E 9:327–334CrossRef
Zurück zum Zitat Papachristodoulou G, Trass O (1984) Rheological properties of coal—oil mixture fuels. Powder Technol 40:353–362CrossRef Papachristodoulou G, Trass O (1984) Rheological properties of coal—oil mixture fuels. Powder Technol 40:353–362CrossRef
Zurück zum Zitat Pishvaei M, Graillat C, Cassgnau P, McKenna TF (2006) Modelling the zero shear viscosity of bimodal high solid content latex: calculation of the maximum packing fraction. Chem Eng Sci 61:5768–5780CrossRef Pishvaei M, Graillat C, Cassgnau P, McKenna TF (2006) Modelling the zero shear viscosity of bimodal high solid content latex: calculation of the maximum packing fraction. Chem Eng Sci 61:5768–5780CrossRef
Zurück zum Zitat Poslinski AJ et al (1988) Rheological behavior of filled polymeric systems II. The effect of a bimodal size distribution of particulates. J Rheol 32:751–771CrossRef Poslinski AJ et al (1988) Rheological behavior of filled polymeric systems II. The effect of a bimodal size distribution of particulates. J Rheol 32:751–771CrossRef
Zurück zum Zitat Probstein RF, Sengun MZ, Tseng T -C (1994) Bimodal model of concentrated suspension viscosity for distributed particle sizes. J Rheol 38:811–829CrossRef Probstein RF, Sengun MZ, Tseng T -C (1994) Bimodal model of concentrated suspension viscosity for distributed particle sizes. J Rheol 38:811–829CrossRef
Zurück zum Zitat Qi F, Tanner RI (2011) Relative viscosity of bimodal suspensions. Korea-Australia Rheology Journal 23:105–111CrossRef Qi F, Tanner RI (2011) Relative viscosity of bimodal suspensions. Korea-Australia Rheology Journal 23:105–111CrossRef
Zurück zum Zitat Qi F, Tanner RI (2012) Random close packing and relative viscosity of multimodal suspensions. Rheol Acta 51:289–302CrossRef Qi F, Tanner RI (2012) Random close packing and relative viscosity of multimodal suspensions. Rheol Acta 51:289–302CrossRef
Zurück zum Zitat Quemada D (1998) Rheological modelling of complex fluids. I The concept of effective volume fraction revisited. Eur Phys J AP 1:119–127CrossRef Quemada D (1998) Rheological modelling of complex fluids. I The concept of effective volume fraction revisited. Eur Phys J AP 1:119–127CrossRef
Zurück zum Zitat Roscoe R (1952) The viscosity of suspensions of rigid spheres. Br J Appl Phys 3:267–269CrossRef Roscoe R (1952) The viscosity of suspensions of rigid spheres. Br J Appl Phys 3:267–269CrossRef
Zurück zum Zitat Rosin P, Rammler E (1933) The laws governing the fineness of powdered coal. J Inst Fuel 7:29–36 Rosin P, Rammler E (1933) The laws governing the fineness of powdered coal. J Inst Fuel 7:29–36
Zurück zum Zitat Santamaría-Holek I, Mendoza CI (2010) The rheology of concentrated suspensions of arbitrarily-shaped particles. J Colloid Interface Sci 346:118CrossRef Santamaría-Holek I, Mendoza CI (2010) The rheology of concentrated suspensions of arbitrarily-shaped particles. J Colloid Interface Sci 346:118CrossRef
Zurück zum Zitat Servais C, Jones R, Roberts I (2002) The influence of particle size distribution on the processing of food. J Food Eng 51:201–208CrossRef Servais C, Jones R, Roberts I (2002) The influence of particle size distribution on the processing of food. J Food Eng 51:201–208CrossRef
Zurück zum Zitat Shapiro AP, Probstein RF (1992) Random packings of spheres and fluidity limits of monodisperse and bidisperse suspensions. Phys Rev Lett 68:1422–1425CrossRef Shapiro AP, Probstein RF (1992) Random packings of spheres and fluidity limits of monodisperse and bidisperse suspensions. Phys Rev Lett 68:1422–1425CrossRef
Zurück zum Zitat Shewan HM, Stokes JR (2015) Analytically predicting the viscosity of hard sphere suspensions from the particle size distribution. J Non-Newtonian Fluid Mech 222:72CrossRef Shewan HM, Stokes JR (2015) Analytically predicting the viscosity of hard sphere suspensions from the particle size distribution. J Non-Newtonian Fluid Mech 222:72CrossRef
Zurück zum Zitat Sudduth RD (1993a) A generalized model to predict the viscosity of solutions with suspended particles. I. J Applied Polymer Sci 48:25–36CrossRef Sudduth RD (1993a) A generalized model to predict the viscosity of solutions with suspended particles. I. J Applied Polymer Sci 48:25–36CrossRef
Zurück zum Zitat Sudduth RD (1993b) A new method to predict the maximum packing fraction and the viscosity of solutions with a size distribution of suspended particles. II. J Appl Polym Sc 48:37–55CrossRef Sudduth RD (1993b) A new method to predict the maximum packing fraction and the viscosity of solutions with a size distribution of suspended particles. II. J Appl Polym Sc 48:37–55CrossRef
Zurück zum Zitat Sudduth RD (1993c) A generalized model to predict the viscosity of solutions with suspended particles. III. Effects of particle interaction and particle size distribution. J Applied Polymer Sci 50:123–147CrossRef Sudduth RD (1993c) A generalized model to predict the viscosity of solutions with suspended particles. III. Effects of particle interaction and particle size distribution. J Applied Polymer Sci 50:123–147CrossRef
Zurück zum Zitat Tanner RI, Qi F, Housiadas KD (2010) A differential approach to suspensions with power-law matrices. J Non-Newtonian Fluid Mech 165:1677–1681CrossRef Tanner RI, Qi F, Housiadas KD (2010) A differential approach to suspensions with power-law matrices. J Non-Newtonian Fluid Mech 165:1677–1681CrossRef
Zurück zum Zitat Vesilind PA (1980) The Rosin-Rammler particle size distribution. Resour Recovery Conserv 5:275–277CrossRef Vesilind PA (1980) The Rosin-Rammler particle size distribution. Resour Recovery Conserv 5:275–277CrossRef
Zurück zum Zitat Wagner NJ, Woutersen ATJM (1994) The viscosity of bimodal and polydisperse suspensions of hard spheres in the dilute limit. J Fluid Mech 278:267–287CrossRef Wagner NJ, Woutersen ATJM (1994) The viscosity of bimodal and polydisperse suspensions of hard spheres in the dilute limit. J Fluid Mech 278:267–287CrossRef
Zurück zum Zitat Wouterse A, Williams SR, Philipse AP (2007) Effect of particle shape on the density and microstructure of random packings. J Phys: Condens Matter 19:406215 Wouterse A, Williams SR, Philipse AP (2007) Effect of particle shape on the density and microstructure of random packings. J Phys: Condens Matter 19:406215
Metadaten
Titel
A simple semiempirical model for the effective viscosity of multicomponent suspensions
verfasst von
Carlos I. Mendoza
Publikationsdatum
17.04.2017
Verlag
Springer Berlin Heidelberg
Erschienen in
Rheologica Acta / Ausgabe 5/2017
Print ISSN: 0035-4511
Elektronische ISSN: 1435-1528
DOI
https://doi.org/10.1007/s00397-017-1011-6

Weitere Artikel der Ausgabe 5/2017

Rheologica Acta 5/2017 Zur Ausgabe

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.