Suspension viscosity

The relative viscosity of the suspension $\eta$ ${\ displaystyle \ eta}$ $\ eta$ (the ratio of the viscosity of the suspension to the viscosity of the dispersion medium ) depends on many factors: temperature T, degree of dispersion , volume fraction $\varphi$ ${\ displaystyle \ varphi}$ $\ varphi$ dispersed phase, shear rate (velocity gradient) ${\dot {\gamma }}$ ${\ displaystyle {\ dot {\ gamma}}}$ ${\ displaystyle {\ dot {\ gamma}}}$ and shear stress time t (in the case of relaxation effects - thixotropy or rheopexy ).

Krieger Formula

The dependence of the relative viscosity of the suspension $\eta$ ${\ displaystyle \ eta}$ $\eta$ on the concentration of the dispersed phase $\varphi$ ${\ displaystyle \ varphi}$ $\varphi$ at a constant shear rate is described by the Krieger formula ^[1] :

$(1)\qquad \eta =\left(1-{\frac {\varphi }{\varphi _{0}}}\right)^{-\varphi _{0}[\eta ]}$ ${\ displaystyle (1) \ qquad \ eta = \ left (1 - {\ frac {\ varphi} {\ varphi _ {0}}} \ right) ^ {- \ varphi _ {0} [\ eta]}}$ $(1)\qquad \eta =\left(1-{\frac {\varphi }{\varphi _{0}}}\right)^{-\varphi _{0}[\eta ]}$

Where $[\eta ]$ ${\ displaystyle [\ eta]}$ $[\eta ]$ - intrinsic viscosity $\varphi _{0}$ ${\ displaystyle \ varphi _ {0}}$ $\varphi _{0}$ - the maximum amount of solid phase that can be introduced into the suspension; while the viscosity of the suspension tends to infinity.

This formula is derived from the following premises ^[2]

A) When the volume fraction of the solid phase tends to zero, the relative viscosity of the suspension tends to unity, and its derivative with respect to the volume fraction of the dispersed phase tends to the characteristic viscosity:

$(2)\qquad {\begin{cases}\eta (0)=1\\{d\eta (0) \over d\varphi }=[\eta ]\end{cases}}$ ${\ displaystyle (2) \ qquad {\ begin {cases} \ eta (0) = 1 \\ {d \ eta (0) \ over d \ varphi} = [\ eta] \ end {cases}}}$ $(2)\qquad {\begin{cases}\eta (0)=1\\{d\eta (0) \over d\varphi }=[\eta ]\end{cases}}$

B) $\eta (\varphi )$ ${\ displaystyle \ eta (\ varphi)}$ $\eta (\varphi )$ must satisfy the Krieger functional equation [1]:

$(3)\qquad \eta {\bigl (}\varphi _{1}+\varphi _{2}{\bigr )}=\eta (\varphi _{1})\eta \left({\frac {\varphi _{2}}{1-\varphi _{1}/\varphi _{0}}}\right)$ ${\ displaystyle (3) \ qquad \ eta {\ bigl (} \ varphi _ {1} + \ varphi _ {2} {\ bigr)} = \ eta (\ varphi _ {1}) \ eta \ left ({ \ frac {\ varphi _ {2}} {1- \ varphi _ {1} / \ varphi _ {0}}} right)}$ $(3)\qquad \eta {\bigl (}\varphi _{1}+\varphi _{2}{\bigr )}=\eta (\varphi _{1})\eta \left({\frac {\varphi _{2}}{1-\varphi _{1}/\varphi _{0}}}\right)$

Where $\varphi _{1}$ ${\ displaystyle \ varphi _ {1}}$ $\varphi _{1}$ and $\varphi _{2}$ ${\ displaystyle \ varphi _ {2}}$ $\varphi _{2}$ - volume fractions of the same component, introduced in parts.

Replacing in equation (3) $\varphi _{1}$ ${\ displaystyle \ varphi _ {1}}$ $\varphi _{1}$ on $\varphi$ ${\ displaystyle \ varphi}$ $\varphi$ , but $\varphi _{2}$ ${\ displaystyle \ varphi _ {2}}$ $\varphi _{2}$ to differential $d\varphi$ ${\ displaystyle d \ varphi}$ $d\varphi$ , we obtain a differential equation whose solution, taking into account the initial conditions (2), is the Krieger formula (1).

A generalization of the Krieger formula to the case of a multicomponent suspension is ^[3] :

$(4)\qquad \eta =\left(1-\sum _{i=1}^{n}{\frac {\varphi _{i}}{\varphi _{0,i}}}\right)^{-{\overline {\varphi }}_{0}[{\overline {\eta }}]}$ ${\ displaystyle (4) \ qquad \ eta = \ left (1- \ sum _ {i = 1} ^ {n} {\ frac {\ varphi _ {i}} {\ varphi _ {0, i}}} \ right) ^ {- {\ overline {\ varphi}} _ {0} [{\ overline {\ eta}}]}}$ $(4)\qquad \eta =\left(1-\sum _{i=1}^{n}{\frac {\varphi _{i}}{\varphi _{0,i}}}\right)^{-{\overline {\varphi }}_{0}[{\overline {\eta }}]}$

Where $[{\overline {\eta }}]$ ${\ displaystyle [{\ overline {\ eta}}]}$ - volumetric characteristic viscosity $[{\overline {\eta }}]={\frac {\sum [\eta _{i}]\varphi _{i}}{\varphi }}$ ${\ displaystyle [{\ overline {\ eta}}] = {\ frac {\ sum [\ eta _ {i}] \ varphi _ {i}} {\ varphi}}}$ ,

${\overline {\varphi _{0}}}$ ${\ displaystyle {\ overline {\ varphi _ {0}}}}$ - average harmonic concentration limit ${\overline {\varphi _{0}}}={\frac {\varphi }{\sum \varphi _{i}/\varphi _{0,i}}}$ ${\ displaystyle {\ overline {\ varphi _ {0}}} = {\ frac {\ varphi} {\ sum \ varphi _ {i} / \ varphi _ {0, i}}}}$ ,

$\varphi$ ${\ displaystyle \ varphi}$ - total volume fraction of solid phase $\varphi =\sum \varphi _{i}$ ${\ displaystyle \ varphi = \ sum \ varphi _ {i}}$ .

Types of flow curves

Fig. 1. Types of Flow Curves

Flow Curves — Shear Stress Plots $\tau$ ${\ displaystyle \ tau}$ on shear rate ${\dot {\gamma }}$ ${\ displaystyle {\ dot {\ gamma}}}$ .

In fig. 1 schematically shows 4 different types of flow curves:

(1) - Newtonian fluid ,

(2) - Bingham plastic ,

(3) - dilatant suspension,

(4) a structurally viscous (pseudoplastic) suspension,

$\tau _{0}$ ${\ displaystyle \ tau _ {0}}$ - yield strength .

Literature

Krieger IM Flow Properties of Latex and Concentrated Solutions. In the book. "Surfaces and Coatings Related to Paaper and Wood." A Symposium, State University College of Forestry at Syracuse University, Syracuse University Press. 1967 P. 25-51.
Barnes HA, Hutton JF, Walters K. An Introduction to Rheology. Rheology series 3, Elsevier. 1988. P. 119-125.
Levinsky A.I. The viscosity of suspensions: the Krieger – Dougherty formula and the Farris effect "// Proceedings of universities. Chemistry and Chemical Technology, 2005. Vol. 48 No. 12.

Notes

↑ Krieger IM In the book. "Surfaces and Coatings Related to Paper and Wood" // Flow Properties of Latex and Concentrated Solutions .. - A Symposium, State University College of Forestry at Syracuse University: Syracuse University Press, 1967. - P. 25-51.
↑ Krieger IM In the book. "Surfaces and Coatings Related to Paper and Wood" // Flow Properties of Latex and Concentrated Solutions .. - A Symposium, State University College of Forestry at Syracuse University: Syracuse University Press, 1967. - P. 25-51.
↑ Levinsky A.I. The viscosity of suspensions: the Krieger – Dougherty formula and the Farris effect // News of universities. Chemistry and chemical technology. - 2005. - T. 48 , No. 12 . - S. 22-25 .

[1] Krieger IM In the book. "Surfaces and Coatings Related to Paper and Wood" // Flow Properties of Latex and Concentrated Solutions .. - A Symposium, State University College of Forestry at Syracuse University: Syracuse University Press, 1967. - P. 25-51.

[2] Krieger IM In the book. "Surfaces and Coatings Related to Paper and Wood" // Flow Properties of Latex and Concentrated Solutions .. - A Symposium, State University College of Forestry at Syracuse University: Syracuse University Press, 1967. - P. 25-51.

[3] Levinsky A.I. The viscosity of suspensions: the Krieger – Dougherty formula and the Farris effect // News of universities. Chemistry and chemical technology. - 2005. - T. 48 , No. 12 . - S. 22-25 .

Suspension viscosity

Krieger Formula

Types of flow curves

Literature

Notes

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