The Rheology Handbook. Thomas Mezger

= exp (αp ⋅ Δp)

      with αp in [1/MPa = MPa-1], at the pressure difference Δp = p - pref [MPa]

      Typical values of αp for liquids are in between +0.005 and +0.05 MPa-1

      (or 5 ⋅ 10-3 MPa-1 ≤ αp ≤ 50 ⋅ 10-3 MPa-1). For water at T < +32 °C, the value of αp is negative, since viscosity decreases with increasing pressure as explained above; for T > +32 °C, αp is

      positive, i. e. viscosity increases now with increasing pressure.

      In order to estimate viscosity values at pressures at which no measuring values are available, proceed as follows:

      1 Select pref (i. e. a pressure, at which an η-value is available), usually is selected:pref = 0.1 MPa (= 1 bar).

      2 Calculate the shift factor ap for another available η(p)-value (using Equation 3.11).

      3 Calculate the coefficient αp (using Equation 3.12).

      4 Calculate the shift factor ap for the desired η(p)-value (using Equation 3.12).

      5 Result: Calculate the desired η(p)-value (using Equation 3.11).

      3.2.1.1.2Example 1: Calculation of viscosity/pressure coefficient and shift factor of a mineral oil, and determination of viscosity values at further pressures

      From a mineral oil is known: η1 = 0.300 Pas (at p1 = 0.1 MPa = 1 bar), and η2 = 2.22 Pas

      (at p2 = 100 MPa = 1000 bar). Desired is the viscosity value η3 at p3 = 75 MPa (= 750 bar).

      1 Here is selected: pref = p1 = 0.1 MPa

      2 ap is calculated for p2 (as ap2): ap2 = η2 (p2) / η1 (pref) = 2.22 Pas / 0.300 Pas = 7.40

      3 αp is calculated, with Δp = Δp21 = p2 – pref:ln (ap) = αp ⋅ Δp, and thus: αp = ln (ap) / Δpαp = ln (7.40) / (100 – 0.1) MPa = 2.00 / 99.9 MPa = 0.02 MPa-1 (= 20 ⋅ 10-3 MPa-1)

      4 ap is calculated for p3 (as ap3), with Δp = Δp31 = p3 – pref:ap3 = exp (αp ⋅ Δp31) = exp [0.02 MPa-1 ⋅ (75 - 0.1) MPa] = e1.5 = 4.48

      5 Result: η3 (p3) = ap3 ⋅ η1 (pref) = 4.48 ⋅ 0.300 Pas = 1.34 Pas

Table 3.6: Pressure-dependent viscosity values, see the example of Chapter 3.6
p [MPa]	0.1	1	10	25	50	75	100
η [Pas]	0.300	0.306	0.366	0.495	0.813	1.34	2.22

      The already mentioned and some further pressure-dependent viscosity values are presented in Table 3.6.

      3.2.1.1.3Example 2: Calculation of the viscosity/pressure coefficient of a crude oil

      From a crude oil is known: η1 = 36 mPas (at p1 = 0.1 MPa = 1 bar),

      and η2 = 45 mPas (at p2 = 10 MPa = 100 bar). Desired is the αp-coefficient.

      1 pref = p1 = 0.1 MPa

      2 ap = η2 (p2) / ηref (p1)= 45/36 = 1.25

      3 αp = ln (ap )/ Δp = ln (1.25) / (10 – 0.1) MPa = 0.0225 MPa-1 (= 22.5 ⋅ 10-3 MPa-1)

      End of the Cleverly section

3.7References

      [3.1]ICA (International Confectionery Association), Viscosity of cocoa and chocolate

      products, Analytical method 46: (2000), Caobisco, Bruxelles

[3.2]Ostwald, Wo. (jun.), Zur Systematik der Kolloide, Kolloid-Z., 1907; Über die Geschwindigkeitsfunktion der Viskosität disperser Systeme, ibid., 1925; Ostwald, Wo., Malss, H., Über Strukturviskosität kritischer Flüssigkeitsgemische, ibid, 1933; Ostwald, Wo., Auerbach, R., Über die Viskosität kolloider Lösungen im Struktur-, Laminar- und Turbulenzgebiet, ibid., 1936

[3.3]Messow, U., Wolfgang Ostwald und der kolloid-disperse Zustand, in: Zur Entwicklung erster Messgeräte der physikalischen Chemie an der Universität Leipzig, Leipziger Univ. Verl., 2013

[3.4]Bingham, E. C., An investigation of the laws of plastic flow, Bull. U.S. Bur. of Standards, 1916; Bingham, E. C., Green, H., Paint – a plastic material and not a viscous liquid, Proc. ASTM, 1919; Fluidity and plasticity, McGraw-Hill, New York, 1922

[3.5]Wassermann, L., From Heraklit to W. Scott Blair, J. Rheology, 04/1991; Historische Aspekte der Lebensmittelrheologie, in: D. Weipert et al., Rheologie der Lebensmittel, Behr’s, Hamburg, 1993

[3.6]Brookfield Inc., Middleboro, product information, www, 2016

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[3.9]Barnes, H. A., Hutton, J. F., Walters, K., An introduction to rheology, Elsevier: Amsterdam, 1989; Barnes, H. A., A handbook of elementary rheology, Univ. of Wales Inst. Non-Newtonian Fluid Mechanics: Aberystwyth, 2000; Barnes, H. A., Viscosity, ibid., 2002

[3.10]Pahl, M., Gleissle, W., Laun, H.-M., Praktische Rheologie der Kunststoffe und Elastomere, VDI, Düsseldorf, 1995

[3.11]Berry, G. C., Fox, T. G., The viscosity of polymers and their concentrated solutions, Adv. Polym. Sci., 1968

[3.12]Malkin, A., Isayev, A. I., Rheology – concepts, methods, and applications, Chem Tec, Toronto, 2006

[3.13]Boger, D. V., Walters, K., Rheological pheno­mena in focus, Elsevier, Amsterdam, 1993

[3.14]Fischer, P., Windhab, E. J., Rheologie der Lebensmittel – Grundlagen und Applikationen, lecture script ETH Zurich, Inst. f. Lebensmitteltechnologie, Zurich, 2003

[3.15]Windhab, E. J., Rheology and microstructure – keys for quality processing of food systems, Proc. 3. Int. Sympos. Food Rheol. & Structure, editor P. Fischer et al., Zurich, 2003

[3.16]Wrana, C., Polymerphysik – Elastomere und ihre anwendungstechnischen Eigenschaften, Springer, Berlin, 2004; Introduction to polymer physics, Lanxess, Leverkusen, 2009

[3.17]Li, X., Cao, H. L., Pan, F. Y., Weng, L. Q., Song, S. H., Huang, Y. D., Preparation of body armour material of Kevlar fabric treated with colloidal silica nanocomposite, J. Plastics, Rubber & Composites, 2008

[3.18]Daniel, F. K., Determinations of mill base compositions for high speed dispersers, JPT 1966

[3.19]Patton, T. C., Paint flow and pigment dispersion, Wiley, New York, 1978 (2nd ed.)

[3.20]Everett, D. H., Basic principles of colloid science, Royal Soc. Chemistry, London, 1988

[3.21]Laba, D. (editor), Rheological properties of cosmetics and toiletries, Dekker, New York, 1993

[3.22]Larson, R. G., The structure and rheology of complex fluids, Oxford Univ. Press, New York, 1999

[3.23]Sangl, R., Streichen von Papier und Karton, in: Blechschmidt, J. (editor): Taschenbuch der Papiertechnik, Hanser, Leipzig, 2013 (2nd ed.)

[3.24]Barnes, H. A., Walters, K., The yield stress myth?, Rheol. Acta, 1985; Barnes, H. A., The “yield stress myth” paper – 21 years on, J. Appl. Rheol. 17, 2007

[3.25]DIN SPEC 143-1 (Fachbericht): Modern rheological test methods, part 1: Determination of the yield point (fundamentals and comparative test methods) , Beuth, Berlin, 2005/2019 (in German: Moderne rheo­logische Prüfverfahren, Teil 1: Bestimmung der Fließgrenze, Grundlagen und Ringversuch)

[3.26]Møller, P. C. F., Mewis,