Summary
The paper deals with laminar flow of power law fluids through granular beds. A critical review of the assumptions concerning the capillary model of the bed, applied by various authors, led us to the conclusion that the derivation of the correlation eq. [13] given byChristopher andMiddleman was based on a too simplified model of the granular bed. Taking advantage of the approach presented in the classical works ofKozeny andCarman (which seems to be partly overlooked by some authors, including our own previous works) a modified correlation equation for power law fluids [21], a corrected formula for shear rate in the bed [29] and for Deborah number [32], as well as corrected correlation equation for fluids exhibiting memory effects [34] were presented.
Zusammenfassung
Diese Arbeit betrifft laminare Strömungen von Potenzgesetzflüssigkeiten durch Kornschüttungen. Eine kritische Prüfung der Annahmen, die von verschiedenen Autoren für das Kapillar-Modell der Schüttung gemacht worden sind, führt uns zu der Folgerung, daß die Herleitung der Korrelationsgleichung [13] nachChristopher undMiddleman auf einem übervereinfachten Modell der Kornschüttung basiert. Unter Nutzbarmachung der Annahmen, die in den klassischen Arbeiten vonKozeny undCarman dargestellt worden sind (sie wurden sowohl von manchen anderen Autoren als auch in unseren früheren Arbeiten nicht beachtet), werden nun eine modifizierte Korrelationsgleichung für die Potenzgesetzflüssigkeiten [21], eine korrigierte Formel für die Schergeschwindigkeit in der Schüttung [29], eine korrigierte Formel für die Deborah-Zahl [32] und eine korrigierte Korrelationsgleichung für Flüssigkeiten, die Gedächtnis-Effekte zeigen [34], angegeben.
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Abbreviations
- A :
-
constant in eq. [9]
- d p :
-
effective particle diameterd p = 6/a (wherea is the specific surface of the bed), m
- f BK :
-
modified friction factor, defined by eq. [1]
- k :
-
power law parameter, N sn/m2
- K :
-
Kozeny constant, defined by eq. [8]
- K 0 :
-
constant depending on the shape of the channel cross-section
- K 1 :
-
constant, defined by eq. [5]
- l :
-
bed height, m
- l e :
-
channel length, m
- n :
-
power law parameter
- Δp :
-
pressure drop due to friction, N/m2
- r h :
-
hydraulic radius, defined by eq. [6], m
- s :
-
bed permeability, defined by eq. [16], m2
- v 0 :
-
mean linear velocity related to an empty crosssection of the column, m/s
- v e :
-
mean linear velocity in the channel, m/s
- \(\dot \gamma _w \) :
-
shear rate at the wall of the channel, s−1
- \(\dot \gamma _w^* \) :
-
shear rate at the wall of the channel calculated according to the formula [29], s−1
- ε :
-
bed porosity
- θ :
-
characteristic time of the fluid, s
- λ :
-
friction factor, defined by eq. [25]
- µ :
-
dynamic viscosity of the fluid, N s/m2
- ψ :
-
parameter, defined by eq. [15], N sn/m1+n
- De :
-
Deborah number, defined by eq. [33]
- De * :
-
Deborah number, defined by eq. [32]
- Re BK :
-
modified Reynolds number, defined by eq. [2]
- Re ′BK :
-
modified Reynolds number, defined by eq. [26]
- Re *BK :
-
modified Reynolds number, defined by eq. [23]
- Re CM :
-
modified Reynolds number byChristopher andMiddleman, defined by eq. [14]
- Re ′CM :
-
modified Reynolds number, defined by eq. [17]
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Kemblowski, Z., Michniewicz, M. A new look at the laminar flow of power law fluids through granular beds. Rheol Acta 18, 730–739 (1979). https://doi.org/10.1007/BF01533348
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DOI: https://doi.org/10.1007/BF01533348