Skip to main content
SpringerLink
Log in

Thermal effects in the capillary rheometer

  • Published:
Rheologica Acta Aims and scope Submit manuscript

Summary

The effect of viscous heating in a capillary rheometer is analysed for a power-law fluid by means of a perturbation expansion based upon a boundary-layer-core structure. This expansion is found to complement the eigenfunction series solution obtained by earlier investigators. A similar analysis is presented for the “work-of-expansion” effect. These two thermal effects are superimposed together with a third perturbation effect due to the pressure dependence of viscosity.

On the basis of the present theory, earlier work in this area is discussed and, in some cases, apparent inaccuracies or inconsistencies are pointed out. A means is indicated for correcting data on the basis of the present theory.

Zusammenfassung

Es wird der Effekt der Erwärmung einer Potenzflüssigkeit infolge viskoser Reibung in einem Kapillar-Rheometer mittels einer Störungsrechnung untersucht, die auf der Unterteilung der Strömung in eine Grenzschicht und einen Kern basiert. Diese Störungsentwicklung ergänzt eine früher von anderen Autoren gefundene Reihenentwicklung mit Hilfe von Eigenfunktionen. Eine ähnliche Untersuchung wird für die thermische Ausdehnungsarbeit durchgeführt. Diese beiden thermischen Effekte sind zusammen einem dritten Störeffekt superponiert, der von der Druckabhängigkeit der Viskosität herrührt.

Aufgrund der vorgelegten Theorie werden verschiedene auf diesem Gebiet früher durchgeführte Arbeiten diskutiert, und es werden in einigen Fällen offensichtliche Ungenauigkeiten und Folgewidrigkeiten aufgedeckt. Schließlich wird eine Methode zur Korrektur von Meßdaten mit Hilfe der vorliegenden Theorie angegeben.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

a :

tube radius

b :

\(\frac{1}{\eta }\frac{{\partial \eta }}{{\partial p}}\); evaluated atT 0 andp = 0 when used in perturbation expansion

C p :

specific heat

f :

\( - \bar \Lambda _1 /(\varepsilon \Lambda _0 )\)

f * :

\(\bar \Lambda _1^ * /(\varepsilon \Lambda _0 )\)

h :

defined by eq. [15]

k :

thermal conductivity

L :

tube length

m :

defined by eq. [8]

m 0 :

m(T0, 0)

n :

power-law index

p :

pressure

Pe:

ρ C p W a/k ⋯ Peclet number

Pr:

C pηa/k ⋯ Prandtl number

Q :

volumetric flow rate

Q 0 :

unperturbed value ofQ in specified-Δp formulation

r :

radial coordinate

Re:

ρ W a/η a ⋯ Reynolds number

T :

temperature

T 0 :

inlet temperature

u :

radial velocity component

u 0 :

0 ⋯ unperturbed radial velocity

w :

axial velocity component

w 0 :

α/γW(1 − ζ γ) ⋯ unperturbed axial velocity

W :

Q/(πa 2) ⋯ average axial velocity

W 0 :

Q 0/(πa 2)

z :

axial coordinate

α :

(3n + 1)/n

α * :

\( - \frac{1}{\eta }\frac{{\partial \eta }}{{\partial T}}\); evaluated atT 0 andp = 0 when used in perturbation expansion

β :

41-n

β * :

\( - \frac{1}{\rho }\frac{{\partial \rho }}{{\partial T}}\)

γ :

(n + 1)/n

\(\dot \gamma \) :

\(\left| {\frac{{\partial w}}{{\partial r}}} \right|\) ... shear rate

\(\dot \gamma _a \) :

4W/a ⋯ apparent shear rate

Δp :

total pressure drop

ΔT :

η a W 2/k ⋯ characteristic temperature difference

ΔT b :

total bulk-temperature rise

ε :

α * ΔT

ζ :

r/a

η :

shear viscosity

η a :

m0 \(\dot \gamma _a^{n - 1} \)

κ :

(1 −ζ)/ζ 1/3

Λ :

—∂p/∂z

Λ 0 :

\(2\alpha ^n \frac{{m_0 }}{a}\left( {\frac{W}{a}} \right)^n \) ... unperturbed value ofΛ

\(\bar \Lambda \) :

z-averaged value ofΛ

µ :

β α n + 1/n

ξ :

z/(a Pe)

ξ L :

L/(a Pe)

ρ :

mass density

τ w :

shear stress at wall

ψ :

streamfunction

ω :

β *T0 (absolute temperature scale)

( )1 :

leading-order effect due to viscous heating

( ) *1 :

leading-order effect due to work-of-expansion

References

  1. Galili, N., R. Takserman-Krozer, Z. Rigbi, Rheol. Acta14, 550 and 816 (1975).

    Google Scholar 

  2. Brinkman, H. C., Appl. Sci. Res.A 2, 120 (1951).

    Google Scholar 

  3. Bird, R. B., S.P.E.J.11, 35 (1955).

    Google Scholar 

  4. Toor, H. L., Trans. Soc. Rheol.1, 177 (1957).

    Google Scholar 

  5. Gerrard, J. E., F. E. Steidler, J. K. Appeldoorn, I.E.C. Fund.4, 332 (1965) and5, 260 (1966).

    Google Scholar 

  6. Choi, S. Y., N. Nakajima, Fifth Int. Congr. Rheol.4, 287 (1970).

    Google Scholar 

  7. Cox, H. W., C. W. Macosko, S.P.E. Antec32, 28 (1974).

    Google Scholar 

  8. Duvdevani, I. J., I. Klein, S.P.E.J.23 (12), 41 (1967).

    Google Scholar 

  9. Kamal, M. R., H. Nyun, Trans. Soc. Rheol.17, 271 (1973).

    Google Scholar 

  10. Nyun, H., Ph. D. Thesis, McGill University (1973).

  11. Cox, H. W., C. W. Macosko Amer. Ind. Chem. Engg. J.20, 785 (1974).

    Google Scholar 

  12. Semjonow, V. Rheol. Acta4, 133 (1965).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sibley School of Mechanical and Aerospace Engineering, Cornell University, 14853, Ithaca, New York, USA

    C. A. Hieber

Authors
  1. C. A. Hieber
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Note: in specified-Δp formulation,W gets replaced byW 0 in definition of Pe, Re,\(\dot \gamma _a \) andξ.

With 7 figures and 7 tables

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hieber, C.A. Thermal effects in the capillary rheometer. Rheol Acta 16, 553–567 (1977). https://doi.org/10.1007/BF01525656

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01525656

Keywords

Search

Navigation