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Non-isothermal flow of polymer melts in a curved tube

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Abstract

The results of a numerical study (using finite differences) of heat transfer in polymer melt flow is presented. The rheological behaviour of the melt is described by a temperature-dependent power-law model. The curved tube wall is assumed to be at constant temperature. Convective and viscous dissipation terms are included in the energy equation. Velocity, temperature and viscosity profiles, Nusselt numbers, bulk temperatures, etc. are presented for a variety of flow conditions.

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Abbreviations

Br :

\(\eta _0 \bar \upsilon ^2 /\lambda t_0 \) — Brinkman number

c :

specific heat, J/kg K

De :

\(2r_p \bar \upsilon \varrho /(\eta _0 \exp ( - kt_0 )(r_p /r_c )^{1/2} )\) — Dean number

E :

dimensionless apparent viscosity, eq. (14d)

G :

dimensionless shear rate, eq. (19)

k :

parameter of the power-law model, °C−1, eq. (7)

\(\dot m\) :

mass flow rate, kg/s

m 0 :

parameter of the power-law model, Pa · sn, eq. (7)

n :

parameter of the power-law model, eq. (7)

Nu :

2r pα/λ — Nusselt number, eqs. (28,31)

p :

pressure, Pa

Pe :

\(\bar \upsilon r_p \varrho c/\lambda \) — Péclet number

P :

—(∂p/∂ϕ)/r c — pressure gradient, Pa/m

\(\dot Q_d \) :

dissipated energy, W, eq. (29)

\(\dot Q_t \) :

total energy, W, eq. (30)

r :

radial coordinate, m

r c :

radius of tube-curvature, m, fig. 1

r p :

radius of tube, m, fig. 1

r t :

variable, m, eq. (6)

R :

dimensionless radial coordinate, eq. (14a)

R c :

dimensionlessr c, eq. (14a)

R t :

dimensionlessr t, eq. (14a)

t :

temperature, °C

\(\bar t\) :

bulk temperature, °C, eq. (27)

t 0 :

inlet temperature of the melt, °C

t w :

tube wall temperature, °C

T :

dimensionless temperature, eq. (14c)

T w :

dimensionless tube wall temperature

T :

dimensionless bulk temperature

u 1 :

variable, s−1, eq. (4)

u 2 :

variable, s−1, eq. (5)

U 1 :

dimensionlessu 1, eq. (18)

U 2 :

dimensionlessu 2, eq. (18)

v :

velocity inϕ-direction, m/s

\(\bar \upsilon \) :

average velocity of the melt, m/s

V :

dimensionlessv, eq. (14b)

\(\bar V\) :

dimensionless\(\bar \upsilon \), eq. (15)

z :

r cϕ — centre length of the tube, m

Z :

dimensionlessz, eq. (14e)

α :

heat transfer coefficient, W/m2 K

\(\dot \gamma \) :

shear rate, s−1, eq. (8)

\(\dot \gamma _0 \) :

\(\bar \upsilon /r_p \) — shear rate, s−1

η :

apparent viscosity, Pa · s, eq. (7)

η 0 :

\(m_0 \dot \gamma _0^{n - 1} \) — apparent viscosity, Pa · s

θ :

angular coordinate, rad, fig. 1

λ :

thermal conductivity, W/m K

ϱ :

melt density, kg/m3

ϕ :

axial coordinate, rad, fig. 1

Δ :

rate of strain tensor, s−1, eq. (8)

(—Δp):

pressure drop, Pa

References

  1. Baurmeister U, Brauer H (1979) VDI-Forschungshefte Nr. 593, VDI-Verlag GmbH, Düsseldorf

  2. Jones RJ (1968) ZAMP 19:301–315

    Google Scholar 

  3. Chang KC, Akiyama M (1970) Int J Heat Mass Transfer 13:471–490

    Google Scholar 

  4. Rieger F, Šesták J (1973) Appl Sci Res 28:89–106

    Google Scholar 

  5. Joseph S, Smith EP, Adler RJ (1975) AIChE J 21:965–979

    Google Scholar 

  6. Kamel MT, Kaloni PN (1977) ZAMP 28:551–576

    Google Scholar 

  7. Rushmore WL, Taulbee DB (1978) Computers and Fluids 6:125–140

    Google Scholar 

  8. Masliyah JH, Nandakumar K (1979) AIChE J 25:478–487

    Google Scholar 

  9. Austin LR, Seader JD (1973) AIChE J 19:85–94

    Google Scholar 

  10. Kalb CE, Seader JD (1974) AIChE J 20:340–346

    Google Scholar 

  11. Zapryanov Z, Christov Ch, Toshev E (1980) Int J Heat Mass Transfer 23:873–880

    Google Scholar 

  12. Akita O, Ito K (1978) Polym Eng Sci 18:951–954

    Google Scholar 

  13. Hsu Chia-Fu, Patankar SV (1982) AIChE J 28:610–616

    Google Scholar 

  14. Carnahan B, Luther HA, Wilkes JO (1969) Applied Numerical Methods, John Wiley, New York

    Google Scholar 

  15. Závadský E, Karniš J (1983) Technical Report No. 47/83, Research Institute for Man-Made Fibres, Svit

  16. Agur EE (1978) PhD Thesis, McMaster University, Hamilton, Canada

  17. Agur EE, Vlachopoulos J (1981) J Appl Polym Sci 26:765–773

    Google Scholar 

  18. Závadský E, Karniš J, Pechoč V (1982) Rheol Acta 21:470–474

    Google Scholar 

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Authors and Affiliations

  1. Ch. des Dents-du-Midi 28, CH-1860, Aigle

    E. Závadský

  2. Research Institute for Man-made Fibres, CS-05921, Svit, Czechoslovakia

    J. Karniš & V. Pechoč

Authors
  1. E. Závadský
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  2. J. Karniš
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  3. V. Pechoč
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Závadský, E., Karniš, J. & Pechoč, V. Non-isothermal flow of polymer melts in a curved tube. Rheol Acta 24, 335–340 (1985). https://doi.org/10.1007/BF01333962

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  • DOI: https://doi.org/10.1007/BF01333962

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