Abstract
An analytical solution is obtained for the stationary temperature profile in a polymeric melt flowing into a cold cavity, which also takes into account viscous heating effects. The solution is valid for the injection stage of the molding process. Although the analytical solution is only possible after making several (at first sight) rather stringent assumptions, the calculated temperature field turns out to give a fair agreement with a numerical, more realistic approach. Approximate functions were derived for both the dissipation-independent and the dissipation-dependent parts which greatly facilitate the temperature calculations. In particular, a closed-form expression is derived for the position where the maximum temperature occurs and for the thickness of the solidified layer.
The expression for the temperature field is a special case of the solution of the diffusion equation with variable coefficients and a source term.
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Abbreviations
- a :
-
\(\frac{\lambda }{{\varrho c}}\) thermal diffusivity [m2/s]
- c :
-
specific heat [J/kg K]
- D :
-
channel half-height [m]
- L :
-
channel length [m]
- m :
-
1/ν
- P :
-
pressure [Pa]
- T :
-
temperature [°C]
- T W :
-
wall temperature [°C]
- T i :
-
injection temperature [°C]
- T A :
-
Br independent part of T
- T B :
-
Br dependent part of T
- T core :
-
asymptotic temperature
- v z(ξ):
-
axial velocity [m/s]
- W :
-
channel width [m]
- x :
-
cross-channel direction [m]
- z :
-
axial coordinate [m]
- Λ(x):
-
gamma function
- γ (a, x):
-
incomplete gamma function
- M(a, b, x):
-
Kummer function
- ε:
-
small parameter
- ϑ(χ):
-
temperature function
- λ:
-
thermal conductivity [W/mK]
- μ:
-
viscosity [Pa · s]
- μ0 :
-
consistency index
- η:
-
power-law exponent
- ϱ:
-
density [kg/m]
- χ:
-
similarity variable
- Br:
-
\(= \frac{{\mu _0 D}}{{\lambda (T_i - T_w )}}\left( {\frac{{(m + 2)\left\langle {\upsilon _z } \right\rangle }}{D}} \right)^{{{(m + 1)} \mathord{\left/{\vphantom {{(m + 1)} m}} \right.\kern-\nulldelimiterspace} m}}\) Brinkman number
- Gz:
-
\(= \frac{{\left\langle {\upsilon _z } \right\rangle D^2 }}{{aL}}\) Graetz number
- ′:
-
\(= \frac{{T - T_w }}{{T_i - T_w }}\)
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Jansen, K.M.B., van Dam, J. An analytical solution for the temperature profiles during injection molding, including dissipation effects. Rheola Acta 31, 592–602 (1992). https://doi.org/10.1007/BF00367013
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DOI: https://doi.org/10.1007/BF00367013