Flow Simulation of Herschel-Bulkley Fluids through Extrusion Dies E. MITSOULW and S. S. ABDALI Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario KIN 6N5 N. C. MARKATOS Computational Fluid Dynamics Unit, Department of Chemical Engineering, National Technical University of Athens, Zographou, Athens, Greece 15773 The flow of viscoplastic materials through extrusion dies has been studied numerically using the finite element method. Rheological data for viscoplastic doiighs have been fitted by the Herschel-Bulkley model, which incorporates a yield stress into the power-law model. Non-isothermal simulations show the extent and shape of yieldedlunyielded regions and the development of temperature field assuming different modes of heat transfer at the boundaries. The results reveal that viscous dissipation causes appreciable temperature rises in the extrudate in agreement with measured values at the extruded material surface. The extrudate swell results show a maximum for a certain range of apparent shear rates also observed experimentally. However, the inelastic simulations based on the Herschel-Bulkley model always under- predict the experimental swelling values. A heuristic approach is also used to determine the Ievel of elasticity required to produce the experimental values. On a CtudiC numtriquement 1’Ccoulement de matCriaux viscoplastiques dans des filDres d’extrusion par une mCthode d’CICments finis. Les donnCes rhCologiques des pltes viscoplastiques ont CtC calCes ii I’aide du modtle Herschel-Bulkley , qui introduit une contrainte de cisaillement dans le modMe de loi de puissance. Les simulations non isothermes mon- trent 1’Ctendue et la forme des regions cisaillies et non cisaillCes ainsi que le dCveloppement du champ de temp6rature en supposant differents modes de transfert de chaleur aux limites. Les rCsultats rtvtlent que la dissipation visqueuse entraine des ClCvations de tempkratures apprCciables dans I’extrudat en accord avec les valeurs mesurkes ii la surface des matCriaux extrudts. Les rksultats sur le gonflement de I’extrudat indiquent un maximum pour une certaine gamme de vitesses de cisaillement apparentes Cgalement observtes expkrimentalement . Cependant, les simulations non Clas- tiques s’appuyant sur le modMe Herschel-Bulkley s’avtrent toujours insuffisantes pour prkdire les valeurs de gonfle- ment expkrimentales. On utilise Cgalement une approche heuristique afin de determiner le degrC d’ClasticitC requis pour produire des valeurs exPCrimentales. Keywords: yield stress, Herschel-Bulkley model, non-Newtonian flow, non-isothermal flow, extrudate swell. n important class of non-Newtonian materials exhibits A a yield stress, which must be exceeded before signifi- cant deformation can occur. Such materials can sustain an applied stress at rest and include greases, slurries, paints, foodstuff and doughs. A list of several materials exhibiting yield was given in a seminar paper by Bird et al. (1983), who have also provided an initial analysis of such materials in simple flow fields. Since then a renewed interest has deve- loped among several researchers for the study of so-called viscoplastic materials (see recent review by Abdali et al., 1992). To model the stress-deformation behaviour, several con- stitutive relations have been proposed and different yield criteria have been used. Two of the most useful and popular models have been the Bingham model, which is a modifica- tion of the Newtonian model to incorporate a yield stress (Bingham, 1922) and the Herschel-Bulkley model, which is the equivalent modification of the power-law model (Her- schel and Bulkley, 1926). In simple shear flow these models take the form (see also Figure 1): Bingham model: 7 = 7,. + p+ for I4>7,. ................ (la) +=o for 1417y ................ (1b) *To whom correspondence should be addressed. Herschel-Bulkley model: ................ r = 7,. + Kj/” for I ~ > T , . (24 ................ +=O for 1417,. (2b) where r is the shear stress, + is the shear rate, 7,. is the yield stress, p is the Newtonian viscosity, K is the-consistency index and n is the power-law index. Note that when the shear stress 7 falls below ry a solid structure is formed (unyielded). Also when the power-law index is unity and the consistency index is equivalent to the viscosity, the Herschel- Bulkley model reduces to the Bingham model. Both models are viscoplastic but inelastic, i.e. they cannot account for viscoelastic phenomena such as stress relaxation and normal stresses exhibited in shear flows. Quite recently, Papanastasiou (1987) proposed a novel con- stitutive equation for materials with yield, where a material parameter controls the exponential growth of stress and which is valid for both yielded and unyielded areas. In simple shear flow, Papanastasiou’s modification to the Bingham model becomes: 7 = 7v [l - exp(-rnj/)] + pj/ ............... (3 where m is the stress growth exponent. It was shown by Papanastasiou (1987) and Ellwood et al. (1990) that this equa- tion closely mimics the ideal Bingham plastic (for m 2 100) and it provides a better approximation to real data of vis- coplastic materials (for m < 100). Papanastasiou’s model THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 71, FEBRUARY, 1993 147