Question 9 of 9: Rheology Fundamentals — Shear Stress, Viscosity, Temperature Effects, and Flow Behaviour Classification

Question 9: Rheology Fundamentals — Shear Stress, Viscosity, Temperature Effects, and Flow Behaviour Classification (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Rheology, its importance, and shear stress. Rheology is the study of how matter deforms and flows under applied stress — for foods, specifically the relationship between the shear stress applied to a fluid or semi-solid and the resulting shear rate (or, for a solid-like food, between stress and strain). It matters in process design because pumps, pipelines, mixers, extruders, coating and filling equipment must all be sized against the actual stress–shear-rate relationship of the product being handled — a pump or pipe sized using a single "viscosity" number measured at the wrong shear rate will be badly under- or over-sized for a shear-thinning or shear-thickening food, and rheological data is equally the basis for relating an instrumental texture measurement back to a consumer's sensory perception of thickness or firmness. Shear stress \(\tau\) is the force per unit area acting TANGENTIALLY (parallel to the surface) across a fluid layer, e.g. the force per unit area a moving plate exerts on the fluid layer beneath it, distinct from a normal (compressive) stress acting perpendicular to a surface.

(b) What viscosity measures. Viscosity measures a fluid's internal resistance to shear flow — physically, the rate at which momentum diffuses between adjacent fluid layers moving at different velocities, which macroscopically appears as the ratio of shear stress to shear rate, \(\mu=\tau/\dot\gamma\) for a Newtonian fluid. It characterizes how much force is needed to sustain a given rate of shearing deformation (equivalently, how "thick" the fluid feels in a given flow), and for a non-Newtonian food that ratio is itself a function of shear rate, which is why a single viscosity value is meaningful only at the shear rate it was measured at.

(c) Effect of temperature on viscosity. Viscosity falls as temperature rises, for essentially every food liquid, because increased thermal (molecular) energy weakens the intermolecular attractive forces and increases the free volume available for molecules to slide past one another, lowering the internal resistance to flow. The dependence is usually described by an Arrhenius-type relation, $$\mu = \mu_0\, e^{E_a/RT},$$ where \(E_a\) is an activation energy for flow; a higher \(E_a\) means the food's viscosity is MORE sensitive to temperature, which is why process design (pumping, heat exchanger pressure drop) must use the viscosity at the ACTUAL process temperature, not a value measured at room temperature.

(d) Bingham plastic, pseudoplastic and dilatant foods. A Bingham plastic food behaves as a rigid solid until an applied shear stress exceeds a finite yield stress \(\tau_0\), after which it flows with a constant (Newtonian) viscosity: \(\tau=\tau_0+\mu_p\dot\gamma\) for \(\tau>\tau_0\), and no flow at all below \(\tau_0\) (e.g. tomato paste, some chocolate systems, which hold their shape under gravity but flow once squeezed or pumped hard enough). A pseudoplastic (shear-thinning) food's apparent viscosity DECREASES as shear rate increases, following the power law \(\tau=K\dot\gamma^{\,n}\) with \(n<1\) (the tomato catsup of Question 3 is a pseudoplastic example) — physically, increasing shear progressively aligns or disentangles elongated molecules/particles, lowering resistance to further shear. A dilatant (shear-thickening) food's apparent viscosity INCREASES with shear rate, the same power law but with \(n>1\) (some concentrated starch suspensions/uncooked cornstarch-in-water systems) — physically, at high shear the densely packed particles jam against one another faster than they can rearrange to accommodate the flow, raising resistance.

(e) Temperature's effect on \(n\) and \(K\). The consistency coefficient \(K\) falls sharply with rising temperature, following essentially the same Arrhenius-type temperature dependence as a Newtonian viscosity (since \(K\) plays the same "overall thickness" role in the power law that \(\mu\) plays for a Newtonian fluid), so a hot food is consistently "thinner" at every shear rate than the same food cold. The flow behaviour index \(n\), by contrast, is comparatively INSENSITIVE to temperature over the normal processing range, because \(n\) reflects the STRUCTURE of the food (the degree of shear-thinning/thickening behaviour arising from particle shape, polymer entanglement, or aggregate structure) rather than the overall energy barrier to flow, and that structural characteristic does not change strongly with a moderate temperature shift — \(n\) instead shifts mainly when the food's underlying STRUCTURE changes (e.g. starch gelatinization, protein denaturation, or a genuine formulation change), not from temperature alone.