Gravity draining of a yield-stress fluid through an orifice
Introduction
The drainage or extrusion of pasty materials is widely applied in industry, for example, in the shaping of ceramics, foodstuffs, cosmetics, and other products. In these processes the material is expelled from a large cylinder (the barrel) through a small orifice, under its own weight in the case of drainage, or by an externally applied pressure in the case of extrusion. The control and optimization of this process requires a good knowledge of the flow characteristics of these highly non-Newtonian materials. Despite its wide applicability, however, the state of knowledge in this field is still rather basic. In this paper we focus on the gravity-driven draining of a yield-stress fluid through an orifice at the bottom of a container. This particular type of flow may be encountered in civil engineering when, for example, a volume of concrete, sewage sludge or batter, is emptied from the bottom of a storage hopper. This problem is also related to the flow induced in the so-called “L-Box” test for concrete and the Bostwick consistometer for foodstuffs or paints, in which a material initially stored in a vertical vessel flows through a gate at the bottom of the vessel. We also expect this work to provide some insight into the extrusion of yield-stress fluids in general.
We note that the word “paste” is used in different fields to describe materials which keep their shape under gravity in the absence of surface tension effects. The expression “yield-stress fluid” is used in rheology to describe a material which exhibits a yield stress but flows as a liquid beyond this yield stress. With these definitions, pastes are similar to yield-stress fluids. Since the stresses induced by gravity increase with sample volume, the yield stress simply defines a critical volume below which the material in a given shape will be unaffected by gravity and beyond which it will be affected. For example, a modelling paste generally keeps its shape under gravity for the small volumes typically used in applications, but would start to flow for much larger volumes. In the following we will use both terms to describe the same systems.
The flow of a fluid through a straight conduit (labelled 1) into another conduit of smaller cross-section (labelled 2) involves some additional viscous dissipation beyond that resulting from the flow in the straight conduits alone. This may be expressed in terms of an additional pressure drop , so that the total pressure drop can be written asHere and are the pressure drops in the two conduits alone. From the momentum balance in a fluid flowing through a cylindrical conduit of radius and length we getin which is the wall stress in conduit i far from the ends. In some previous work (Boger, 1977) a distinction is drawn between “entrance” and “exit” losses, but here we treat the extra dissipation as a single effect due to the change of cross-section which affects the flow field both before and after the orifice.
For a Newtonian fluid of viscosity flowing in steady state in a straight cylindrical conduit, integration of the momentum equation gives the wall stress aswhere is the mean flow velocity. In this case, calculations (Sampson, 1891, Hasegawa et al., 1997) for two asymptotic geometries—flow through a conduit of constant radius containing an orifice of negligible thickness, and flow through a long conduit with a stepwise change in radius—led to the expression is referred to as the entrance correction, and was calculated to be 0.589, while experimental values of range from 0.589 to 1.08 (Boger, 1977). By comparing this result with Eq. (2) it follows that the viscous dissipation due to the change of cross-section is of the same order as the viscous dissipation in a tube of radius R and length L.
The behaviour of yield-stress fluids is described quite well by the Herschel–Bulkley model (Adams et al., 1994, Adams et al., 1997a, Coussot, 2005): andHere is the shear rate and K and n are material parameters. Let us consider the steady flow of a yield-stress fluid through a tube of length L and radius R. Integration of the expression for the shear rate determined from (2), (5) allows one to derive an expression for the flow rate as a function of pressure drop (Bird et al., 1982). In dimensionless form, this may be written aswhere , , and . It is worth noting no flow occurs if the pressure drop is below a critical value equal to . Note that this equation assumes no wall slip, an effect which frequently occurs with such pasty materials (Yilmazer and Kalyon, 1989).
Similarly in the case of extrusion of a paste, the existence of the yield stress implies that there should exist a certain critical pressure drop below which no flow is possible. Because the behaviour of such fluids is strongly nonlinear, it is very difficult to find analytical solutions for flows in complex geometries. The entrance correction for the flow of a Bingham fluid (for which the exponent in Eq. (5)) through an orifice has been determined from numerical simulations to be (Abdali et al., 1992) , where , is the radius of the upstream cylinder and the average velocity through the orifice. Benbow and Bridgwater (1993) modeled extrusion flow of a yield-stress fluid as a homogeneous compression between parallel plates. With this analogy they estimated the minimum die entry pressure drop to bein which is the uniaxial yield stress (equal to , according to Adams et al., 1997b) and and A are the upstream and downstream cross-sectional areas, respectively. For cylindrical conduits one gets . Refined expressions involving conical entry angles were determined by Horrobin and Nedderman (1998) from large-deformation elastic–plastic finite element calculations. Benbow and Bridgwater (1993) modified the expression (7) by adding a term depending on to account for the role of velocity. Alternatively, Basterfield et al. (2005) assumed that the flow through the orifice is equivalent to the flow in a conical duct with an angle around 45° and for cylindrical conduits found to be given bywith . This reduces to Eq. (7) as approaches zero. Efforts to verify the above formulae experimentally have mainly been concerned with pastes having a very high yield stress (Basterfield et al., 2005), for which it was difficult to determine the rheological parameters separately.
Here we focus on the gravity-driven flow of simple yield-stress fluids through an orifice. There are two fundamental differences between this flow and pressure-driven extrusion through an orifice: first, the pressure increases with depth from the fluid surface to the orifice at the bottom of the container as a result of the weight of the fluid above, and second, the flow just after the orifice is unconstrained so that, in particular, the pressure there is equal to the ambient pressure. In contrast, in conventional extrusion or flow through an orifice the pressure is imposed upstream, far from the orifice, and decreases downstream. However, as discussed below, it is likely that some characteristics observed here can be extrapolated to conventional extrusion. We first establish a baseline by studying gravity-driven drainage with a viscous Newtonian fluid, then focus on the flow of a yield-stress fluid whose rheological properties have been well characterised by independent rheometric measurements.
Section snippets
Experimental technique and materials
The experimental set up consists of a vertical Plexiglas vessel with a square cross-section of side length . The bottom plate of the vessel has a thickness and a hole of radius R in its centre. For some experiments a tube of length l and inner radius R was used instead of a simple hole. Initially the hole is blocked with a piece of tape. The vessel is filled with the experimental fluid, after which the hole is opened so that the fluid can flow out. We monitored the height h as a
Drainage of a Newtonian fluid
Fig. 2 shows a semi-logarithmic plot of the surface height h as a function of time t for the drainage of the Newtonian glycerol solutions. The height decreases exponentially with time for a range of different hole diameters, vessel sizes and fluid viscosities, deviating from this behaviour at longer times as h approaches zero. The exponential decrease implies that . Since in general the potential energy is proportional to h while viscous effects are proportional to , this result suggests
Conclusion
Our results provide an expression for the critical pressure drop associated with the stoppage of flow in the drainage, and by extension in the extrusion, of yield-stress fluids. We find where when the downstream length l is small (on the order of and when l is much larger than In developing these expressions from our data, we have assumed that the viscous dissipation in our flows was analogous to that for shear flow through a tube. The application of these
Notation
upstream and downstream cross-sectional areas Bingham parameter: ratio of the yield stress to the viscous stress e thickness of the vessel bottom plate g acceleration due to gravity h fluid height in the vessel I dimensionless pressure parameter parameters of the Herschel–Bulkley model l length of the additional downstream tube L conduit length length of the tube leading to a pressure drop entrance correction p pressure ambient pressure pressure drop pressure drop due to a change in
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