POWDER FLOW IN ROTATIONAL MOLDING

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2 POWDER FLOW IN ROTATIONAL MOLDING R. Pantani, M. d' Amore, G. Titomanlio Department of Chemical and Food Engineering, University of Salerno, Fisciano (SA), Italy R. Mauri, Department of Chemical Engineering, The City College of CUNY, New York, NY INTRODUCTION Rotational molding is a widely used technology to manufacture hollow industrial plastic products [1]. In this process, a thermoplastic powder is rotated in an oven around two perpendicular axes, so that the powder eventually melts and adheres to the internal surface of the mold. In order to improve the quality of the product, a better understanding of the heat and mass transport during the process is required. For a preliminary analysis of the system, it is advisable to study the simplest possible mold shape. The first step in modeling the process is to study the movement of the powder in the system. In this work, a series of experimental observations was performed using a rotating Pyrex cylinder partially filled with a powder of known characteristics. Images taken with a camera were computer analyzed by means of an image acquisition system. A simple phenomenological model was then developed, based on the assumption that the powder behaves as a power-law fluid, obtaining theoretical predictions which describe main experimental features EXPERIMENTAL A cylinder (0. 14m in diameter, DAm long) having Pyrex surfaces was partially filled with 150mL of sand whose physical characteristics are known [2]. This sand is a type B powder in the Geldartis classification so that short-range interparticle forces are small and the study of the powder dynamics is easier. A constant cylinder rotational velocity was ensured by a computer controlled 500 W electrical engine. The velocity could be varied from 5 to 100 rpm. Images were taken by both a photocamera for snapshooting the flow lines of the powder (exposing time 1/30 s, stop 4), and a digital high resolution videocamera for powder motion processing. This latter was computer-aided so as to have possibility of measuring the velocity of single particles. At very low rotation speed, the material was raised beyond its angle of repose, subsequently collapsing to a lower angle (slumping bed). Then, increasing the speed up to 30rpm, the surface became substantially steady (Fig. I - rolling bed) and the measured angle of elevation of this surface was close to the angle of repose of the powder. Finally, at higher speeds (still lower than the critical speed at which centrifugal forces predominates on gravity forces) the free surface assumed an elongated S-shape (cascading bed). The measured angle at the steepest part of this surface resulted to be close to the angle of approach. As shown in fig. I, three zones can be clearly distinguished: a top zone of rapidly flowing powder; a middle zone in which the material is substantially steady; a bed of material rotating with the cylinder.. The separation line between the zone in which the material is going up and the zone in which it is going down (zero-velocity line) is located close to halfway between cylinder and free surface, closer to the latter. In all the conditions examined, zero-velocity line was always at about 60% of the powder depth, and its position was found substantially independent on cylinder rotation speed. Powder velocity at the free surface (downwards) increases with cylinder rotation speed. 141

3 Figure 1: r..xperimental ohservation of Ihe powder profile d1lring rolation in the cylinder. Material: Sand x: w=30rpm. R=7cm, L=.fOcm. Volume o/powders=150cm J THEORETICAL ANALYSIS Since our goal here is to describe powder kinetics main features, we assume that the powder behave like a fluids 6.7 Therefore, assuming that the motion of the powder takes place within a thin layer of thickness d«r, with d/dr»d/d(r8), at leading order the equation of motion can be written in terms of the non-dimensional distance s from the wall, as: r = R( I - s ~), ( I ) dr -d = -pdgsin B., s where g is the gravity field and r the shear stress. In this approximation we have assumed that the width L of the cylinder is large, so that edge effects can be neglected and the motion of the powder, at leading order, can be assumed to be one-dimensional, along the circumpherential direction. Equation (2) can be easily integrated, since r =0 at.1'= 1, so that we obtain: r=pdgsinb(l-s) (3) We assume here that shear stress is given by the power-law expression, r = 3-.1 dvl n. dv = 3-. dv n,i (4) d n + 1 ds ds d n + 1 d~ where v is the velocity in the circumpherential direction, while 11 and n are constant power-law parameters. At this point, we have to impose the boundary condition at the wall. Now, in general, powders tend to slip, depending on the coarseness of both powder and wall, and on the local pressure field. However, for lack of experimental data, here we have assumed the following slip function v(s = 0) = Vn = a( O)(tl R wlrere rlifo'<1f/ (5) a(o) = ~ lcos(o)/ COS(If/) elsewlrere where \If (no-slip angle) was measured to be about 20 Equations 4 and 5 lead to the following expression for velocity profile (2) 142

4 (6) I where A =(P(J""gSin(O»);;;_I_"+1 (7) " awr 11+2 (if the fluid is Newtonian - i.e. n=o - A" represents the ratio between the free-fall velocity of the powder layer and its imposed velocity at the walls - In steady-state conditions, the integral of velocity over the thickness is zero at each angle, I.e. I ' r ( 11+ ljl Q=Ld "ds=ldrwll-a. (8) 1--- J=O~A.= By substituting eq. 7 in 8 one obtains the following expression for powder profile (d) ((2 3).. 1 )y.., d =..!!..!..- (aw R)" "_ (9) I pgsin(o) The position within the powder layer where the powder velocity is zero can be found from eq.6 1 )::: s'= 1- ( 1- A.. For s<s*, velocity is positive (as cylinder wall) and vice versa for s>s* Since s* IS experimentally measured to be higher than 0.5, n must be negative. Total powder volume (V) is given by "= r"d(r-!i.)do (II) Jo 2 In order to compare the predictions of this model with our experimental results, we have chosen the following values of the physical parameters: L=40cm R=7cm co=30rpm TJ/p=4000cm 2 /sec (12) V=150cm' n=-2.38 dm=lcm where TJ/p and n were fitted comparing theoretical predictions of powder profile (eq. 9) with theexperimental observation that bed maximum depth (dm) is about Icm. When n=-2.38 the zero-velocity curve located at s*=0.59 (fig.2a), consistently with experimental observations. Substituting the values (12) of the physical parameters into Eq. (9), we find the thickness of the powder layer, plotted in fig. 2b. (10) o ' ' Figure 2: a) Velocity profile at a fixed angle (8=55 ). b) 771ickness oj the powder layer d=d(b). CONCLllSIONS We have studied the motion of a powder with known physical characteristics (such as angle of repose and angle of approach), as it partially fills a rotating cylinder. A series of 143

5 experimental observations was performed, showing that three zones can be clearly distinguished: a top zone of rapidly flowing powder, a middle zone in which the material is substantially stationary, and a bed of material rotating with the cylinder. A very simple phenomenological model was then developed, based on the assumption that the powder behaves as a power-law fluid, obtaining theoretical predictions which describe main experimental data features. References 1. Titomanlio, 1988, Stampaggio Rotazionale, CLUP. 2. Brown and Richards, 1966, Principles of Powder Mechanics, Pergamon Press, New York 3. Rietema K., 1991, The Dynamics of Fine Powders, Elsevier, London 4. Jaeger, H.M., S.R. Nagel and R.P. Behringer, 1996, Granular solids, liquids and gases, Rev. Mod. Phys. 68,