Page 231 - Process Modelling and Simulation With Finite Element Methods

218 Process Modelling and Simulation with Finite Element Methods

vp = pv2u+ pg
v-u=o

where p is the fluid viscosity and p its density, and all other symbols have their
usual fluid flow interpretation. Equations (6.2) are dimensional, pseudo-
stationary, and inertia-free. As they are also linear, they have been the subject of
exhaustive analysis. Ockendon and Ockendon [3] is a good reference for the
area. Homsy et al. [4] provides several excellent visualizations of the
“pathologies” of viscous flow with vanishingly small Reynolds number. My
attention to the problem of an orifice plate was drawn by Professor Dugdale [5],
who arrived at the solution to (6.2) in the vicinity of a sharp-edged orifice by
requiring the condition of optimum energy dissipation within the orifice itself,
ignoring the dissipation on all other boundaries of the vessel. His argument is
that since the orifice is so small, and all of the flow is forced through the orifice,
nearly all of the energy must be dissipated through it, gives a dimensional
argument that for a three-dimensional orifice with characteristic opening length
a, the energy dissipation rate E must satisfy

PQ2
E=c- =W=QAp
a3
where Q is the volumetric flow rate and W is the rate of working. c is an
unknown constant of proportionality that Dugdale calculates theoretically on the
basis of the extremum of the energy absorption or can be found experimentally
by measuring Q and pressure loss. In a two dimensional system, the analogous
dimensional argument makes E’ the dissipation loss per unit length and Q’ the
cross-sectional area flow rate, giving rise to the scaling argument

(6.4)

Dugdale reports experiments with molasses determining c in the range of 3.17 to
3.30. His theoretical result was 3.0. Bond [6] gives an argument of the
similarity of orifice plates to Hagen-Poiseuille pipeflow in a pipe of length 2ka,
where a is the orifice radius, and his pressure drop equated to k=O.631, implying
c=3.21.
One of us has been interested for some time in the drag on close fitting
particles in tubes. For the same rationale leading to (6.3) or (6.4), close fitting
particles in tubes have drag controlled by the gap width. Zimmerman for thin
discs [7] (broadside motion) and for spheres [S] sedimenting in cylindrical tubes,
reports on the rapid growth of drag as the particle is taken as having larger radius
(smaller gap width a). By using perturbation methods in small particle radius (1-
a) and summing the series expansion, it is possible to determine the nature of the