Page 259 - Gas Adsorption Equilibria

5. Oscillometry 245

The ratios of moments of inertia in the numerator and the denominator on the
r.h.s. of this equation need to be related to measurable quantities of the motion
of the pendulum, i. e. the angular frequency and the logarithmic
decrement ofthe corresponding damped harmonic oscillations. To achieve
this we again have to consider the equation of motion of the pendulum (EOM)
(5.18). It includes on its r.h.s. the torque or damping moment exerted by the
fluid flow on the pendulum. Physically this is caused by the internal friction
of the fluid, i. e. in Newtonian fluids to which we are here restricted, by the
velocity gradient in the fluid’s flow perpendicular to the surface of the sorbent
loaded disk. Basically, the torque (M(t)) is an unknown quantity. However,
since the motion of the disk and the fluid flow initiated by it are strongly
coupled systems, in addition to (5.18) there should be a relation between the
amplitude of the pendulum and the torque (M(t)) rooted in the equations
of motion of the fluid and its boundary conditions, i. e. the shape and physical
properties of the disk. Assuming rotationally symmetric flow of the sorptive
fluid, M(t) can be represented as a surface integral

with the velocity field of the (instationary) fluid flow represented as

Here is the dynamic viscosity of the fluid and being unit vectors in r-
and z-direction of cylindrical coordinates with the wire of the pendulum as z-
axis and is the unit vector in the azimuthal direction. The quantities (O)
and (df) indicate the surface of the pendulum and a vectorial differential of it,
respectively.

Restricting ourselves to slow motions of the pendulum, i. e. small Reynolds
numbers