The
rotameter is a very precise mass flow sensor. As shown
in the Figure, a rotameter consists of a ball or float
in a tapered tube. When the ball in the tube is fixed
at a certain height, the forces on the ball are in equilibrium.
There are three forces: the weight of the ball, the buoyancy
of the ball, and the force that the moving fluid places
on the ball. The third of the forces is called drag and,
as described in fluid mechanics texts, scales with the
square of the fluid velocity.
Since
the tube is tapered, more flow area is available as the
ball moves higher, and the local velocity of the fluid in
the tube is a function of the ball’s position in the
tube. The drag force on the ball is therefore also a function
of the ball’s position in the tube. At different flow
rates, the equilibrium between drag and gravity will produce
different positions for the ball. The float moves in the
tube until the upward force on the float due to the flow
is exactly balanced by the relative weight of the float.
The following equation provides an exact relationship for
the drag on the ball as a function of the flow velocity,
the coefficient of drag, the cross sectional area of the
ball, and the density of the fluid.
The
following equation provides an exact relationship
for the drag on the ball as a function of the flow
velocity, v, the coefficient
of drag, , Cd,
the cross sectional area of the ball, Ab,
and the density of the fluid, ρf
.
The weight of the ball is ρbgVb
and the buoyancy force on the ball is ρfgVb,
where g is the acceleration due to gravity and Vb
is the volume of the ball. A simple force balance
on the ball produces an expression for the velocity
of the flow, v. For a gas,
ρf < ρb.
The rotameter readings are taken at the center of
the ball.
The
rotameter is a very precise mass flow sensor. As shown
in the Figure, a rotameter consists of a ball or float
in a tapered tube. When the ball in the tube is fixed
at a certain height, the forces on the ball are in equilibrium.
