Coriolis ﬂowmeters represent the state-of-the-art in mass ﬂow measurement. While incredibly versatile and accurate, their internal operation can be diﬃcult to understand. Put into very simple terms, a Coriolis ﬂowmeter works by shaking one or more tubes carrying the ﬂowing ﬂuid, then precisely measuring the frequency and phase of that shaking. The back-and-forth shaking is driven by an electromagnetic coil, powered by an electronic ampliﬁer circuit to shake the tube(s) at their mechanical resonant frequency. Since this frequency depends on the mass of each tube, and the mass of the tubes depends on the density of the ﬂuid ﬁlling the ﬁxed volume of the tubes, the resonant frequency becomes an inverse indication of ﬂuid density (Note 1) , whether or not ﬂuid is ﬂowing through the tubes.
As ﬂuid begins to move through the tubes, the inertia of the moving ﬂuid adds another dimension to the tubes’ motion: the tubes begin to undulate (Note 2) , twisting slightly instead of just shaking back and forth. This twisting motion is directly proportional to the mass ﬂow rate, and is internally measured by comparing the phase shift (θ) between motion at one point on the tube versus another point on the tube: the greater the undulation, the greater the phase shift between these two points’ vibrations.
Note 1: In fact, this density-measuring function of Coriolis ﬂowmeters is so precise that they often ﬁnd use primarily as density meters, and only secondarily as ﬂowmeters!
Note 2 : An interesting experiment to perform consists of holding a water hose in a U-shape and gently swinging the hose back and forth like a pendulum, then ﬂowing water through that same hose while you continue to swing it. The hose will begin to undulate, its twisting motion becoming visually apparent.
The Coriolis force
In physics, certain types of forces are classiﬁed as ﬁctitious or pseudoforces because they only appear to exist when viewed from an accelerating perspective (called a non-inertial reference frame). The feeling you get in your stomach when you accelerate either up or down in an elevator, or when riding a roller-coaster at an amusement park, feels like a force acting against your body when it is really nothing more than the reaction of your body’s inertia to being accelerated by the vehicle you are in. The real force is the force of the vehicle against your body, causing it to accelerate. What you perceive is merely a reaction to that force, and not the primary cause of your discomfort as it might appear to be.
Centrifugal force is another example of a “pseudoforce” because although it may appear to be a real force acting on any rotating object, it is in fact nothing more than an inertial reaction. Centrifugal force is a common experience to any child who has ever played on a “merry-go-round:” that perception of a force drawing you away from the center of rotation, toward the rim. The real force acting on any rotating object is toward the center of rotation (a centripetal force) which is necessary to make the object radially accelerate toward a center point rather than travel in a straight line as it normally would without any forces acting upon it. When viewed from the perspective of the spinning object, however, it would seem there is a force drawing the object away from the center (a centrifugal force).
Yet another example of a “pseudoforce” is the Coriolis force, more complicated than centrifugal force, arising from motion perpendicular to the axis of rotation in a non-inertial reference frame. The example of a merry-go-round works to illustrate Coriolis force as well: imagine sitting at the center of a spinning merry-go-round, holding a ball. If you gently toss the ball away from you and watch the trajectory of the ball, you will notice it curve rather than travel away in a straight line. In reality, the ball is traveling in a straight line (as viewed from an observer standing on the ground), but from your perspective on the merry-go-round, it appears to be deﬂected by an invisible force which we call the Coriolis force.
In order to generate a Coriolis force, we must have a mass moving at a velocity perpendicular to an axis of rotation:
The magnitude of this force is predicted by the following vector equation :
If we replace the ball with a ﬂuid moving through a tube, and we introduce a rotation vector by tilting that tube around a stationary axis (a fulcrum), a Coriolis force develops on the tube in such a way as to oppose the direction of rotation just like the Coriolis force opposed the direction of rotation of the rotating platform in the previous illustration:
To phrase this in anthropomorphic terms, the ﬂuid “ﬁghts” against this rotation because it “wants” to keep traveling in a straight line. For any given rotational velocity, the amount of “ﬁght” will be directly proportional to the product of ﬂuid velocity and ﬂuid mass. In other words, the magnitude of the Coriolis force will be in direct proportion to the ﬂuid’s mass ﬂow rate. This is the basis of a Coriolis mass ﬂowmeter.
A demonstration of this Coriolis force may be made by modifying the nozzles on a rotary lawn sprinkler so they point straight out from the center rather than angle in one direction. As water squirts through the now-straight nozzles, they no longer generate a rotational reaction force to spin the nozzle assembly, and so the nozzles remain in place (this much should be obvious). However, if someone were to try rotating the nozzle assembly by hand, they would discover the Coriolis force opposes the rotation, acting to keep the nozzle assembly from rotating. The greater the mass ﬂow rate of water through the nozzles, the stronger the inhibiting Coriolis force. Instead of a rotating lawn sprinkler, you are now the proud owner of an anti-rotating lawn sprinkler that actually ﬁghts any attempt to rotate it:
This is a very non-intuitive concept, so it deserves further explanation. The “anti-rotating” sprinkler doesn’t just fail to rotate on its own – it actually opposes any attempt to rotate from an external force (e.g. a person trying to push the tubes by hand).
This opposition would not occur if the tubes were merely capped oﬀ at the ends and ﬁlled with stagnant water. If this were the case, the tubes would simply be heavy with the water’s weight, and they would rotate freely about the axis just like any pair of heavy metal tubes would (whether hollow and ﬁlled with water, or solid metal). The tubes would have inertia, but they would not actively oppose any external eﬀort to rotate.
Having liquid water move through the tubes is what makes the diﬀerence, and the reason becomes clear once we imagine what each water molecule experiences as it ﬂows from the center (axis of rotation) to the nozzle at the tube tip. Each water molecule originating from the center begins with no lateral velocity, but must accelerate as it travels farther along the tube toward the circumference of the tips’ rotation where the lateral velocity is at a maximum. The fact that new water molecules are continually making this journey from center to tip means there will always be a new set of water molecules requiring acceleration from center velocity (zero) to tip velocity (maximum). In capped tubes ﬁlled with stagnant water, the acceleration would only occur in getting the tubes’ rotation up to speed – once there, the lateral velocity of each water molecule sitting stagnant inside the tubes would remain the same. However, with water ﬂowing from center to tip, this process of acceleration from zero velocity to tip velocity must occur over and over again (continually) for each new water molecule ﬂowing through. This continual acceleration of new mass is what generates the Coriolis force, and what actively opposes any external force trying to rotate the “anti-rotating” sprinkler.
As you might guess, it can be diﬃcult to engineer a tubing system capable of spinning in circles while carrying a ﬂowstream of pressurized ﬂuid. To bypass the practical diﬃculties of building a spinning tube system, Coriolis ﬂowmeters are instead built on the principle of a ﬂexible tube that oscillates back and forth, producing the same eﬀect in a cyclic rather than continuous fashion. The eﬀect is not unlike shaking a hose (Note) side to side as it carries a stream of water:
Note : The Coriolis force generated by a ﬂowing ﬁre hose as ﬁreﬁghters work to point it in a diﬀerent direction can be quite signiﬁcant, owing to the high mass ﬂow rate of the water as it ﬂows through the hose and out the nozzle!
The Coriolis force opposes the direction of rotation. The greater the mass ﬂow rate of water through the hose, the stronger the Coriolis force. If we had a way to precisely measure the Coriolis force imparted to the hose by the water stream, and to precisely wave the hose so its rotational velocity held constant for every wave, we could directly infer the water’s mass ﬂow rate.