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# Mass flow meter

A mass flow meter, also known as inertial flow meter and coriolis flow meter, is a device that measures how much fluid is flowing through a tube. It does not measure the volume of the fluid passing through the tube; it measures the amount of mass flowing through the device.

Volumetric flow metering is proportional to mass flow rate only when the density of the fluid is constant. If the fluid has varying density, or contains bubbles, then the volume flow rate multiplied by the density is not an accurate measure of the mass flow rate.

In a mass flow meter the fluid is contained in a smooth tube, with no moving parts that would need to be cleaned and maintained, and that would impede the flow.

## Operating principle

There are two basic configurations: the curved tube flow meter and the straight tube flow meter. This article discusses the curved tube design.

The animations on the right do not represent an actually existing coriolis flow meter design. The purpose of the animations is to illustrate the operating principle, and to show the connection with rotation.

Fluid is being pumped through the mass flow meter. When there is mass flow, the tube twists slightly. The arm through which fluid flows away from the axis of rotation must exert a force on the fluid, to increase its angular momentum, so it bends backwards. The arm through which fluid is pushed back to the axis of rotation must exert a force on the fluid to decrease the fluid's angular momentum again, hence that arm will bend forward.

In other words, the inlet arm is lagging behind the overall rotation, and the outlet arm leads the overall rotation.

The animation on the right represents how curved tube mass flow meters are designed. When the fluid is flowing, it is led through two parallel tubes. An actuator (not shown) induces a vibration of the tubes. The two parallel tubes are counter-vibrating, to make the measuring device less sensitive to outside vibrations. The actual frequency of the vibration depends on the size of the mass flow meter, and ranges from 80 to 1000 vibrations per second. The amplitude of the vibration is too small to be seen, but it can be felt by touch.

When no fluid is flowing, the vibration of the two tubes is symmetrical, as shown in the animations.

The animation on the right represents what happens during mass flow. When there is mass flow, there is some twisting of the tubes. The arm through which fluid flows away from the axis of rotation must exert a force on the fluid to increase its angular momentum, so it is lagging behind the overall vibration. The arm through which fluid is pushed back towards the axis of rotation must exert a force on the fluid to decrease the fluid's angular momentum again, hence that arm leads the overall vibration.

The inlet arm and the outlet arm vibrate with the same frequency as the overall vibration, but when there is mass flow the two vibrations are out of sync, the inlet arm is behind, the outlet arm is ahead. The two vibrations are shifted in phase with respect to each other, and the degree of phase-shift is a measure for the amount of mass that is flowing through the tubes.

## Density and volume measurements

The mass flow of a u-shaped coriolis flow meter is given as: $Q_m=frac\left\{ K_u -I_uomega^2 \right\}\left\{2Kd^2\right\}tau$

where $K_u$ is the temperature dependent stiffness of the tube, $K$ a shape-dependent factor, $d$ the width, $tau$ the time lag, $omega$ the vibration frequency and $I_u$ the inertia of the tube. As the inertia of the tube depend on its contents, knowledge of the fluid density is needed for the calculation of an accurate mass flow rate.

If the density changes too often for manual calibration to be sufficient, the coriolis flow meter can be adapted to measure the density as well. The natural vibration frequency of the flow tubes depend on the combined mass of the tube and the fluid contained in it. By setting the tube in motion and measuring the natural frequency, the mass of the fluid contained in the tube can be deduced. Dividing the mass on the known volume of the tube gives us the density of the fluid.

Such an instantaneous density measurement in turn allow us to calculate the flow in volume per time, by dividing mass flow with density.

## Calibration

Both mass flow and density measurements depend on the vibration of the tube. This depends on the rigidity of the tube which in turn depend on its temperature. Calculations must therefore take the temperature of the fluid into account.