The vortex flow meter and the calculations behind its working principles are a result of nearly 150 years of development. Over that time, we saw an evolution in fluid dynamics that allows us to produce accurate flow measurements using vortex flow meter formulas.

Vortex flow meter calculation involves a number of formulas and equations. The principle of fluid oscillation makes the calculation of flow rate possible by measuring the frequency of vortices produced in the system. The K factor improves the accuracy of this calculation by calibrating the output signal to the specific device.

This article explains the formulas and equations behind vortex flow meters. Knowing how to use these is not always required to install, read, or troubleshoot vortex flow meters. However, ifm provides this information as part of a comprehensive resource for flow meters. 

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The Strouhal number

The Strouhal number produces a dimensionless constant that describes oscillating flow mechanisms. It's applicable to fluids and gasses: St=f*Lv

Where:

• St is the Strouhal number
• f is the frequency of vortex shedding
• L is the characteristic length
• v is the flow velocity

History

In 1878, Vincenc Strouhal, observed the rhythmic hum of telegraph wires. From his observations, he derived an equation describing flow mechanics and applied it to fluids and gasses. The equation below produces a dimensionless constant known as the Strouhal number( St.) 

The Strouhal number is also a function of the Reynolds number( Re), which we'll discuss next.

Vortex flow meter sizing calculation

The vortex flow meter sizing calculation is Re=v*Dэ

Where:

• Re is the Reynolds number.
• v is the fluid velocity.
• D is the inner pipe diameter.
• э is the kinematic viscosity (absolute viscosity divided by the fluid density.) 

The Reynolds number

The Reynolds number is a function of velocity, as seen in the equation above. Increasing velocity will increase the Reynold’s number.

Vortex flowmeters typically need a Re value greater than 10,000. However, it is important to note that the St remains constant across a range of Re from about 102 to 107. This allows for the assumptions necessary to derive the vortex flow meter flow calculation.

Decreasing the size of the meter can increase the velocity through the meter. This would increase the Reynolds number across the meter. This is similar to how more water flows through a hose when you decrease the size of the hose nozzle.

Depending on your fluid properties, you may need to decrease the pipe size upstream and downstream from the meter. This increases the Re across the meter.

Varying the pipe diameter will cause some additional pressure drop across the meter. Most installations will require 10D line size upstream and 5D line size downstream from the meter. This varies depending on if your configuration contains bends or valves in the pipe. 

History

In 1883, Osbourne Reynolds popularized the use of a constant derived from the inertial and viscous properties of fluids to predict the flow patterns of a fluid. This determines if the flow of a liquid will be laminar or turbulent.

This constant was officially coined as the Reynolds number in 1908. The Reynolds number and its calculation are important factors in sizing a vortex flow meter. It is the next step in deriving the calculations necessary to predict the vortices and determine the flow rate. 

Vortex flow meter equation

The vortex flow meter equation uses the principle of fluid oscillation, which applies to fluids and gasses. The principle measures the predictable vortices produced within the meter. This in turn allows a vortex flow meter to measure the volumetric flow rate through a system.

Calculations are used to derive the fluid flow rate from those measurements. That use essentially makes it the vortex flow meter equation.

Fluid oscillation principle

The principle of fluid oscillation uses the following formula to determine the frequency of vortex generation: f=St*vD

Where:

• f is the frequency of vortex generation
• St is the Strouhal number
• v is the fluid velocity
• D is the width of the bluff body or shedder bar

The vortex flow meter derivation uses this formula. You can use this formula to find the flow rate as a function of the frequency range of vortices: Q=f*K*Dv

Where:

• Q is the volumetric flow rate
• f is the frequency of the vortex shedding within the meter (which is the number of vortices recorded over a unit of time)
• K is the vortex shedding constant( also known as the k factor,) which is specific to the meter
• D is the width of the bluff body or shedder bar within the meter
• v is the fluid velocity

You can multiply volumetric flow rate by fluid density to obtain the mass flow rate from the volumetric flow rate: ṁ=Q*𝛒

Where:

• ṁ is the mass flow rate.
• Q is the volumetric flow rate.
• 𝛒 is the fluid density.

This equation only works if the fluid density is constant. This is dependent on the fluid composition as well as the temperature and pressure of the environment. There are also a wide range of mass flow meters that return mass flow rate as the output signal. 

History and the Von Kármán effect

In 1912, Theodore von Karman coined the term Karman Vortex Street. It describes a the pattern of alternating swirling vortices seen when an obstruction is introduced into a stream.

Karman observed that fluid flowing past an obstruction was flowing faster, generating more vortices (and vice versa). He built upon the work done by earlier physicists along with his observations to derive the equation necessary for the development of the vortex flow meter.

What is a k factor in a vortex flow meter?

A K factor in a vortex flow meter is a dimensionless unit that is specific to the design of each flow meter. It is used to convert the vortex frequency into the flow rate.

The K factor will be supplied for each individual flow meter by the manufacturer. It will be accurate for a range of Reynolds numbers that correspond to the viscosity of the process fluid.

The turbine flow meter works using the same equations as the vortex flow meter, but the output signal is determined by the number of revolutions of the turbine instead of the number of vortices.

The vortex flow meter k factor calculation is Q=K*Os

Where:

• Q is the flow rate.
• K is the specific K factor for the meter being used.
• Os is the output signal

The output signal is a set of pulses corresponding to the frequency of vortices. The K factor is the number of pulses per unit of volume of the process fluid.

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