Variable Area Flowmeter - Calculation

Theoretical calculation of a variable area flow meter

A variable area flow meter basically consists of:

• A conically shaped pipe that is mounted vertically and in which the cross-section of the pipe increases towards the top.
• A float that can move freely in the pipe
  The flow of the medium pushes the float upwards and gravity pulls it downwards.

Calculation of the individual forces:

• Flow force F_W
  The flow force F_W, which pushes the float upwards, depends on the flow rate and the aerodynamic properties (shape) of the float. It is calculated using the formula known from aerodynamics.
  F_W= 0,5 ⋅ ρ_R ⋅ v² ⋅ C_W ⋅ A
  ρ_R: Density of the medium in the annular gap between the float and the measuring tube. [kg/m³].
      This can deviate from the normal medium density (ρ_M) in compressible media (gases).
      As ρ_R is difficult to calculate, the normal medium density (ρ_M) is often used in practice.
      The difference is taken into account by the empirically determined C_W-value.
  v: Flow velocity of the medium in the annular gap[m/s]
  C_W: Drag coefficient (depending on the shape and surface properties of the float),
       the surface of the measuring tube and the Reynolds number [dimensionless]
  A: Cross-sectional area of the float (projection area = largest diameter) [m²]
  
  
• Weight force of the float F_G
  F_G= ρ_S ⋅ V_S ⋅ g
  ρ_S: Density of the float [kg/m³]
  V_S: Volume of the float [m³]
  g: Acceleration due to gravity 9.81 [m/s]
  
  
• Buoyancy of the float F_{A
  A}= ρ_M ⋅ V_S ⋅ g
  ρ_M: Density of the medium [kg/m³]
  V_S: Volume of the float [m³]
  g: Acceleration due to gravity 9.81 [m/s]
  
  
• Relationship between flow velocity and volume flow
  The flow velocity v of the variable area flow meter is directly related to the volume flow Q
  Q= v⋅A_R
  v: Flow velocity of the medium in the annular gap [m/s]
  A_R: Free cross-sectional area between float and measuring tube, at the position of the float [m²].
  This is the gap through which the medium can flow.
  This area changes with the height of the float.
  

Summary of the individual forces:
In order to determine the theoretical volume flow of a variable area flow meter in the “floating state”, i.e. a certain height in the measuring tube, the above formulas of the individual forces are used in the basic equation.

F_W = F_G - F_A

The result is:

0,5 ⋅ ρ_R ⋅ v² ⋅ C_W ⋅ A = (ρ_S ⋅ V_S ⋅ g) - (ρ_M ⋅ V_S ⋅ g)

If v² is replaced by the above formula for the volume flow Q, the final formula for calculating the volume flow is obtained after conversion and simplification:

Practical calculation of a variable area flow meter

This paragraph is currently being revised.

Calculation of a scale at new conditions

The scale of the variable area flowmeters is designed in the factory for a specific fluid with properties (density, viscosity) and the operating conditions (pressure, temperature) at which the the device is used.

Most manufacturers provide design software for variable area flowmeter-conversion of a scale at new conditions, which can be used to specify the optimum device for the measurement task

Example for calculation software: http://sizing.heinrichs.eu/programs/schwebekoerper/de

The most common task for the user is to adapt the existing scale to the new conditions when operating conditions change or a different fluid is used. or when using a different fluid, to adapt the existing scale to the new conditions. Changes in the material properties and operating conditions of gases or liquids have different influences.

• Gases:
  Gases are compressible, i.e. they change their volume and thus density when the pressure and temperature conditions change. Because of the low gas density and low viscosity, a change in density and viscosity with different gases usually has little influence on the measurement.
• Liquids:
  Liquids are normally non-compressible, i.e. they do not or only slightly change their volume and thus their density with changes in pressure and temperature.
  A change in pressure and temperature does not normally have to be taken into account for the same fluid. However, other viscosity or density values for different liquids have a great influence on the measurement. It may also be necessary to take into account changes in the viscosity of the liquid caused by a change in temperature.

The following rules of thumb can be used to calculate a factor with which an existing measuring scale can be converted to new operating conditions or material properties.

Factor calculation for gases:

• Calibrated scale in standard volume (e.g. Nm³/h) New scale values = factor x calibrated scale values

  

  p _cal = Absolute pressure of calibrated scale, p _new = Absolute pressure of new scale, T _cal = Temperature of the calibrated scale in Kelvin, T _new = Temperature of the new scale in Kelvin, d _cal= Density of the gas of the calibrated scale, d _new= Density of the gas of the new scale
  Kelvin = 273 + °C
  
  
• Calibrated scale in operating volume (e.g. m³/h) New scale values = factor x calibrated scale values

  

  p_cal = Absolute pressure of calibrated scale, p_new = Absolute pressure of new scale, T_cal = Temperature of the calibrated scale in Kelvin, T_new = Temperature of the new scale in Kelvin, d_cal= Density of the gas of the calibrated scale, d_new= Density of the gas of the new scale
  Kelvin = 273 + °C

Example of an Nm³/h air scale calibrated at 2 bar_abs.. New operating pressure 8 bar_abs.