The key to the success of a target flowmeter is the measurement of the drag force.
The drag force Fd is given by the drag equation of incompressible flow:
where V is flow velocity, is the density of the fluid, A is the projected area of the target, and Cd is the drag coefficient to be determined experimentally based on the flow conditions and the geometry of the drag element. For flat plate and sphere, the drag coefficients typically are:
Please note that the 1.28 drag coefficient is for flat plates that are perpendicular to the flow direction and the drag coefficient for spheres is related to the Reynolds Number.
For a given design, A and Cd are constant. If the density of the liquid is also constant, then the drag force Fd is solely a function of V2.
With strains at certain points measured by strain gages, the drag force can be calculated by a beam-bending formula (cantilever beam with an end load). The flow velocity can consequently be obtained.
Suppose the strain gage is attached at the front and/or back surface(s) of the extension rod () at location . The stain measured will be
where is the stress at the cross section along the x direction, M is the bending moment, y is the coordinate perpendicular to the rod's longitudinal (x) direction, E is Young's (elastic) modulus of the rod material, and I is the rod's cross section's area moment of inertia. (Please refer to the beam theory for further details. The area moment of inertia can be found in Area of the Mathematics section.)
The bending moment M of a cantilever beam with length L under a concentrated end load P is
where x is measured from the fixed end. In this case, L is the length of the extension rod, and P is the drag force Fd.
The strain becomes
and the drag force can be expressed in terms of strain:
Plug the above formula into the drag equation we will have the following expression:
This formula for V is valid under the following assumptions:
- The cross section of the extension rod is much smaller than the diameter of the drag element.
- The drag element is much smaller than the inside diameter of the pipe.
Otherwise, further calibration is needed to determine the correction factors.