Turbine flow meters:
A design tool and
improved method of linearising flow
By Martin Cuthbert
MEng (Hons), Webtec Products Ltd.
This paper was presented at the Power, Process and Technology
Division meeting at the Institution of Mechanical Engineers in London on 19th
November 1997. Martin was awarded the B.F.P.A. 1997 Prize for Young Engineers
for this innovation.
Synopsis
Turbine flow meters
are inherently non-linear. A computer at the point of display using a plot of
K factor / Flow, often corrects this non-linearity. This method is accurate to
~1% of indicated reading for the particular oil and temperature upon which the
meter was calibrated. Flow meter accuracy deteriorates if oil of a different viscosity
or the same oil at a different temperature is used.
This project investigates
a new method of linearising flow, common to a range of different size turbine
flow meters, that is compatible with oils of different viscosity over a range
of temperatures. Tests were carried out using oil and using water and the results
were plotted against two dimensionless groups: modifications of Strouhal Number
and Reynolds Number. The resulting curves for different size flow meters are
very similar and are all described by a common mathematical equation. The outcome
of this investigation is a method of linearisation that successfully compensates
for changes in fluid viscosity and is common to a range of flow meters. The
result, in comparison with traditional K factor / Flow linearisation, is improved
accuracy, greater flexibility, and a valuable design tool.
1. Introduction
This report is a résumé
of research carried out over a period of eighteen months, the majority of which
was for the author's final year investigative project at Sheffield University.
Final testing and validation of this method is now underway prior to commercial
release by Webtec Products Ltd. at the end of this year.
2. Turbine flow meters
The turbine flow meter,
also known as a propeller meter, belongs to the family of volumetric flow meters.
The basic design consists of a turbine, mounted on bearings, located longitudinally
within a machined flow tube. Turbine flow meters tend to be bi-directional and
their design includes flow straighteners on either side of the rotor. The rotation
is measured by electric or magnetic pickup of the blades as they pass a fixed
point. The rotation of the turbine is approximately proportional to the volume
flow rate, in general the number of pulses per unit fluid volume over a 5:1 flow
range is constant to within a +/- 0.25%. Rotors have two or usually more blades
and can be of varying designs.
The LT range of flow
meters consists of three sizes T1, T2, and T3, these cover flows from 4-125, 15-400
and 25-800 litres per minute (l.p.m) respectively. The flow meters are all of
a similar design, namely a machined aluminium body housing a stainless steel six-bladed
turbine. The turbine is mounted on stainless steel bearings and fitted between
two six-armed flow straighteners. The turbine rotor is of a helical or optimised
angle design. Pickup is via a transducer screwed to the top of the block. A thermister
is housed next to the transducer and sits flush with the inner wall of the turbine
block.
See:
Figure
1 - An LT type turbine flow meter
The volume flow rate
of a liquid through the flow meter is proportional to the angular velocity of
the turbine. As each blade of the turbine passes the transducer it generates a
pulse. Thus the rotation of the turbine results in a stream of pulses, one revolution
of the turbine being equal to six pulses for a six-bladed turbine, as used in
the LT range. The meter factor is defined as the number of pulses measured by
the turbine transducer divided by the flow that generated those pulses and typically
has the units pulses per litre or ppl.
At constant flow
any resistance offered by bearing friction, magnetic detent and viscous drag
is balanced by the driving torque of the fluid impacting the blades of the turbine.
However the factors attributable to resisting the flow and hence the rotation
of the turbine are non-linear. As a result the inaccuracies caused by these
factors become of greater importance at lower flows and even more so for smaller
flow meters; those less than two inches in diameter.
With the advent
of the microprocessor it is no longer necessary that turbine flow meters demonstrate
linear behaviour so long as the non-linear behaviour is repeatable. In this
way the curve of meter factor / flow curve can be stored and the correct flow
calculated. A typical curve is shown in Figure 2.
See:
Figure
2 - A typical K factor / flow curve
3. Shortcomings of
the K factor / Flow curve
The single greatest
influence on the shape of the linearisation curve is the viscosity of the fluid
being used. This is of particular importance when dealing with mineral oils; the
fluid used the majority of the time in hydraulic power transmission. As the oil
temperature decreases, the kinematic viscosity increases. The increase in viscosity
results in a more rounded curve; i.e. the initial slope becomes less steep and
the linear section of the curve becomes shorter.
The inaccuracy that
can result from oil being tested at different temperatures can be of the order
of 4% of indicated reading at low flows. Figure 3 shows T2 tested using ISO
32 oil at three different temperatures and plotted on a graph of K factor /
flow.
See:
Figure
3 - K factor / flow at three different temperatures (viscosity) T2
It is also important
to notice that the three flow meters are of different dimensions and hence have
completely different K factors. The respective curves whilst similar are not the
same, the main difference being in the initial steep section and ëhumpí of the
curve as can be seen in Figure 4.
See:
Figure
4 - K factor / flow for three sizes of turbine flow meter tested on ISO 32 at
one temperature
4. Solutions
Hochreiter [1] using
dimensional analysis predicted the relationship between the meter factor and fluid
viscosity, shown in equation below.
Equation 1
Where f is frequency,
Q is flow, 
is kinematic
viscosity, and 
denotes
function.
Hochreiter reported
that this equation worked to a limited extent, giving a single smooth curve
for fluids of different viscosity but the curve did begin to separate at low
flows. The curve does not take into account non-viscous retarding forces and
was hence deemed at the time to have little practical use. However this principle
has since been adopted by a major flow-measurement company to plot a Universal
Viscosity Curve (U.V.C.); namely K factor against frequency / kinematic viscosity.
This method was applied to the data from flow meters T1, T2, and T3 as can be
seen in Figure 5.
See:
Figure
5 - Data from three different size turbine flow meters plotted on the Universal
Viscosity Curve
4.2 One curve for
different size flow meters
Whilst the U.V.C. successfully
amalgamated the results from three different viscosity fluids into one smooth
curve, it however had no affect on the relationship between the curves for different
size flow meters. The next goal was to find one common curve for different size
flow meters.
In Figure 5 the
differences between the curves in the x direction were due to the three meters
having different internal dimensions. Thus the use of Reynolds number (Re) [2]
in place of ëfrequency / kinematic viscosityí brought the curves in line.
Equation 2
The
differences on the y-axis were due to each flow meter having a very different
meter factor (number of pulses per litre of flow). Another dimensionless group,
Strouhal number (St), was seen as a good equivalent to the meter factor but
included an additional constant to remove dimensional differences.
Equation 3
Figure 6 shows the
three curves, similar both in shape and x/y values.
See:
Figure
6 - Data from three different size turbine flow meters plotted on a graph of
St_m / Re
A further modification
was made to compensate for the differences in pitch of the three turbines, the
resulting group was then renamed St_m.
Equation 4
Where 
is density, V is velocity, dbk is block diameter, dbd
is blade diameter, 
is
dynamic viscosity, is
kinematic viscosity, Q is volume flow rate, f is frequency, Pf is
pitch factor.
4.3 Best of both
worlds
The use of Reynolds
number on the x-axis brought with it certain problems. The original k factor /
flow curve used an iterative process in order to calculate flow. The U.V.C., as
it only contained flow on one of the two axes, was simply a look-up chart not
requiring any iteration. The St_m / Re method once again contained flow on both
axes and hence required iteration. To avoid this Reynolds number was modified;
flow was substituted with frequency and the group was renamed Re_m.
Equation 5
The curves when plotted
using Re_m / St_m were very similar in shape. As a result it was possible, by
varying the constants, to fit one mathematical equation, a sixth order polynomial,
to all three curves. The data and fitted curves plotted on a Re_m / St_m graph
can be seen in Figure 7.
See:
Figure
7 - Data and fitted curves for three different size turbine flow meters
6. A design tool
The ability to plot
very similar graphs for different size turbines allows the design engineer to
predict the effect of changes on certain parameters. Should the need arise to
alter the pitch of the turbine then the pitch factor can be adjusted and the new
curve accurately predicted prior to any tests being carried out. In the same way
adjustments could be made to other dimensions and the effect on the curve could
be immediately seen.
7. Conclusion
The research to date
has gone a considerable way to reducing the differences between linearisation
curves for different size turbine flow meters. Research is continuing to further
improve on this solution with the ultimate aim of plotting the data from different
size flow meters in one smooth curve.
References
- Hochreiter, H.M.
- Dimensionless correlation of coefficients of turbine type flow meters, Trans.
ASME, October 1958, pg. 1363-68.
- Massey, B. S.
- Mechanics of Fluids - Sixth Edition, Chapman & Hall, London, 1992.
Acknowledgements
Dr S.B.M. Beck - Lecturer
in Thermodynamics and Fluid Mechanics, Sheffield University.
Mr John Price - Head
of Research and Development at Webtec Products Ltd, St. Ives.
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