A new method is available to convert relative index test data into absolute flow and absolute efficiency data, without measuring absolute flow. This method is NOT intended as a replacement for absolute flow testing. In fact, this new method uses the results of homologous model tests or at least the results of absolute flow testing on other, similar prototype turbines. However, use of this method has illustrated that it provides the ability to make such relative to absolute conversions within the uncertainty limits of the test codes. Further, it is a significantly more accurate method to reduce relative index test data in its relative form.

The existing methods for conversion of relative data all first seek to calculate an absolute efficiency profile. They then seek to calibrate the relative flow measuring system, such as the Winter-Kennedy piezometer taps, to measure absolute flow values. Among its unique differences, this new method reverses the process by first calibrating the Winter-Kennedy taps, to convert their relative flow readings into absolute flow values, and then using that data to calculate the absolute efficiencies.

**Index testing**

An index test is a relative efficiency test of a hydraulic turbine. The efficiency is relative because the volumetric flow rate is measured by indexing it against another parameter, which in turn is measured in its own real terms or units. One such commonly used parameter is the Winter-Kennedy piezometer system.

Relative index data allows for the determination of a relative efficiency profile, as well as the optimum three-dimensional (3D) cam surface to input into the governor to control the tilt angle of the blades of a Kaplan turbine relative to wicket gate opening and head. Recently, several experts have developed load-sharing computer programs to optimize total powerhouse efficiency. However, the performance data of the units must be based on absolute flow and absolute efficiency to provide the most benefit from these programs. Therefore, there is a greatly increased need for turbine performance data to be on an accurate, absolute basis. Unfortunately, particularly for Kaplan turbines, there are only a few techniques to measure absolute flow. All of these are expensive and time-consuming to use.

Several techniques have been developed and used in the past to employ turbine model data to convert relative flow and relative efficiency data into absolute data. However, none of these has the level of accuracy required by the new load-sharing computer programs. Consequently, the author has developed a new method that uses a turbine model "hill curve" (a graph showing a composite of turbine performance) or other similitude prototype data to convert index test data into absolute flow and efficiency data. This method has a high degree of accuracy both to reduce index data in its relative form and to convert relative data to absolute data, while reflecting the individual performance differences of each machine.

**Traditional conversion
methods**

The most common method for using a turbine model hill curve to convert relative index data into absolute data is to first identify the peripheral speed coefficient, phi (Φ), corresponding to the prototype test head (and synchronous speed). Then, the peak efficiency of the model at that phi, which is the x-axis of the turbine hill curve, is equated to the peak relative efficiency from the index test. From that point, the relative efficiency data may be converted into absolute efficiency data. Also, because relative efficiency is equal to power over the square root of the Winter-Kennedy difference, the square root of the difference can be determined at the point of peak relative efficiency. Then, using the below equation, the Winter-Kennedy calibration constant (k) can be calculated.

Equation 1:

Q = k(D)^{1/2}

where:

– Q is the volumetric flow rate in a penstock;

– k is the
Winter-Kennedy calibration constant; and

– D is the piezometric difference
between two taps.

When this method is used on the data from a single turbine, there are three general sources of error. First, to avoid having two unknowns for this equation, a fixed and exact square root, an exponent value of 0.5 must be assumed. Second, the peak relative efficiency may not occur at the same power as predicted by the hill curve, and there are no means to account for this variation. Third, it generally is not known whether an efficiency step-up is to be applied to the efficiency value of the turbine model hill curve.

In addition, as a method to be employed to prepare data for use with the load-sharing computer programs, its big drawback is that it computes the same peak absolute efficiency for every unit in a family. This method does not retain differences in individual turbines in its conversion, which are needed in using the load-sharing programs.

Other methods have been proposed for the relative to absolute conversion. However, all of these are modifications of this basic method, and all suffer the same sources of inaccuracy or uncertainty.

**Improving the
Winter-Kennedy calibration equation**

The first step in developing this new conversion method is to use a new, more accurate form of the calibration equation for the Winter-Kennedy piezometers. The principle of the Winter-Kennedy piezometer system, invented in 1933, is that for a flow regime that is curvilinear or in a circular path, there is an increase in the angular momentum with increasing radial distance. This increased angular momentum exerts an increased force or pressure on the outer flow boundary compared to that at the flow boundary on the inner radius.

Consequently, a piezometer tap on the outer radius of a spiral or semi-spiral
case will register a higher piezometric pressure than a tap on the inner radius.
If these taps are on the same radial plane from the center of fluid rotation,
the piezometric difference, D, will be nominally proportional to the square of
the weight flow rate and hence the square of the volumetric flow rate, Q. These
piezometric pressures will be decreased by the individual velocity heads across
the orifices of each piezometer tap. However, because velocity head is also a
function of the square of the velocity and consequently the square of the
volumetric flow rate, the square root of the piezometric difference still is
nominally proportional to the flow rate. These factors result in the classical
calibration equation of Q = k(D)^{0.5}.

When calibrated using any method of measuring absolute flow rates, the
exponent of this piezometric difference rarely is found to be an exact square
root, i.e., 0.5. In fact, the test codes allow a range of 0.48 to 0.52. However,
the author recently has found that an individual Winter-Kennedy exponent to D
actually is a variable. Thus, a new, more accurate calibration equation is
derived.^{2}

Equation 2:

Q = kD^{m+a(logD)}

where:

– Q is the volumetric flow rate in a penstock;

– k is the
Winter-Kennedy calibration coefficient and is equal to 10b;

– D is the
piezometric difference between the two taps; and

– m and a are the
coefficients of a quadratic equation, log_{10}Q =
a(log_{10}D)^{2} + m(log_{10}D) + b, derived by plotting
log_{10}Q versus log_{10}D.

It is hypothesized that the reason for this slight variability of the
Winter-Kennedy exponent is that the center of fluid rotation is not exactly
coincident with the center of mechanical rotation. Therefore, this new form of
the calibration equation accommodates minor, nonlinear effects, in the relation
of log_{10}Q to log_{10}D.

**Principle of the new
method**

For years, experts have known that matching individual values of efficiency from the hill curve in any manner is not sufficiently accurate to convert relative data into absolute performance data. Efficiency curves tend to be quite radical in shape. Therefore, it was concluded that any improved method of comparison would need to be on the basis of comparing whole shapes or whole curves over the full performance range of either the hill curve or index test data. Further, such comparative curves would need to be smooth and well-behaved.

An evaluation of all possibilities showed the shape of the flow-versus-power curve for an individual turbine tends to be very smooth and well-behaved, with a gradual change in slope with a change in power. This led to the new basic methodology of calibrating the Winter-Kennedy piezometers in absolute flow terms such that the shape of the relative flow-versus-power curve from the index test matches the shape of the prototype flow-versus-prototype power curve derived from the model hill curve. With such a calibration of the Winter-Kennedy piezometers, the relative efficiency data then could be converted into absolute efficiency data.

This method may be used on any hydraulic reaction turbine, including Francis and, in particular, Kaplan turbines. A hill curve depicts the optimum performance of the model. For a Kaplan turbine, it depicts the model as though it were operating with its optimum 3D blade-to-gate cam. Consequently, curves must be matched by comparing flow versus power derived from the tangent or optimum relative efficiency curve in the index test data with the prototype flow-versus-prototype power curve derived from the turbine model hill curve. It is noted that because the hill curve is in terms of turbine output, the index test data used to derive this Winter-Kennedy calibration also must be in terms of turbine, not generator, output.

**Applying the new
method**

To use this new method, the index test data first is reduced in the conventional manner. The one difference is that the new calibration equation of the Winter-Kennedy piezometers is introduced to convert relative flow into absolute flow so that absolute efficiencies may be calculated. Any Winter-Kennedy calibration equation could be used as a starting point because this method is an iterative one that ultimately derives the best-fit calibration equation. However, as with almost any iterative process, the closer the starting point is to the ultimate answer, the fewer the iterations that are required.

The next step is to plot the data for the fixed blade angles versus turbine output (see Figure 1). Next, a tangent curve is drawn connecting each fixed blade profile. The index test data for the example given here is taken from a test conducted in December 2004 on Unit 4 at the U.S. Army Corps of Engineers' 810-MW Lower Granite Lock and Dam. During this test, flow rates also were measured using scintillation and acoustic time of flight absolute methods.

At this point, the new methodology enters the picture. The turbine output and flow rate for the individual points of tangency are determined. Then the piezometric differential for the flow value of each tangent point is back-calculated using whatever Winter-Kennedy calibration had been used in that iteration.

Next, the curve of prototype power and prototype flow is derived from the hill curve (see Figure 2). As noted previously, this is a well-behaved curve. In actuality, this curve of prototype power-versus-prototype flow does not need to be derived from a homologous model. Prototype data from a unit of the same family, resized data from a different size prototype based on the same design, or even data from a prototype of the same specific speed may be used. Obviously, however, the accuracy of the end result depends on the pedigree of the prototype flow to power curve.

At this point, there is a curve of turbine output versus the Winter-Kennedy
differential from the index test and a separate curve of turbine output versus
absolute flow from the model hill curve. Consequently, a Winter-Kennedy
differential may be calculated for each value of absolute flow rate. These are
then plotted on a log_{10}Q versus log_{10}D graph, and a
quadratic equation is calculated by the trend line, log_{10}Q =
a(log_{10}D)^{2} + m(log_{10}D) + b. This becomes a new
Winter-Kennedy calibration equation, Q = 10^{b}D^{m+a(logD)}.
This equation is then taken back to the beginning of the spreadsheet and entered
to recompute the absolute flow rates of the original index test data, and the
entire process repeated.

Only six to eight iterations were required for the solution to converge, and the process took only a few hours to perform manually on a spreadsheet.

**Accuracy of the new
method**

The reason for the accuracy of this new method is that it matches the shape of a curve over the full range, rather than attempting to match specific points of relative data to absolute data. Again, it is matching the shape of the power to flow curve, and this shape is unique to each individual turbine. For instance, if there is an efficiency step up in the model to prototype, this curve is shifted to the right and/or down. However, its shape remains essentially the same.

Figure 3 shows the absolute flow rate versus turbine output resulting from this new method for the five points of tangency, plotted on top of the prototype absolute flow rate versus prototype turbine output resulting from the hill curve. There is nearly perfect agreement, showing the iterative process does correctly converge.

The accuracy of this new method may be judged from two different aspects. The first is how accurate it is as a method to convert relative flow data into absolute flow data. The second is how much it improves the conventional method of reducing relative index test data.

With regards to the accuracy of converting relative data into absolute flow
data, one judgment would be by comparing the Winter-Kennedy calibrations.
Because absolute flow also had been measured during this prototype index test by
scintillation and acoustic time of flight, Winter-Kennedy calibrations were
calculated by both methods as Q = 11,739D^{0.4954} and Q =
11,847D^{0.4982}, respectively. These are compared with this new
method's calibration of Q = 11,426^{D0.5665 – 0.1342(logD)} (see Figure
4).

Figure 5 shows another comparison of accuracy. This was done by comparing the resulting tangent curves to the fixed blade efficiency profiles from scintillation and acoustic time of flight to the tangent curve shown on Figure 1.

The tangent curve or efficiency profile from this new method differs from that derived from measuring absolute flow by scintillation by less than 1 percent and from acoustic time of flight by less than 2 percent. Further, the efficiency profile from this new method almost exactly matches the shape of the other two efficiency profiles, showing it reflects the individual characteristics of this particular machine.

A separate improvement in accuracy became evident in applying this new technique: the accuracy of reducing the relative index test data itself was obviously improved. In the traditional method of index test data reduction, the relative efficiency profiles of the individual fixed blades commonly have some excessively high or low peaks, such that a tangent curve will have meandering deflections. This causes the placement of the tangent curve to be at the discretion of the data reduction engineer. Before applying this new method to this particular data set, the original index test data reduction showed two of the five blade profiles were low in comparison to the tangent curve drawn through the other three.

By the final iteration (see Figure 6), all the peaks came to form an ever more well-defined, smooth tangent curve. This figure is a customary final product of an index test with the lower curves of flow to power being read on the bottom and right axes. The reason an original misalignment often was encountered in the past now can be observed to be due to treating the Winter-Kennedy piezometric differential as a fixed, exact square root, rather than as a defined variable exponent. This result of the smoothing of the tangent curve increases the accuracy of identifying the points of tangency and the resulting blade-to-gate cam curve.

A paper detailing how to apply this method and a spreadsheet containing data
from its application at Lower Granite is available on the
Internet.^{2}

**Notes**

- Sheldon, Lee H., "A New Form of the Calibration Equation for the Winter-Kennedy Piezometer System," Waterpower XV Technical Papers CD-Rom, HCI Publications, Kansas City, Mo., 2007.
- http://actuationtestequipment.com, see Relative flow and index data converted into absolute flow and test data.

*Lee Sheldon, P.E., is a consulting engineer.*