*A new method, called the dynamic orifice model, allows for waterhammer calculations to be performed in the initial design phase of a hydroelectric project, when many characteristics of the system are not yet well-defined. This method is particularly applicable when the system is equipped with a reaction turbine, such as a Francis unit.*

**By Helena M. Ramos**

This article has been evaluated and edited in accordance with reviews conducted by two or more professionals who have relevant expertise. These peer reviewers judge manuscripts for technical accuracy, usefulness, and overall importance within the hydroelectric industry.

Because of their long pipe systems, hydroelectric facilities are subject to safety and economic concerns caused by severe hydraulic transients. The basis of any study designed to analyze waterhammer, a pressure transient caused by rapid closure of a turbine’s wicket gates, is abnormal operation under extreme conditions. When the pipe system is equipped with a reaction turbine with low inertia and low specific speed (such as a Francis unit), runaway can easily occur during abnormal conditions, inducing dangerous overpressures. For example, a serious safety problem can occur after a full load rejection when the overspeed effect causes maximum overpressure, which can be attained in as little as 3 seconds. The total duration of wicket gate closure, although an important parameter, is not the most important parameter in this case.

An efficient model for transient analysis enables evaluation of the operational behavior of the hydroelectric system. This analysis may lead to unconventional solutions by eliminating conservative protection devices, such as a surge tank. In fact, during the initial design phase (civil works conception), the characteristics of the various aspects of the system are not yet well-defined or completely available. To overcome such lack of knowledge, the author developed a computational model based on a new methodology, called the dynamic orifice concept.1,2,3 Using this concept, the specific speed allows characterization of the turbine behavior that will influence transients along the conveyance system.

Previously, two levels of tools were available for hydraulic transient analysis: complete models based on the full set of machine equations for each unit type (but often not available from turbine manufacturers) and classical methods based on gate effect or on empirical data. The dynamic orifice model was developed to fill the gap between these two approaches.

**Overpressure estimation**

For design of hydraulic conveyance structures, the general specifications of the equipment and the overpressures along the penstocks must be determined. Most formulas are based on the start-up (or inertia) time of the water mass at full load and involve multiple variables, including: maximum head variation; turbine gross head; water mass at full load (calculated using pipe length, flow velocity, and gravitational acceleration); closing time; and KC, a factor that depends on the turbine specific speed (Ns) (calculated using rated wheel speed, rated turbine power, and rated head or the head of best efficiency). Typically, KC values are 1.3 to 1.5. Due to the runner overspeed effect in reaction turbines, maximum head variation will be a function of the inertia of the rotating masses or the unit starting time.

The above method requires that a great deal of data be available on system operation. The dynamic orifice model is more flexible and general and is based on a small number of parameters. The objective of this model is to provide a method to analyze the extreme condition generated by runaway and/or guide vane closure as the most unfavorable condition with regard to hydraulic circuit design (overpressures). It includes prediction of overspeed effect under runaway conditions on hydro transient response all the way along the conveyance system. The dynamic orifice model can be a useful tool for an integrated computer analysis of any multi-component systems (i.e., reservoir, total pressurized penstock, mixed circuit composed of a canal and a forebay with free surface flow and a penstock with pressure flow, and typical response of any type of turbine-generator groups).

The dynamic orifice model is based on the concept of the turbine acting as a hydraulic resistive component where the head lost by the flow is characterized by a dimensionless formula that has a dynamic discharge coefficient. This coefficient is composed of two terms: a gate factor and a runner overspeed factor. The gate factor defines the maximum turbine discharge for a given head and speed as a function of the gate opening. The runner overspeed factor modifies the discharge coefficient as a function of the runner speed. The coupled response depends on the specific speed, as well as the rated turbine speed (NR), discharge (QR), and head (HR).

With low specific speed turbines, the discharge decreases with runner speed. Conversely, for high specific speed turbines, the discharge may increase with speed. This behavior has a significant effect on the transient response of the conveyance system after a full load rejection and must be taken into account in the simulation of extreme operational conditions. Both the overspeed factor of the turbine discharge coefficient and the turbine hydraulic torque are based on dimensionless relationships and on a few simple parameters and are characterized by heuristic equations that are approximations of the real characteristic curves.

For impulse turbines, the transient discharge is decoupled from the runner speed and the turbine discharge does not change with wheel speed as long as the gate or nozzle opening remains constant. For this type of turbine, the overspeed factor is equal to unity and the model contains only the pure orifice model.

When the dynamic orifice is coupled to flow modeling along the conveyance system, the model can simulate:

- - Pressure and discharge variations and the head envelopes along the pipelines, as well as the head variation along the canal and forebay for mixed circuits;
- - Transient pressure and discharge variations in the powerhouse for full-load rejection and/or guide vane closure (or nozzle closure); and,
- - Rotational speed variation of groups.

The runaway condition of turbine-generator groups is characterized by two dimensionless parameters, also known as runaway discharge and rotating speed coefficients. These are:

Equation 1

where:

- QRW is turbine discharge at runaway speed;

- QR is rated turbine discharge;

- NRW is turbine runaway speed; and,

- NR is rated turbine speed.

These parameters depend on the turbine type.1 A low specific speed Francis turbine will have αR 1, and a high specific speed propeller or Kaplan turbine will have αR > 1.

In practice, turbine manufacturers know the values of these parameters through turbine tests and can provide them for different specific speeds under rated head.

Based on published information2,3 and on data from turbine manufacturers, average values of αR and ßR (see Figure 1) can be estimated:

Equation 2

According to data obtained from case studies, for low-inertia units in small hydroelectric plants equipped with reaction turbines, ßR is about 2 +/-20%.

As an example of the results obtained using systematic computer simulations, Figure 2 reveals the dimensionless maximum upsurge or overpressure values induced by full load rejection at the downstream end of a single uniform penstock. For low specific speed Francis turbine units or for αR < 1, the overspeed effect will potentially induce greater overpressures compared to the gate effect. In fact, as αR approaches 1, the flow reduction induced by turbine overspeed does not modify the maximum overpressure because waterhammer depends only on the gate effect, including the time of gate closure.

When turbine runaway speed is attained in a short time interval, the overpressure due to overspeed can be evaluated using the following modified Joukowsky formula:

Equation 3

where:

- HM is maximum head variation in meters;

- Ho is gross head in meters;

- Hw = cV/2gH0 = Tw/Tc, when Tc≈TE (typically hw < 1 for high-head systems), where Tw is the startup or inertia time of the water mass in seconds and Tc is the guide vane closing time in seconds;

- TE is the elastic time constant in seconds; and,

- Tm is machine starting time (sec).

This formula was obtained by adjusting the values of dimensionless upsurge for different system characteristics.

The dynamic orifice model shows the influence of the group’s inertia on the overspeed and pipe-fluid elasticity through TE/Tm.

For each TW/TC value, the maximum overpressure due to overspeed can (for low αR) exceed the maximum Michaud value corresponding to the critical partial gate closure.

**Rotating mass equation**

The unbalanced torque between turbine and generator changes in accordance with the angular momentum equation for the rotating mass:

Equation 4

where:

- TH is net hydraulic turbine torque;

- TG is electromagnetic resistant torque;

- I is total polar moment of rotational mass inertia;

- ω is angular velocity of the rotating mass;

- d is the derivative symbol; and,

- t is time in seconds.

After a full load rejection, the electromagnetic resistance torque of the unit can be set equal to zero. The polar moment has a significant influence on the speed variation of the rotating mass of the turbine-generator units.

For low inertia units, the turbine runner speed increases rapidly after a full load rejection and can attain runaway conditions.

**Analysis of results**

This new hydropower model has the advantage that the dynamic runaway condition induced by any type of reaction turbine units can be easily evaluated. This technique, together with computer modelling of other components of the system, enables an integrated hydro transient analysis.

Figures 3 and 4 show some characteristic parameter variation - such as turbine discharge (Q), head (H), and rotational speed (N) - applied to an experimental hydroelectric lab facility for a simultaneous occurrence of a full load rejection and wicket gate closure for two types of Francis turbine units (i.e., one low and one medium specific speed turbine).

The dynamic behavior was compared with predictions using the new developed model and lab tests.1

**Conclusions**

Empirical formulae or simplified models for hydro transient analysis in systems with long hydraulic circuits, based only on the wicket gate closure time, cannot be accurate enough for design purposes. However, it can be difficult to obtain the complete set of turbine characteristic curves in the early stages of each design, depending on the available manufacturer’s data.

From the practical operational point of view, application of the dynamic orifice technique, in real case studies, has enabled more reliable and economical layout solutions. This technique is a powerful aid for hydro transient analysis, for better understanding the type of system response and the powerhouse design dependence of hydraulic phenomena parameters.

The allowable penstock pressure, guide vane manoeuvre time, and polar inertia moment can be specified before the full machine characteristics are known to control excessive overpressures, especially for the situation of critical full load rejection in a low specific Francis turbine speed.

The feasibility of a hydroelectric project depends on civil construction costs and environmental impacts, requiring accurate computational simulations and cost-benefit studies to mitigate these factors. With this integrated model, more confident and efficient solutions can be selected from among different alternatives at an early but important design stage.

**Notes**

1 Ramos, Helena, *Simulation and Control of Hydrotransients at Small Hydroelectric Power Plants*, PhD Thesis, Technical University of Lisbon, Portugal (in Portuguese), 1995.

2 Ramos, Helena, and A.B. Almeida, “Dynamic Orifice Model on Waterhammer Analysis of High and Medium Heads of Small Hydropower Schemes,” *Journal of Hydraulic Research*, Volume 39, No. 4, 2001, pages 429-436.

3 Ramos, Helena, and A.B. Almeida, “Parametric Analysis of Waterhammer Effects in Small Hydropower Schemes,” *Journal of Hydraulic Engineering*, Volume 128, No. 7, 2002, pages 689-697.

Helena Ramos is an associate professor in the civil engineering department of the Technical University of Lisbon in Portugal.

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