*The author proposes a new method for calculating water startup time in a spiral case of a hydraulic turbine. Water startup is the dynamic parameter used in determining stability of the generating system that the governor in a hydropower plant must be able to control.*

**By Lee H. Sheldon**

Water startup time (or water running up time in the United Kingdom) is the name of a metric used to describe the dynamics of a water conveyance system in a hydropower plant. It describes the time to completely change velocity or flow rate of an enclosed fluid column from one end to the other. In hydropower, it is defined as the time, with an instantaneous gate opening, to accelerate the fluid column from zero to rated discharge, when operating under rated head. For a hydropower plant, the water conveyance system includes the intake, penstock, spiral or semi-spiral case, wheel case and draft tube. For all existing spiral cases, this method also applies to verifying the conservative estimates used by the previous method in calculating water startup times.

The counterpart to this dynamic is mechanical startup time or "mechanical running up time." This is defined as the time to accelerate the rotating elements, such as generator rotor, shaft and turbine runner from zero to synchronous speed under the application of rated power.

The ratio of mechanical to water startup time is the primary parameter in determining the stability of the generating system that the governor must be able to control.

Because the definition of water startup time requires the impossibility of instantaneous full gate opening, this metric cannot be measured on a prototype generating unit, making it a theoretical value that must be calculated. Also, the only way the definition of mechanical startup time, which requires the application of rated power, could be accomplished would be to apply that power to the generator while it is acting as a motor. Because this is impractical, mechanical startup time cannot be measured on a prototype and is also only a theoretically calculated value.

However, by application of Newton's Second Law, an equation based on the rotating inertia can be derived to determine mechanical startup time. The only uncertainty in this calculation is that the resulting formula does not account for the power diverted to overcome windage and friction. Both of these mechanical losses are complex functions of rotational speed during the startup time. Friction is a function of velocity to the first power, and windage is a function of velocity to the third power. (The efficiency of the generator acting as a motor is assumed to be 100%).

## Water startup time

The theoretical calculation of water startup time has a higher degree of uncertainty and, for one component in the fluid column (the spiral or semi-spiral case) it is little more than a guess. The general equation for water startup time is adapted from the basic equation for water hammer or pressure transient, which in turn is also adapted from Newton's Second Law. For rigid column water hammer theory in which a valve is completely closed faster than the time in which a pressure transient wave can travel twice the length of the fluid column, the magnitude of the pressure rise is:

Equation 1

H_{w} = -(L/g)dV/dt

where:

- H_{w} is increase in head due to the pressure transient, in feet;

- L is length of the fluid column, in feet;

- g is acceleration of gravity, in feet/second2;

- V is velocity, in feet/second; and

- t is time, in seconds.

The term "rigid column" means the boundaries of the fluid column do not expand under a pressure increase, and the minus sign denotes that deceleration of the fluid causes a positive water hammer. This equation may be integrated to yield:

Equation 2

t = LV/gH_{w}

If head is now steady state and static, and dropping the negative sign, an expression for water startup time Tw with units in seconds is obtained as:

Equation 3

T_{w} = LV/gH

This equation is then compared to Newton's Second Law to reveal:

Equation 4

dF = d(mV)/dt or dt = d(mV)/dF

where:

- F is force, in pounds; and

- m is mass, in slugs.

This indicates that LV represents the integrated fluid momentum, while gH represents the integrated force. With this expression, a time may be calculated for any change in flow conditions. To standardize its use, the conditions are defined as the time to accelerate the flow from zero to rated discharge, with an instantaneous gate opening, when the fluid column is under rated head. Rated head is defined as the lowest head at which the turbine can drive the generator to produce the generator nameplate rating, called rated power. Rated discharge is the flow rate when the generator is producing rated power while operating under rated head.

## Uncertainties in calculating water startup times

Even with this seemingly precise definition, there are several sources of uncertainty in calculating a theoretical water startup time:

- Rigid column water hammer theory assumes a perfect or ideal fluid; that is, an inviscid fluid, in which no head loss occurs. Therefore, whether net or gross rated head is intended by the standard is not defined;

- Generator nameplate ratings are derived under different criteria, such as the different classes of insulation used in the windings;

- Different hydraulic turbines have different efficiencies at full gate under rated head. This causes them to discharge more or less water at their generator rated condition; and

- All of the foregoing assumes that the fluid stays confined within the boundaries. That is, whatever flow rate enters at one end of a length of a flow component discharges at the other.

This last uncertainty is the subject of this article. The Law of Continuity provides that:

Equation 5

Q = AV

where:

- Q is volumetric flow rate, in feet^{3}/second; and

- A is cross-sectional area, in feet^{2}.

Substituting this law into the numerator of water startup time shown in Equation 3, the numerator becomes Q(L/A). In this manner, if Q is constant throughout the fluid conveyance system, all the different sections of all the different sizes of intake, penstock and draft tube can be equated to a single size by the following geometric relation:

Equation 6

L/A = l_{1}/a_{1} + l_{2}/a_{2} + l_{3}/a_{3} + ...

where:

- a is cross-sectional area of the individual component of the conveyance system, in feet^{2}; and

- l is length of the individual component of the conveyance system, in feet.

This formula provides a geometric equivalence and allows water startup time to be calculated for the entire conveyance system from forebay to tailrace. The uncertainty occurs with the spiral or semi-spiral case because, although a ln/an can be calculated for either type of case, Q is not confined but is continually exiting from the inner radius, through the distributor's wicket gates. (This is like calculating a pressure transient for a penstock with a series of holes along its length.) This condition is not defined for the standard method of calculating theoretical water startup time. Therefore, by convention, ln/an is calculated for the geometry of the spiral or semi-spiral case, and 50% of that value is used in calculating water startup time. The rationale for this approach is not documented in any available literature.

This 50% value assumes any molecule of water has an equal chance of entering the distributor at any radial position along the spiral length. However, even a casual inspection shows this is not realistic. There is a bias depending on where a stream line starts from the entrance. A flow stream line at a spiral case entrance at the innermost radius is most likely to enter the distributor first, while that furthest away is most likely to enter the distributor last, around the case at the splitter vane. For projects with very long penstocks, this uncertainty causes a relatively small error in the total water startup time. However, as the penstocks become shorter or for projects where the intake leads directly to the semi-spiral case, the uncertainty becomes significant.

## Traditional method of calculating water startup time

To clarify the terms spiral case and semi-spiral case, a spiral case is a design for higher-head turbines, usually Francis. It is circular in cross-sectional area and wraps completely around the periphery of the unit. A semi-spiral case is a design used for lower-head turbines, such as Kaplans. It is rectangular in cross-sectional area and wraps a little more than three-quarters of the way around the periphery.

To provide for an analytic approach to evaluating startup time, the concept of stream tubes will be used. To accomplish this, only a simplified spiral case configuration with uniform geometry will be considered for this analysis. As Figure 1 shows, the spiral case splitter vane, at the end of the spiral case, is at the same radial angle as the start of the spiral, and the flow cross section is rectangular and of constant height. The inner circle represents the wicket gate stem circle. Because flow is considered to leave the spiral case through the wicket gates, an upstream stay vane circle is omitted in this simplified geometry.

The analysis will be done in two parts. First, a traditional Q(L/A) will be calculated based on the simplified geometry only, as shown in Figure 1 (at left). Second, a new Q(L/A) will be calculated based on the lengths of individual constant area stream tubes, as shown in Figure 1 (at right). A ratio of the two results will be a more accurate percentage of the traditional geometric calculated Q(L/A) to use in computing a water startup time.

For the first part, the spiral case is divided into four quadrants (see Figure 1, left side). It could be divided into more, but four is considered sufficient for the level of accuracy of this simplified geometry. The first quadrant extends from a spiral angle, , of 0 to π/2. At its flow centerline, it has a maximum radius of r + d/2 (where r is the radius of the wicket gate stem circle in feet and d is the diameter of the penstock inlet to spiral case in feet) and a minimum radius of r + 3d/8, for an average radius of r + 7d/16. If the height is considered to be one unit, the maximum area, at = 0, is, (1)d, and a minimum area at = π/2 is 3(1)d/4, for an average area of 7(1)d/8. The centerline length over area of that spiral quadrant would be the average radius times the swept angle, over the average area. Therefore:

Equation 7

l_{1}/a_{1} = (r + 7d/16)(π/2)/[7(1)d/8]

The other three quadrants may be treated the same and, when added together, yield:

Equation 8

Σl_{n}/a_{n} = (704/105) πr/d + π = π [6.705r/d + 1]

Therefore:

Equation 9

Q(L/A) = πQ [6.705r/d + 1]

## Stream tube method of calculating water startup times

Figure 1 (at right) shows the same simplified spiral case, but for stream lines that form stream tubes. These tubes start at = 0 with uniform flow where they are each of the same width and cross-sectional area and they each carry the same incremental flow rate. There may be any number of wicket gates, but they are oriented such that a pair straddles the splitter vane. The number of stream tubes equals the number of wicket gates.

As depicted, the centerline of the innermost stream tube terminates in the middle of the first complete pair of wicket gates so that the stream tube discharges through an equivalent full opening for a pair of gates. At the of that terminus, the second innermost centerline and stream tube falls into the void space left by the first stream tube. This process continues around the 360 degrees of the spiral case as the outer radius of the case narrows to the inner circle.

The radius of the centerline of the innermost stream tube is constant and is equal to r + d/2k. The swept angle is equal to 2π/k, and the area for one unit depth for all stream tubes is equal to (1)d/k. Therefore:

Equation 10

l_{1}/a_{1} = (r + d/2k)(2π/k)/[(1)d/k] = π (2r/d + 1/k)

where:

- k is the number of wicket gates or stream tubes.

The length over area of the second innermost stream tube, counting upstream from the wicket gates, then becomes:

Equation 11

l_{2}/a_{2} = (r + d/2k)(2π/k)/[(1)d/k] + (r + 3d/2k)(2π/k)/[(1)d/k] = π (2r/d + 1/k) + π (2r/d + 3/k)

Likewise, the length over area of the third innermost stream tube becomes:

Equation 12

l_{3}/a_{3} = π (2r/d + 5/k) + π (2r/d + 3/k) + π (2r/d + 1/k)

When all the stream tubes are added together, the result is an infinite series of:

Equation 13

Σl_{n}/a_{n} = π[2kr/d + (1/k)Σk1(2k - 1)]

Next, the series summation term is evaluated as:

Equation 14

Σ^{k}1(2k - 1) = k^{2}

Therefore:

Equation 15

Σl_{n}/a_{n} = π[2kr/d + (1/k)k^{2}] = π[2kr/d + k]

Finally, the equal incremental flow rate in each stream tube, designated q, is such that the total flow rate is equal to Q, or Q = kq.

Based on this:

Equation 16

LV = Q(L/A) = Σ(q)l_{n}/a_{n} = Q/k[2kr/d + k] = πQ[2r/d + 1]

## Comparison of calculated results

A ratio of the results of stream tube divided by the traditional calculation then yields:

Equation 17

ratio = πQ [2(r/d) + 1]/ πQ[6.705r/d + 1] = (2r/d + 1)/( 6.705r/d + 1)

To evaluate this ratio, a numerical value for the ratio r/d needs to be determined. The only source of such information would be the designs of actual spiral cases. This introduces an additional uncertainty of mixing real geometric data with the simplified geometry used in this analysis. However, with no other alternative, published data on spiral case dimensions is utilized.^{1} Thus, the ratio of the wicket gate circle to the diameter of the penstock at the inlet is found to be a function of the specific speed, Ns, as seen in Figure 2. It ranges from r/d = 0.72 at N_{s} = 100 up to r/d = 1.31 at N_{s} = 20. Figure 2 also shows both the stream tube and traditional calculations of L/A. The corresponding L/A ratios range from 41.8% at Ns = 100 down to 37.0% at N_{s} = 20, with an average of 39.3%. This is rounded to an even 40% or 0.4.

## Conclusions

In other words, 0.4 appears to be a more accurate multiplier than 0.5 for the L/A ratio of a spiral case calculated using the traditional geometric method. Further conclusions are that the L/A ratio increases as specific speed increases, because the ratio of r/d increases. However, as specific speed increases, the head decreases. This results in virtually no change in the water startup time of a spiral case with specific speed.

An even more far-reaching conclusion is that the governing stability of hydropower projects, at least in this regard, has been conservatively designed. This means that grid systems, with a high proportion of hydropower, probably have a little more reserve stability to absorb electrical energy from various new green resources. The corollary of this is that continuing to take hydropower facilities out of production at a time when additional generation is being sought from green resources is counterproductive.

## Recommendation

How applicable these results are to the water startup times of semi-spiral cases is uncertain. Most of the stream tubes in semi-spiral cases would behave in a similar manner, and therefore the multiplier of 0.4 would appear to apply. However, it is known that where semi-spiral cases have multiple intakes, the volumetric flow rate in each intake is significantly different. Therefore, more sophisticated analysis is recommended to determine how applicable these results from a simplified spiral case are to an actual semi-spiral case.

## Note

^{1}Selecting Hydraulic Reaction Turbines, Engineering Monograph No. 20, U.S. Department of Interior's Bureau of Reclamation, Denver, Colo., 1976.

**Lee Sheldon, P.E., is a consulting engineer and also affiliated with Black & Veatch.**