Turbine Waterhammer Theory

Four quadrant data is defined by the following dimensionless parameters (“R” subscript refers to rated values). Reference Wylie, et al, 1993Wylie, E.B., V.L. Streeter & L. Suo, Fluid Transients in Systems, Prentice Hall, Englewood Hills, New Jersey, 1993..

• Head Loss Ratio:

(1)

• Torque Ratio (unbalanced torque):

(2)

• Flow Ratio:

(3)

• Speed Ratio:

(4)

• Function Relating Head Loss to Flow and Speed:

(5)

• Function Relating Torque to Flow and Speed:

(6)

• Four Quadrant Data Angle:

(7)

• Integer function for Fh and Fb indices:

(8)

For the Speed change related to net torque, we start with Eq. 7 - 47 from Wylie (similar to Eq. 7 - 20, without the negative sign)

	(9)
	(10)
	(11)
	(12)
	(13)
	(14)
	(15)
	(16)
	(17)
	(18)
	(19)

This is a modified version Eq 7-22 from Wylie. Wylie integrates over two time steps, to be consistent with the “staggered grid” solution of the Method of Characteristics grid. The above result is based on integration over only one time step for use in the “complete grid”.

It’s convenient, at this point, to lump constants together and simplify.

	(20)
	(21)
	(22)
	(23)
	(24)
	(25)
	(26)
	(27)
	(28)
	(29)
	(30)

Eq. 30 is a modified version of Wylie's Eq. 7-26. Wylie uses Eq. 7-20 (for pumps) but Equation 30 above uses Eq. 7-47 (for turbines).

Head loss can be described using a similar procedure, along with the characteristic equations.

	(31)
	(32)
	(33)
	(34)
	(35)
	(36)
	(37)
	(38)
	(39)
	(40)
	(41)
	(42)
	(43)
	(44)

Eq. 44 comes from Wylie Eq. 7-19. Wylie includes a valve loss term in Eq. 44, which is not included here. Also, signs are reversed to be consistent with turbine head loss, as opposed to pump head rise.

At this point, FH and FT can be solved using Newton Raphson, in 2x2 matrix form.