The turbine specific speed is a quantity derived from dimensional analysis. For a
specific turbine type. (Francis, Kaplan, Pelton), the turbine efficiency will.
Turbine specific speed concepts The turbine specific speed is a quantity derived from dimensional analysis. For a specific turbine type (Francis, Kaplan, Pelton), the turbine efficiency will be primarily a function of specific speed. The specific speed should be a dimensionless quantity, yet often (as in the book) it is presented in dimensional form. These notes outline the derivation and use of the specific speed. Neglect, for the moment, the effect of head losses on the turbine power. The power will then be given by ˙ T = ηT ρ V˙ g H (1) W Obviously the turbine should have as large an efficiency ηT as possible. In general, ηT will depend on the specific geometrical configuration of the turbine system, as well as the flow rate V˙ , the head H, and the turbine rotation rate ω. For specific values of V˙ , H, and ω, an optimum geometrical design would exist which would optimize the turbine efficiency. Determination of this optimum design would be performed using either experimental methods or (more recently) numerical CFD simulations, and work of this type has led to the development of the Francis, Kaplan, and Pelton designs of common hydroelectric use. Alternatively, given a specific turbine design (i.e., Francis, Kaplan, Pelton), one would anticipate that there would be a specific set of operating conditions V˙ , H, and ω which would optimize the turbine efficiency. The basic concept of the turbine specific speed is to identify the optimum operating conditions for a given turbine design. This identification process can be developed via simple dimensional analysis, coupled with an inviscid (i.e., ideal) model of fluid mechanics. Say that we have developed an optimized turbine design (i.e., maximized ηT ) for the specific conditions V˙ 1 , H1 , and ω1 . This design would have associated it a characteristic size D1 . You could view D1 as the turbine runner (or rotor) diameter, yet it is not important to precisely connect D1 to some actual dimension of the turbine; the length D1 is simply meant to represent the overall size (or scale) of the turbine. Since the design is optimized for the specific conditions, we can state that ˙ T,1 = ηT,opt ρ V˙ 1 g H1 W
(2)
where ηT,opt is the optimum (i.e., maximized) efficiency. Say we change the head to some new value H2 : we want to estimate the corresponding conditions V˙ 2 and ω2 which will maintain the optimum efficiency of the turbine. Or, perhaps, we scale the turbine to a new characteristic size D2 : what are the corresponding new values of ω2 , V˙ 2 , H2 which maintain ηT,opt ?
If the conditions at state 2 give the same efficiency as state 1, then Eq. (2) would imply that ˙ T,2 W V˙ 2 H2 = ˙ T,1 W V˙ 1 H1
(3)
The volumetric flow rate V˙ will be proportional to a characteristic velocity in the turbine, V, times a characteristic flow area. The flow area, in turn, would be proportional to the square of the characteristic size, D2 . Therefore, V˙ 2 D2 V2 = 22 (4) D1 V1 V˙ 1 If we neglect viscous effects (i.e., friction losses), Bernoulli’s equation would show that V2 ∼ gH 2 and this implies that ( )2 ( )1/2 V˙ 2 D2 H2 = · D1 H1 V˙ 1
(5)
(6)
Therefore, for the same turbine design operating at the two optimized states 1 and 2, we would expect that ( )2 ( )3/2 ˙ T,2 W D2 H2 = · (7) ˙ T,1 D1 H1 W Now consider the rotation speed of the turbine, ω. If R represents the radius of the turbine runner and U the velocity at this radius, then ω = U/R, in radians per s. For two turbines of the same design, one would expect that U2 /U1 = V2 /V1 and R2 /R1 = D2 /D1 . Using again the head relation for the characteristic velocity V, Eq. (5), we get ( )1/2 ω2 D1 H2 = · (8) ω1 D2 H1 The size ratio D2 /D1 can be eliminated between Eqs. (7) and (8), and after rearranging, ( )1/2 ˙ T,1 ω1 W 5/4
(H1 )
( )1/2 ˙ T,2 ω2 W =
5/4
(H2 )
= constant ≡ Nsp
(9)
The quantity Nsp is referred to as the specific speed of the turbine. Understand that Nsp , as defined above, is not a dimensionless quantity – we would need to appropriately include ρ and g to cancel out the units (see the text). The important point of Eq. (9) is that a turbine operating at it’s optimum design conditions would have a constant value of Nsp . Figure 3.9 in the book presents a plot of turbine efficiency ηT as a function of specific speed Nsp . The
units of Nsp , which are not shown on the plot, would ˙ , correspond to the conventional definition in which W ω, and H are in hp, RPM, and ft, respectively. The plot shows characteristic curves for the three main designs of water turbines: Pelton (impulse), and Francis and Kaplan (reactive). Please pay attention to the fact that Nsp is presented on a log scale. Figure 3.9 can be used to select an appropriate turbine for a given hydropower application. In most hydropower design problems one would typically know beforehand the available head H. The total available flowrate V˙ would also be known beforehand, which would correspond to some fraction of the total river flow. If this flow is sent through a single ˙ T produced by the turbine, turbine, then the power W assuming an efficiency of 1 and no head losses, could be estimated from Eq. (2). Figure 3.9 could then be used, in conjunction with the head H and the es˙ T , to determine the corresponding timated power W rotational speeds of Pelton, Francis, and Kaplan turbines operating at their maximum efficiency, i.e., ω=
H 5/4 Nsp,opt ˙ 1/2 W T
Experience would then tell the engineer which of the calculated rotation rates, for the three turbine types, is best suited for practical use. Of course, none of the three ω values calculated via this procedure may be feasible. An alternative approach is to specify beforehand the desired rotation rate ω of the turbine. The power produced by the three turbine types, operating at optimum efficiency, would then be obtained by 5/2 2 ˙T = H W Nsp,opt ω2
(11)
The flow rate through the turbine could then be calculated from Eq. (2), and the number of required turbine units would be obtained from the total available flowrate divided by the flow through a single turbine.
In general, impulse turbines are most suitable for high head situations, for which the head can be converted into a high speed jet. Francis turbines are (10) used for intermediate heads, and Kaplan for the lowest heads.
