Franco Podio, Gilfredo Cavagnolo and Enrico Maria Cipriano
November 13, 2012

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In many instances, the plant was rebuilt because the configuration had changed from horizontal to vertical or complete replacement was more convenient.
— Pelton: Minimum parameters needed are the power, number of jets and speed. Studies carried out in the 1980s3 provide a relationship between the total turbine weight (including the main inlet valve) and the parameters of power, number of jets and rated speed. A similar relationship was discovered with regard to supply costs from 2000 to 2010 (see Figure 2).
— Francis and reversible single stage: Minimum parameters needed are power and speed. From the 1980s analysis,^{3} a relationship between total weight and these parameters was found. A more recent analysis was carried out on Francis machines by evaluating a larger number of cases, and the validity of the previous model was confirmed (see Figure 3).
— Pumps and reversible multiple stage: Minimum parameters needed are power, number of stages and speed. Again, in the 1980s analysis,^{3} a relationship between weight and these parameters was found. Because of the peculiarity of the machines and the small number of ENEL plants in which they are installed, the analysis of the historical results could not be confirmed.
— Kaplan: Minimum parameters are power and speed. From 2000 to 2010, by using the parameters obtained from plant renovation, it was possible to perform an analysis comparable to those of the other machines, obtaining a similar expression.
Generators
The synchronous generator (vertical axis) rotor weight may be expressed as:^{4}
R = 50 x (A/n^{0.5})^{0.74}
where:
— A is apparent power in kVA;
— n is rated speed in rpm; and
— R is rotor weight in tons.
The generator total weight estimate depends on the following relationship:^{5}
W = 107.25 x (A/n^{0.5})^{0.74}
A further analysis was made starting from a draft of an ENEL study from the 1990s. In a diagram of the cost by power unit in function of the power and of the cost by weight unit in function of the total weight of a series of vertical machines, ENEL found the following relationships:
Cost/MVA = k x A^{0.428}
Cost/kg = h x M^{0.229}
where:
— Proportion coefficients k and h are a function of the rotating speed.
Since:
M/A = (k/h)^{1.297} x A^{0.428}/M^{0.229}
Then:
M_{(t)} = K_{j} x A_{(MVA)}^{0.742}
where:
— K_{j} is a function of k and h and the speed. This parameter can be represented by K_{j} = 1837 x n^{0.808}
The total mass of the generator can be written again as:
M_{(t)} = 1837 x (A/n^{1.09})^{0.742} ≈ (A/n)^{0.74}
This expression places the total mass in relationship with the ratio A/n and consequently with the ratio P/n (torque), as it is for the hydraulic machines. By applying this relationship to the recent years' supplies, a better data interpolation results (see Figure 4).
Secondary parameters include inertia, temperature class, power factor, short circuit ratio, subtransient reactance and efficiency. An accurate analysis of the secondary parameters can be found.^{6}
As a consequence of analyses carried out over many years, ENEL can summarize the simple method used to evaluate the costs as seen below.
For hydraulic machines, the functions found with interpolation are:
Francis: Weight_{F} = K_{F} x (Pmt/n)^{0.6}
Kaplan: Weight_{K} = K_{K} x (Pmt/n)^{0.6}
Pelton: Weight_{P} = K_{P} x j x (Pmt/j/n)^{0.56}
Reversible single stage: Weight_{RS} = K_{RS} x (Pmt/n)^{0.6} = 1.4 Weight_{F}
Reversible multistage and pumps: Weight_{RP} = K_{RP} x s x (Pmp/s/n)^{0.54}
where:
— Pmt is turbine power in kW;
— n is the rated speed in rpm;
— j is the number of jets;
— Pmp is mechanical pump power in kW; and
— s is number of stages for reversible machines or pumps.
The total weight is expressed in tons.
For electric machines, ENEL assumed as fundamental parameters the apparent power, speed and axis configuration (vertical or horizontal) and set the value of K = 40 for horizontal axis and K = 50 for vertical axis generators.
The total machine weight can be obtained as follows:
Weight_{SG} = K x (An/n^{0.5})^{0.74} x 2.145
For those models having the purpose of evaluating the total cost of the machines with the variation in power in a large field, it is necessary to consider that the specific cost by unit of weight or power is not a constant but increases with the decrease in power of the machine.
This function formulation has been deduced from what can be analyzed in previously published information.^{7}
In a first approximation, this function was considered equal for the hydraulic machinery and electric machinery.
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