Problem Statement
A turbine is to operate under a head of 30 m at 300 r.p.m. The discharge is 10 m³/s. If the efficiency is 90%, determine : (i) specific speed of the machine, (ii) power generated, and (iii) types of the turbine.
Given Data & Constants
- Head, \(H = 30 \, \text{m}\)
- Speed, \(N = 300 \, \text{r.p.m.}\)
- Discharge, \(Q = 10 \, \text{m}^3/\text{s}\)
- Overall efficiency, \(\eta_o = 90\% = 0.90\)
- Density of water, \(\rho = 1000 \, \text{kg/m}^3\)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)
Solution
(ii) Power Generated (Shaft Power)
First, we calculate the total power available from the water (Water Power), and then use the overall efficiency to find the shaft power.
(i) Specific Speed of the Machine (\(N_s\))
The specific speed is calculated using the rotational speed, the shaft power generated, and the head.
(iii) Type of the Turbine
The type of turbine is determined by the range in which its specific speed falls.
Since our calculated value of 205 falls squarely within the Francis turbine range, this is the appropriate turbine type.
(i) Specific speed of the machine: \( \approx 205 \)
(ii) Power generated: \( \approx 2648.7 \, \text{kW} \)
(iii) Type of the turbine: Francis Turbine
Explanation of Specific Speed
Specific Speed (\(N_s\)) is a crucial, dimensionless parameter used to classify turbomachinery. It represents the speed at which a geometrically similar turbine would have to run to produce one unit of power (e.g., 1 kW) under one unit of head (e.g., 1 m).
Its primary importance is in turbine selection. Different turbine designs are efficient under different combinations of head and flow rate. The specific speed value effectively summarizes this relationship:
- Low Specific Speed (e.g., < 60): Indicates a high-head, low-flow application. This is the domain of the Pelton (impulse) turbine.
- Medium Specific Speed (e.g., 60 - 300): Indicates a medium-head, medium-flow application. This is ideal for a Francis (reaction) turbine.
- High Specific Speed (e.g., > 300): Indicates a low-head, high-flow application. This requires a Kaplan or Propeller (axial-flow) turbine.
By calculating the specific speed for the given conditions, we can confidently select the most efficient type of turbine for the job.
