Short Answer:

The specific speed of a turbine is the speed at which a geometrically similar turbine would run if it were scaled to develop one unit of power under one unit of head. It is a dimensionless parameter used to compare different types of turbines and to select the most suitable turbine for a given head and discharge. The specific speed helps in determining whether a turbine is of impulse or reaction type.

Mathematically, it is expressed as:
Ns = N × √P / H^(5/4)
where, Ns = Specific speed, N = Speed of turbine (rpm), P = Power developed (kW), and H = Head (m).

Detailed Explanation :

Specific Speed of a Turbine

The specific speed of a turbine is an important characteristic parameter used in the design and selection of hydraulic turbines. It provides a relationship between the speed, power, and head of a turbine and enables comparison between turbines of different sizes and types. It helps engineers select the right type of turbine for a given site condition, based on available head and flow rate.

1. Meaning of Specific Speed:
   Specific speed can be defined as the speed of a geometrically similar turbine that would produce one unit of power when working under one unit of head.It is a dimensionless number but is often expressed in a numerical form that depends on chosen units. It provides a standardized way to compare the performance of turbines, regardless of their actual physical dimensions.

In simpler terms, the specific speed indicates how the speed of a turbine changes with different heads and power outputs. It helps to understand the nature of flow within the turbine and guides the designer in choosing the correct turbine type—Pelton, Francis, or Kaplan—for a particular head and discharge.

2. Mathematical Formula:
   The formula for the specific speed of a turbine is given by:

Where:

•  = Specific speed of the turbine
•  = Rotational speed of turbine (in revolutions per minute, rpm)
•  = Power produced by the turbine (in kilowatts, kW)
•  = Net head available at the turbine (in meters)

This formula indicates that the specific speed depends on the turbine’s actual speed, the power it generates, and the available head.

3. Significance of Specific Speed:
   Specific speed is used to classify turbines based on their speed and the type of energy conversion that occurs inside them. It helps in selecting the turbine best suited for a given water head and discharge. The significance of specific speed can be summarized as follows:

• It is used to compare turbines of different sizes and capacities.
• It helps to select the right turbine for a particular head and discharge condition.
• It provides information about the shape and flow characteristics of the turbine.
• It assists in predicting turbine performance under various load and head conditions.

A turbine with low specific speed will have high head and low discharge (e.g., Pelton wheel), while a turbine with high specific speed will have low head and high discharge (e.g., Kaplan turbine).

4. Types of Turbines Based on Specific Speed:
   The value of specific speed helps to classify turbines into different types:

• Low Specific Speed (Below 50):
  These turbines are used for high heads and low discharges. Example: Pelton wheel turbine (Impulse turbine).
• Medium Specific Speed (50 to 250):
  These are used for medium heads and medium discharges. Example: Francis turbine (Reaction turbine).
• High Specific Speed (Above 250):
  These turbines are suitable for low heads and high discharges. Example: Kaplan and propeller turbines (Reaction turbines).

Thus, specific speed directly indicates the type of turbine suitable for a given site depending on available head and discharge conditions.

5. Importance in Turbine Design and Selection:
   Specific speed is a key factor in turbine selection for hydroelectric power plants. It determines the type of runner, shape of blades, and design of casing. For instance:

• Sites with high heads and low discharge require Pelton turbines, which have small runners rotating at low speeds.
• Sites with medium heads require Francis turbines, which are medium-speed reaction turbines.
• Sites with low heads and large discharge need Kaplan turbines, which are high-speed axial-flow turbines.

Hence, specific speed provides engineers with a direct clue to match turbine type with site characteristics, ensuring efficient power generation.

6. Dimensional Analysis:
   Specific speed is derived from the laws of similarity between turbines. If turbines are geometrically similar and operate under dynamically similar conditions, their performance can be compared using dimensionless parameters such as discharge coefficient, power coefficient, and specific speed.

Although the expression of specific speed contains dimensional quantities, it becomes dimensionless when derived using consistent units and proper scaling laws.

7. Example:
   Suppose a turbine develops 1000 kW power under a head of 100 meters and runs at a speed of 300 rpm.
   Then,

 

Hence, the specific speed is 30, which indicates that it is a Pelton wheel turbine suitable for high head and low discharge conditions.

Conclusion:

In conclusion, the specific speed of a turbine is a very important factor in hydraulic turbine design and selection. It indicates how the turbine will behave under different head and discharge conditions. It helps classify turbines into impulse and reaction types and ensures the right turbine is selected for a particular site. By knowing specific speed, engineers can design turbines that operate efficiently, economically, and reliably under given water flow conditions.