Discharge Through Runner Calculator

Volume Flow Rate Formula:

\[ Q = \frac{\pi}{4} \times (D_o^2 - D_b^2) \times V_{fi} \]

Volume Flow Rate (Q):

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1. What is the Volume Flow Rate Formula?

The volume flow rate formula calculates the discharge through a runner using the cross-sectional area and flow velocity. It is essential for turbine design and fluid dynamics analysis in hydraulic systems.

2. How Does the Calculator Work?

The calculator uses the volume flow rate equation:

\[ Q = \frac{\pi}{4} \times (D_o^2 - D_b^2) \times V_{fi} \]

Where:

\( Q \) — Volume flow rate (m³/s)
\( D_o \) — Outer diameter of runner (m)
\( D_b \) — Diameter of hub (m)
\( V_{fi} \) — Flow velocity at inlet (m/s)
\( \pi \) — Mathematical constant pi (approximately 3.14159)

Explanation: The formula calculates the annular area between the outer runner and hub, then multiplies by the flow velocity to determine the volume flow rate.

3. Importance of Volume Flow Rate Calculation

Details: Accurate volume flow rate calculation is crucial for turbine efficiency analysis, hydraulic system design, and performance optimization in fluid machinery.

4. Using the Calculator

Tips: Enter outer diameter and hub diameter in meters, flow velocity in m/s. All values must be positive, and outer diameter must be greater than hub diameter.

5. Frequently Asked Questions (FAQ)

Q1: What is the significance of the hub diameter?
A: The hub diameter accounts for the central obstruction in the flow path, providing a more accurate calculation of the effective flow area.

Q2: How does flow velocity affect the volume flow rate?
A: Volume flow rate is directly proportional to flow velocity - doubling the velocity doubles the flow rate, assuming constant area.

Q3: What are typical units for volume flow rate?
A: The SI unit is cubic meters per second (m³/s), but other common units include liters per second (L/s) or cubic feet per second (ft³/s).

Q4: When is this formula applicable?
A: This formula is specifically designed for calculating discharge through turbine runners with annular flow passages.

Q5: What assumptions are made in this calculation?
A: The formula assumes uniform flow velocity across the annular area and neglects friction losses and turbulence effects.