Discharge Through Pipe Given Pressure Gradient Calculator

Formula Used:

\[ Q = \frac{\pi}{8\mu} \times R^4 \times \frac{dp}{dr} \]

Discharge in Pipe (Q):

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1. What is the Discharge Through Pipe Formula?

The discharge through pipe formula calculates the volumetric flow rate of fluid through a pipe based on the pressure gradient, pipe radius, and fluid viscosity. This formula is derived from the Hagen-Poiseuille equation for laminar flow in circular pipes.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ Q = \frac{\pi}{8\mu} \times R^4 \times \frac{dp}{dr} \]

Where:

\( Q \) — Discharge in pipe (m³/s)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \mu \) — Dynamic viscosity (Pa·s)
\( R \) — Pipe radius (m)
\( \frac{dp}{dr} \) — Pressure gradient (N/m³)

Explanation: The formula shows that discharge is directly proportional to the fourth power of the pipe radius and the pressure gradient, and inversely proportional to the fluid viscosity.

3. Importance of Discharge Calculation

Details: Accurate discharge calculation is crucial for designing piping systems, determining flow rates in hydraulic systems, and optimizing fluid transport in various engineering applications.

4. Using the Calculator

Tips: Enter dynamic viscosity in Pa·s, pipe radius in meters, and pressure gradient in N/m³. All values must be positive and non-zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of validity for this formula?
A: This formula is valid for laminar flow conditions (Reynolds number < 2000) in circular pipes with Newtonian fluids.

Q2: How does pipe radius affect the discharge?
A: Discharge is proportional to the fourth power of the radius, meaning doubling the radius increases discharge by 16 times.

Q3: What units should be used for input values?
A: Use SI units: viscosity in Pa·s, radius in meters, pressure gradient in N/m³, which gives discharge in m³/s.

Q4: Can this formula be used for turbulent flow?
A: No, this formula is specifically derived for laminar flow conditions. Different equations are needed for turbulent flow.

Q5: What factors affect the accuracy of this calculation?
A: The accuracy depends on precise measurement of viscosity, exact pipe dimensions, and maintaining laminar flow conditions.