The diameters of an impeller of a centrifugal pump at inlet and outlet are 20 cm and 40 cm respectively. Determine the minimum speed for starting the pump if it works against a head of 25 m.

Pump Minimum Starting Speed Calculation

Problem Statement

Given Data & Constants

Inlet diameter, $D_1 = 20 \, \text{cm} = 0.2 \, \text{m}$
Outlet diameter, $D_2 = 40 \, \text{cm} = 0.4 \, \text{m}$
Head, $H = 25 \, \text{m}$
Acceleration due to gravity, $g = 9.81 \, \text{m/s}^2$

Solution

1. Principle of Minimum Starting Speed

For the pump to start delivering water, the centrifugal head generated by the rotating impeller must be at least equal to the head it is working against. At the point of starting, there is no flow, and the water in the impeller rotates as a forced vortex.

The head generated by a forced vortex is given by:

$$ H = \frac{u_2^2 - u_1^2}{2g} $$

2. Express Tangential Velocities in Terms of Speed (N)

The tangential velocities $u_1$ and $u_2$ are functions of the rotational speed $N$ (in r.p.m.) and the diameters.

$$ u_1 = \frac{\pi D_1 N}{60} \quad \text{and} \quad u_2 = \frac{\pi D_2 N}{60} $$

3. Solve for the Minimum Speed (N)

Substitute the expressions for $u_1$ and $u_2$ into the head equation.

$$ H = \frac{1}{2g} \left[ \left(\frac{\pi D_2 N}{60}\right)^2 - \left(\frac{\pi D_1 N}{60}\right)^2 \right] $$ $$ H = \frac{1}{2g} \left(\frac{\pi N}{60}\right)^2 (D_2^2 - D_1^2) $$ $$ \text{Rearranging to solve for N:} $$ $$ N^2 = \frac{2gH \times 60^2}{\pi^2 (D_2^2 - D_1^2)} $$ $$ N = \sqrt{\frac{2 \times 9.81 \times 25 \times 3600}{\pi^2 (0.4^2 - 0.2^2)}} $$ $$ N = \sqrt{\frac{1765800}{\pi^2 (0.16 - 0.04)}} = \sqrt{\frac{1765800}{\pi^2 (0.12)}} $$ $$ N = \sqrt{\frac{1765800}{1.18435}} \approx \sqrt{1490915} $$ $$ N \approx 1221 \, \text{r.p.m.} $$

Final Result:

The minimum speed for starting the pump is approximately $1221 \, \text{r.p.m.}$

Explanation of the "Forced Vortex" Concept

When the pump motor is switched on, but before the water starts to flow, the impeller spins the water trapped inside it. This creates a condition known as a forced vortex, where the entire fluid mass rotates as a solid body. In a forced vortex, the pressure increases with the square of the radius from the center of rotation.

This pressure difference between the outlet ($D_2$) and the inlet ($D_1$) of the impeller creates a "centrifugal head." Pumping can only begin when this internally generated head is large enough to overcome the external head of the system (25 m in this case). If the speed is too low, the centrifugal head will be insufficient to push water out against the existing pressure.

The diameters of an impeller of a centrifugal pump at inlet and outlet are 20 cm and 40 cm respectively. Determine the minimum speed for starting the pump if it works against a head of 25 m.

Problem Statement

Given Data & Constants

Solution

1. Principle of Minimum Starting Speed

2. Express Tangential Velocities in Terms of Speed (N)

3. Solve for the Minimum Speed (N)

Explanation of the "Forced Vortex" Concept

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