A single-acting reciprocating pump has piston diameter 15 cm and stroke length 30 cm. The centre of the pump is 5 m above the water level in the sump. The diameter and length of the suction pipe are 10 cm and 8 m respectively. The separation occurs if the absolute pressure head in the cylinder during suction stroke falls below 2.5 m of water. Calculate the maximum speed at which the pump can run without separation. Take atmospheric pressure head = 10.3 m of water.

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Maximum Pump Speed to Prevent Separation

Problem Statement

A single-acting reciprocating pump has piston diameter 15 cm and stroke length 30 cm. The centre of the pump is 5 m above the water level in the sump. The diameter and length of the suction pipe are 10 cm and 8 m respectively. The separation occurs if the absolute pressure head in the cylinder during suction stroke falls below 2.5 m of water. Calculate the maximum speed at which the pump can run without separation. Take atmospheric pressure head = 10.3 m of water.

Given Data & Constants

  • Piston diameter, \(D = 15 \, \text{cm} = 0.15 \, \text{m}\)
  • Stroke length, \(L = 30 \, \text{cm} = 0.3 \, \text{m}\)
  • Static suction head, \(h_s = 5 \, \text{m}\)
  • Suction pipe diameter, \(d_s = 10 \, \text{cm} = 0.1 \, \text{m}\)
  • Suction pipe length, \(l_s = 8 \, \text{m}\)
  • Separation pressure head, \(H_{\text{sep}} = 2.5 \, \text{m}\) (absolute)
  • Atmospheric pressure head, \(H_{\text{atm}} = 10.3 \, \text{m}\)
  • Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)

Solution

1. Condition for Separation

Separation occurs at the beginning of the suction stroke when the pressure in the cylinder is at its minimum. The absolute pressure head in the cylinder at this point is given by:

$$ H_{cyl} = H_{atm} - h_s - h_{as} $$ Where \(h_{as}\) is the acceleration head in the suction pipe. To avoid separation, \(H_{cyl}\) must be greater than or equal to \(H_{sep}\). The maximum speed occurs when \(H_{cyl} = H_{sep}\).

2. Calculate the Maximum Allowable Acceleration Head (\(h_{as}\))

We rearrange the formula to find the maximum acceleration head the system can tolerate before separation occurs.

$$ H_{sep} = H_{atm} - h_s - h_{as} $$ $$ h_{as} = H_{atm} - h_s - H_{sep} $$ $$ h_{as} = 10.3 - 5 - 2.5 = 2.8 \, \text{m} $$

3. Relate Acceleration Head to Pump Speed (\(\omega\))

The formula for acceleration head at the beginning of the stroke is:

$$ h_{as} = \frac{l_s}{g} \times \frac{A}{a_s} \times \omega^2 r $$ $$ \text{Where:} $$ $$ \text{Crank radius, } r = L/2 = 0.3/2 = 0.15 \, \text{m} $$ $$ \text{Cylinder area, } A = \frac{\pi}{4}D^2 = \frac{\pi}{4}(0.15)^2 \approx 0.01767 \, \text{m}^2 $$ $$ \text{Suction pipe area, } a_s = \frac{\pi}{4}d_s^2 = \frac{\pi}{4}(0.1)^2 \approx 0.007854 \, \text{m}^2 $$

Now, we solve for \(\omega^2\):

$$ \omega^2 = \frac{h_{as} \cdot g \cdot a_s}{l_s \cdot A \cdot r} $$ $$ \omega^2 = \frac{2.8 \times 9.81 \times 0.007854}{8 \times 0.01767 \times 0.15} = \frac{0.2155}{0.0212} \approx 10.165 $$ $$ \omega = \sqrt{10.165} \approx 3.188 \, \text{rad/s} $$

4. Convert Angular Velocity (\(\omega\)) to Speed (N)

$$ N = \frac{\omega \times 60}{2 \pi} = \frac{3.188 \times 60}{2 \pi} \approx 30.44 \, \text{r.p.m.} $$
Final Result:

The maximum speed at which the pump can run without separation is approximately \(30.4 \, \text{r.p.m.}\)

Explanation of Separation (Cavitation)

Separation, or cavitation, occurs when the pressure of a liquid drops below its vapor pressure. At this low pressure, the liquid begins to boil, forming small vapor bubbles. In a reciprocating pump, this is most likely to happen at the beginning of the suction stroke for two reasons:

  1. The piston is lifting the water against the static suction head (\(h_s\)).
  2. The piston is accelerating rapidly, which requires a large force (and thus a large pressure drop, \(h_{as}\)) to accelerate the water in the suction pipe.

The combination of these two effects causes the pressure in the cylinder to be at its lowest point. If this pressure falls below the liquid's vapor pressure (given here as a head of 2.5 m absolute), bubbles form. These bubbles collapse violently when they move to higher pressure zones, causing noise, vibration, and severe damage to the pump components. Therefore, there is a strict maximum speed limit for any given pump installation to ensure the pressure never drops this low.

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