A single-acting reciprocating pump running at 30 r.p.m., delivers 0.012 m³/s of water. The diameter of the piston is 25 cm and stroke length is 50 cm. Determine : (i) The theoretical discharge of the pump, (ii) Co-efficient of discharge, and (iii) Slip and percentage slip of the pump.

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Reciprocating Pump Performance Analysis

Problem Statement

A single-acting reciprocating pump running at 30 r.p.m., delivers 0.012 m³/s of water. The diameter of the piston is 25 cm and stroke length is 50 cm. Determine : (i) The theoretical discharge of the pump, (ii) Co-efficient of discharge, and (iii) Slip and percentage slip of the pump.

Given Data & Constants

  • Speed, \(N = 30 \, \text{r.p.m.}\)
  • Actual discharge, \(Q_{\text{act}} = 0.012 \, \text{m}^3/\text{s}\)
  • Piston diameter, \(D = 25 \, \text{cm} = 0.25 \, \text{m}\)
  • Stroke length, \(L = 50 \, \text{cm} = 0.5 \, \text{m}\)

Solution

(i) The Theoretical Discharge of the Pump (\(Q_{\text{th}}\))

First, we calculate the area of the piston. Then, we use the formula for theoretical discharge for a single-acting pump.

$$ \text{Area of piston, } A = \frac{\pi}{4} D^2 = \frac{\pi}{4} (0.25)^2 \approx 0.049087 \, \text{m}^2 $$ $$ Q_{\text{th}} = \frac{A \cdot L \cdot N}{60} $$ $$ Q_{\text{th}} = \frac{0.049087 \, \text{m}^2 \times 0.5 \, \text{m} \times 30 \, \text{r.p.m.}}{60} $$ $$ Q_{\text{th}} \approx 0.01227 \, \text{m}^3/\text{s} $$

(ii) Co-efficient of Discharge (\(C_d\))

The co-efficient of discharge is the ratio of the actual discharge to the theoretical discharge.

$$ C_d = \frac{Q_{\text{act}}}{Q_{\text{th}}} $$ $$ C_d = \frac{0.012}{0.01227} \approx 0.978 $$

(iii) Slip and Percentage Slip of the Pump

Slip is the difference between the theoretical and actual discharge. Percentage slip expresses this as a fraction of the theoretical discharge.

$$ \text{Slip} = Q_{\text{th}} - Q_{\text{act}} $$ $$ \text{Slip} = 0.01227 - 0.012 = 0.00027 \, \text{m}^3/\text{s} $$ $$ \text{Percentage Slip} = \frac{\text{Slip}}{Q_{\text{th}}} \times 100 = \frac{0.00027}{0.01227} \times 100 $$ $$ \text{Percentage Slip} \approx 2.2\% $$
Final Results:

(i) Theoretical Discharge: \( Q_{\text{th}} \approx 0.01227 \, \text{m}^3/\text{s} \) (or 12.27 L/s)

(ii) Co-efficient of Discharge: \( C_d \approx 0.978 \)

(iii) Slip: \( 0.00027 \, \text{m}^3/\text{s} \), Percentage Slip: \( \approx 2.2\% \)

Explanation of Key Terms

Theoretical Discharge (\(Q_{\text{th}}\)): This is the volume of fluid that would be displaced by the piston in a perfect pump with no leakage. It's calculated based purely on the pump's geometry (piston area, stroke length) and its speed.

Co-efficient of Discharge (\(C_d\)): This is a measure of the pump's volumetric efficiency. It compares the actual volume of fluid delivered to the theoretical volume. A value less than 1 indicates that some fluid is lost, typically due to internal leakage (slip).

Slip: This is the difference between the theoretical and actual discharge, representing the volume of fluid that "slips" past the piston or valves back to the low-pressure side during each cycle. Positive slip (as seen here) is normal. Negative slip can occur in some situations, such as with a long suction pipe and low delivery head, where fluid inertia pushes more water out than the piston displaces.

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