The diameter and stroke length of a single-acting reciprocating pump are 100 mm and 200 mm respectively. The lengths of suction and delivery pipes are 10 m and 30 m respectively and their diameters are 50 mm. If the pump is running at 30 r.p.m. and suction and delivery heads are 3.5 m and 20 m respectively, find the pressure head in the cylinder : (i) at the beginning of the suction and delivery stroke, (ii) in the middle of suction and delivery stroke, and (iii) at the end of the suction and delivery stroke. Take atmospheric pressure head = 10.3 m of water and co-efficient of friction = .009 for both pipes.

Reciprocating Pump Full Cycle Analysis

Problem Statement

Given Data & Constants

Cylinder diameter, $D = 100 \, \text{mm} = 0.1 \, \text{m}$
Stroke length, $L = 200 \, \text{mm} = 0.2 \, \text{m}$
Speed, $N = 30 \, \text{r.p.m.}$
Static suction head, $h_s = 3.5 \, \text{m}$
Static delivery head, $h_d = 20 \, \text{m}$
Suction pipe: $l_s = 10 \, \text{m}$, $d_s = 50 \, \text{mm} = 0.05 \, \text{m}$
Delivery pipe: $l_d = 30 \, \text{m}$, $d_d = 50 \, \text{mm} = 0.05 \, \text{m}$
Atmospheric pressure head, $H_{atm} = 10.3 \, \text{m}$
Friction factor, $f = 0.009$
Acceleration due to gravity, $g = 9.81 \, \text{m/s}^2$

Solution

1. Calculate Common Parameters

$$ \text{Crank radius, } r = L/2 = 0.2/2 = 0.1 \, \text{m} $$ $$ \text{Angular velocity, } \omega = \frac{2 \pi N}{60} = \frac{2 \pi \times 30}{60} = \pi \approx 3.142 \, \text{rad/s} $$ $$ \text{Cylinder area, } A = \frac{\pi}{4}D^2 = \frac{\pi}{4}(0.1)^2 \approx 0.007854 \, \text{m}^2 $$ $$ \text{Pipe area, } a = \frac{\pi}{4}d^2 = \frac{\pi}{4}(0.05)^2 \approx 0.001963 \, \text{m}^2 $$

2. Calculate Head Losses and Gains

Acceleration Head (Suction):

$$ h_{as} = \frac{l_s}{g} \frac{A}{a} \omega^2 r = \frac{10}{9.81} \times \frac{0.007854}{0.001963} \times (\pi)^2 \times 0.1 \approx 4.0 \, \text{m} $$

Acceleration Head (Delivery):

$$ h_{ad} = \frac{l_d}{g} \frac{A}{a} \omega^2 r = \frac{30}{9.81} \times \frac{0.007854}{0.001963} \times (\pi)^2 \times 0.1 \approx 12.0 \, \text{m} $$

Friction Head (Suction & Delivery - at mid-stroke):

$$ \text{Velocity in pipe, } v = \frac{A}{a} \omega r = \frac{0.007854}{0.001963} \times \pi \times 0.1 \approx 1.257 \, \text{m/s} $$ $$ h_{fs} = \frac{4 f l_s v^2}{2 g d_s} = \frac{4 \times 0.009 \times 10 \times (1.257)^2}{2 \times 9.81 \times 0.05} \approx 0.579 \, \text{m} $$ $$ h_{fd} = \frac{4 f l_d v^2}{2 g d_d} = \frac{4 \times 0.009 \times 30 \times (1.257)^2}{2 \times 9.81 \times 0.05} \approx 1.737 \, \text{m} $$

3. Calculate Absolute Pressure Head in Cylinder

(i) At the Beginning of Strokes:

$$ H_{suction, begin} = H_{atm} - h_s - h_{as} = 10.3 - 3.5 - 4.0 = 2.8 \, \text{m} \, (\text{abs}) $$ $$ H_{delivery, begin} = H_{atm} + h_d + h_{ad} = 10.3 + 20 + 12.0 = 42.3 \, \text{m} \, (\text{abs}) $$

(ii) In the Middle of Strokes:

$$ H_{suction, middle} = H_{atm} - h_s - h_{fs} = 10.3 - 3.5 - 0.579 = 6.221 \, \text{m} \, (\text{abs}) $$ $$ H_{delivery, middle} = H_{atm} + h_d + h_{fd} = 10.3 + 20 + 1.737 = 32.037 \, \text{m} \, (\text{abs}) $$

(iii) At the End of Strokes:

$$ H_{suction, end} = H_{atm} - h_s + h_{as} = 10.3 - 3.5 + 4.0 = 10.8 \, \text{m} \, (\text{abs}) $$ $$ H_{delivery, end} = H_{atm} + h_d - h_{ad} = 10.3 + 20 - 12.0 = 18.3 \, \text{m} \, (\text{abs}) $$

Final Results:

Beginning of Stroke: Suction $ \approx 2.8$ m (abs), Delivery $ \approx 42.3$ m (abs)

Middle of Stroke: Suction $ \approx 6.22$ m (abs), Delivery $ \approx 32.04$ m (abs)

End of Stroke: Suction $ \approx 10.8$ m (abs), Delivery $ \approx 18.3$ m (abs)

Explanation of Pressure Dynamics

The pressure inside the pump cylinder is constantly changing throughout the suction and delivery strokes. The absolute pressure head is determined by the atmospheric pressure, the static head (lift), and the dynamic heads caused by acceleration and friction.

Suction Stroke: The pressure in the cylinder is below atmospheric pressure. The static lift ($h_s$) and dynamic heads ($h_{as}, h_{fs}$) are subtracted from the atmospheric head.
Delivery Stroke: The pressure in the cylinder is above atmospheric pressure. The static lift ($h_d$) and dynamic heads ($h_{ad}, h_{fd}$) are added to the atmospheric head.
Acceleration Head ($h_a$): This is required to accelerate the water column. It is maximum at the start of a stroke and zero at mid-stroke. It opposes the piston's motion at the beginning (making suction pressure lower and delivery pressure higher) and aids it at the end (making suction pressure higher and delivery pressure lower).
Friction Head ($h_f$): This is required to overcome pipe friction. It is zero at the start and end of a stroke (when velocity is zero) and maximum at mid-stroke when velocity is highest. It always opposes the flow.