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The pressure of a three-piston, single-acting pump is 800 kPa during its delivery stroke. Its plunger diameter is 220 mm and it has a stroke length of 380 mm. Calculate: a) The total force required on the piston. b) The work done during 10 delivery strokes.

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2 parts
a

30410.4 N\text{30410.4 N}

b

346678.56 J\text{346678.56 J}

Step-by-step solution

6 steps

Here's how to solve the problem:

Part a· 3 steps

The total force required on the piston.

  1. Convert the given values to SI units.

    • Pressure, P=800 kPa=800×103 PaP = 800 \text{ kPa} = 800 \times 10^3 \text{ Pa}
    • Plunger diameter, D=220 mm=0.220 mD = 220 \text{ mm} = 0.220 \text{ m}

  2. Calculate the cross-sectional area of one piston.

    The area (AA) of a circular piston is given by the formula A=πD24A = \frac{\pi D^2}{4}.

    A=π(0.220 m)24A = \frac{\pi (0.220 \text{ m})^2}{4} A=π×0.0484 m24A = \frac{\pi \times 0.0484 \text{ m}^2}{4} A=0.0121π m2A = 0.0121\pi \text{ m}^2 A≈0.038013 m2A \approx 0.038013 \text{ m}^2

  3. Calculate the force required on one piston.

    The force (FF) is calculated using the formula F=P×AF = P \times A.

    F=(800×103 Pa)×(0.038013 m2)F = (800 \times 10^3 \text{ Pa}) \times (0.038013 \text{ m}^2) F=30410.4 NF = 30410.4 \text{ N}

    The total force required on one piston is 30410.4 N\boxed{\text{30410.4 N}}.

Part b

346678.56 J\text{346678.56 J}

The work done during 10 delivery strokes.

  1. Convert the stroke length to SI units.

    • Stroke length, L=380 mm=0.380 mL = 380 \text{ mm} = 0.380 \text{ m}

  2. Calculate the work done by one piston during one delivery stroke.

    Work done (WW) is calculated using the formula W=F×LW = F \times L.

    Wone_stroke=(30410.4 N)×(0.380 m)W_{one\_stroke} = (30410.4 \text{ N}) \times (0.380 \text{ m}) Wone_stroke=11555.952 JW_{one\_stroke} = 11555.952 \text{ J}

  3. Calculate the total work done by the three-piston pump during 10 delivery strokes.

    Since there are three pistons and 10 delivery strokes, the total work done is:

    Wtotal=Wone_stroke×number of pistons×number of strokesW_{total} = W_{one\_stroke} \times \text{number of pistons} \times \text{number of strokes} Wtotal=11555.952 J×3×10W_{total} = 11555.952 \text{ J} \times 3 \times 10 Wtotal=346678.56 JW_{total} = 346678.56 \text{ J}

    The work done during 10 delivery strokes is 346678.56 J\boxed{\text{346678.56 J}}.

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