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A single-acting reciprocating compressor operates at 150 rpm with initial air conditions of 97.9 kPaa and 27 °C, discharging air at 379 kPaa to a cylindrical tank. The bore and stroke are 355 mm and 381 mm, respectively, with a percentage clearance of 5%.

Given that the surrounding air is at 100 kPaa and 20 °C and the compression and expansion process follows the equation \(pv^{1.3} = C\), determine the following:

a) Free air capacity in CFM

b) Power in kW

Answer :

To solve this problem, we have to follow several steps for both parts: determining the Free Air Capacity (CFM) and the power in kW.

Given Data:

  • Initial pressure, [tex]P_1 = 97.9 \, \text{kPa}[/tex]
  • Initial temperature, [tex]T_1 = 27^\circ C = 300.15 \, \text{K}[/tex]
  • Discharge pressure, [tex]P_2 = 379 \, \text{kPa}[/tex]
  • Atmospheric pressure, [tex]P_{atm} = 100 \, \text{kPa}[/tex]
  • Atmospheric temperature, [tex]T_{atm} = 20^\circ C = 293.15 \, \text{K}[/tex]
  • Bore, [tex]D = 355 \, \text{mm} = 0.355 \, \text{m}[/tex]
  • Stroke, [tex]L = 381 \, \text{mm} = 0.381 \, \text{m}[/tex]
  • Clearance volume percentage, [tex]C = 5\% = 0.05[/tex]
  • Speed, [tex]N = 150 \, \text{rpm}[/tex]
  • Compression and expansion process: [tex]pv^{1.3} = C[/tex]

(a) Free Air Capacity (CFM)

First, compute the swept volume ([tex]V_s[/tex]) and clearance volume ([tex]V_c[/tex]):

[tex]V_s = \frac{\pi}{4} D^2 L[/tex]
[tex]V_c = C \times V_s[/tex]

Next, determine the volume at the start of compression ([tex]V_1[/tex]) and discharge ([tex]V_2[/tex]):

[tex]V_1 = V_s + V_c[/tex]
[tex]\frac{V_1}{V_2} = \left(\frac{P_2}{P_1}\right)^{1/n} \text{, where } n = 1.3[/tex]

Calculate [tex]V_2[/tex], when rearranged:

[tex]V_2 = \frac{V_1}{\left(\frac{P_2}{P_1}\right)^{1/n}}[/tex]

The swept volume during discharge (9) is [tex]V_s - V_2[/tex], and converting to cubic feet per minute (CFM):

[tex]\text{Free Air Delivery (FAD)} = \text{Volumetric Efficiency} \times N \times V_s \times \frac{1}{35.315} \text{ (Convert \(m^3/min\) to CFM)}[/tex]

Volumetric efficiency is calculated considering clearance volumes.

(b) Power in kW

The indicated power for compression is given by:

[tex]\text{Indicated Power } = \frac{N}{60} \times \text{Number of Cylinders} \times \left( \frac{P_1V_1^n - P_2V_2^n}{n-1} \right)[/tex]
Then convert from [tex]\text{J}[/tex] to [tex]\text{kW}[/tex]:

Finally, compute real power considering mechanical efficiency.

Note, each of these steps involves substituting known quantities and solving the mathematical expressions. The final result provides FAD in CFM and required power in kW.

It's important to ensure unit consistency and proper conversion, especially from cubic meters to cubic feet and from pressure/volume units. Complex equations and intermediate results should be thoroughly checked for accuracy.

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