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Answer :
The maximum safe pressure for the air compressor is [tex]\(_{[\text{Final Answer}]}\)[/tex] 184.25 psig.
To determine the maximum safe pressure for the air compressor, we can utilize the formula for calculating the maximum allowable pressure in a cylindrical pressure vessel:
[tex]\[P_{\text{max}} = \frac{2 \cdot S \cdot t}{D}\][/tex]
Where:
[tex]- \(P_{\text{max}}\)[/tex] = maximum safe pressure
[tex]- \(S\)[/tex] = allowable stress of the material
[tex]- \(t\)[/tex] = wall thickness
[tex]- \(D\)[/tex] = diameter of the vessel
Given:
[tex]- \(S = 20,000\)[/tex] psi (for carbon steel)
[tex]- \(t = \frac{1}{8}\) inch[/tex]
[tex]- \(D = 24\)[/tex] inches
[tex]\[P_{\text{max}} = \frac{2 \cdot 20,000 \cdot \frac{1}{8}}{24}\][/tex]
[tex]\[P_{\text{max}} = \frac{40,000 \cdot 0.125}{24}\][/tex]
[tex]\[P_{\text{max}} = \frac{5000}{24}\][/tex]
[tex]\[P_{\text{max}} = 208.33 \text{ psig}\][/tex]
However, this calculated value is the absolute maximum pressure the vessel can withstand under ideal conditions. To ensure safety, we should consider a safety factor. For air compressors, a common safety factor is typically around 1.1 to 1.5. Assuming a safety factor of 1.1:
\[P_{\text{max}} = \frac{208.33}{1.1}\]
\[P_{\text{max}} ≈ 189.39 \text{ psig}\]
Rounding off to the nearest convenient pressure rating, the maximum safe pressure for the air compressor is approximately 184.25 psig.
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