1. **State the problem:** We need to find the input horsepower (HP) required to drive a compressor delivering 5,099 SCFM at 139.6 psia with an overall efficiency of 83%. 2. **Formula used:** The power required for a compressor can be estimated using the formula: $$\text{Power input} = \frac{\text{Power output}}{\text{Efficiency}}$$ 3. **Calculate power output:** Power output depends on the flow rate, pressure, and other factors. For air compressors, power output (in HP) can be approximated by: $$\text{Power output} = \frac{144 \times P_1 V}{(k-1) \times \eta_c} \left[ \left(\frac{P_2}{P_1}\right)^{\frac{k-1}{k}} - 1 \right]$$ where: - $P_1$ = inlet pressure (psia) - $P_2$ = outlet pressure (psia) - $V$ = volumetric flow rate (ft³/min) - $k$ = specific heat ratio for air (approximately 1.4) - $\eta_c$ = compressor efficiency (decimal) 4. **Given data:** - $V = 5,099$ SCFM - $P_2 = 139.6$ psia - $P_1 = 14.7$ psia (atmospheric pressure, assuming inlet at atmospheric) - $k = 1.4$ - $\eta_c = 0.83$ 5. **Calculate the pressure ratio:** $$\frac{P_2}{P_1} = \frac{139.6}{14.7} = 9.5$$ 6. **Calculate the term inside the brackets:** $$\left(9.5\right)^{\frac{1.4-1}{1.4}} - 1 = 9.5^{0.2857} - 1$$ Calculate exponent: $$9.5^{0.2857} \approx 1.933$$ So: $$1.933 - 1 = 0.933$$ 7. **Calculate power output:** $$\text{Power output} = \frac{144 \times 14.7 \times 5099}{(1.4-1) \times 0.83} \times 0.933$$ Calculate denominator: $$1.4 - 1 = 0.4$$ Calculate numerator: $$144 \times 14.7 \times 5099 = 10,799,404.8$$ So: $$\text{Power output} = \frac{10,799,404.8}{0.4 \times 0.83} \times 0.933 = \frac{10,799,404.8}{0.332} \times 0.933$$ Calculate division: $$\frac{10,799,404.8}{0.332} \approx 32,530,783.13$$ Multiply by 0.933: $$32,530,783.13 \times 0.933 = 30,345,000$$ 8. **Convert to horsepower:** Since the units are in ft-lb/min, convert to HP by dividing by 33,000: $$\text{HP} = \frac{30,345,000}{33,000} = 919.55$$ 9. **Adjust for efficiency:** $$\text{Input HP} = \frac{919.55}{0.83} = 1107.47$$ **Final answer:** The input horsepower required is approximately **1107.47 HP**.