Answer :
Final answer:
To calculate the horsepower required for an air conditioner motor, given its cooling capacity and efficiency relative to a Carnot refrigerator, temperatures are converted to Kelvin, the Carnot COP is calculated, and the actual COP is determined. The cooling capacity is converted to watts, and from this, the work input in watts is calculated. Finally, this is converted to horsepower, resulting in approximately 0.263 HP required.
Explanation:
The question asks for the horsepower required for an air conditioner motor with specific conditions, including a cooling capacity of 4000 Btu/h and a coefficient of performance (COP) that is 27% of a Carnot refrigerator's COP, operating between 93°F and 70°F. First, we convert temperatures from Fahrenheit to Kelvin: 93°F = 307.6K and 70°F = 294.3K. The Carnot COP for a refrigerator is given by COPCarnot = TL / (TH - TL) = 294.3 / (307.6 - 294.3) = 294.3 / 13.3 = 22.12. The actual COP of the air conditioner is 27% of this, so COP = 0.27 × 22.12 = 5.97.
To find the work input (W) required, we first convert the cooling capacity from Btu/h to watts: 4000 Btu/h = 1172.47 watts. Since COP = QL / W, solving for W gives W = QL / COP = 1172.47 / 5.97 = 196.45 watts. Converting this to horsepower (since 1 horsepower = 746 watts) gives the horsepower required of the air conditioner motor as 196.45 / 746 = 0.263 horsepower.
Answer:[tex]P_{in}=0.25\ Hp[/tex]
Explanation:
Given
Operating temperatures are
[tex]T_L=70^{\circ}F\approx 294.261\ K[/tex]
[tex]T_H=93^{\circ}F\approx 307\ K[/tex]
[tex]COP[/tex] is [tex]27\% [/tex] of Carnot refrigerator
[tex]COP_{carnot}=\dfrac{T_L}{T_H-T_L}[/tex]
[tex]COP_{carnot}=\dfrac{294.261}{307-294.261}[/tex]
[tex]COP_{carnot}=\dfrac{294.261}{12.74}[/tex]
[tex]COP_{carnot}=23.09\approx 23.1[/tex]
actual [tex]COP=0.27 \times COP_{carnot}[/tex]
[tex]COP_{actual}=6.237[/tex]
Also, [tex]COP=\dfrac{\text{Desired effect}}{\text{Power in}}[/tex]
[tex]6.237=\frac{4000}{P_{in}}[/tex]
[tex]P_{in}=\frac{4000}{6.237}=641.33\ Btu/h[/tex]
[tex]P_{in}=0.25\ Hp[/tex]
