Answer :
Final answer:
The horsepower required for the air conditioner motor would be determined by first calculating the Carnot Coefficient of Performance (COP) using the given temperatures, then applying the given efficiency of 27% to find the real COP. Using this COP and the cooling capacity, we can calculate the required work input which can then be converted to horsepower.
Explanation:
Let's first understand the role of an air conditioner and the meaning of the Carnot refrigerator. An air conditioner operates as a heat engine but in the reverse order, it moves heat from a cold environment to a hotter one, thus cooling the lower temperature environment. A Carnot refrigerator represents an ideal heat-processing engine operated in reverse, which sets an upper limit to the performance any real refrigerator can achieve.
The coefficient of performance (COP) for refrigeration normally ranges from 2 to 6, with smaller temperature differences achieving higher values. In this question, however, the coefficient of performance for your air conditioner is described as being 27% of that of a Carnot refrigerator operating between the same two temperatures.
To find the horsepower required for the air conditioner motor, first convert the working temperatures to the Kelvin scale. 91°F = 305.9 K and 77°F = 298.2 K. The Carnot COP can be calculated using the ratio Th/(Th-To) where Th is the hot temperature and To is the cold temperature. Applying the given 27% efficiency to the Carnot COP will let us estimate the real COP of the air conditioner. Knowing the air conditioner's cooling capacity (4000 BTUs/hr = 1055055.85 Joules/hr), we can then compute the necessary work input in joules by manipulating the COP equation. To convert the obtained work input from watts to horsepower, use the conversion factor 746 Watts = 1 Horsepower.
Reminder: Be cautious with unit conversions throughout the process to ensure accuracy.
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