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By removing energy through heat transfer from a room, a window air conditioner maintains the room at 20°C on a day when the outside temperature is 28°C.

(a) Determine, in kW per kW of cooling, the minimum theoretical power required by the air conditioner.

(b) To achieve the required rates of heat transfer with practical-sized units, air conditioners typically receive energy by heat transfer at a temperature below that of the room being cooled and discharge energy by heat transfer at a temperature above that of the surroundings. Consider this effect by determining the minimum theoretical power, in kW per kW of cooling, required when [tex]T_C = 16°C[/tex] and [tex]T_H = 32°C[/tex], and determine the ratio of the power for part (b) to the power for part (a).

Answer :

(a) The minimum theoretical power required by the air conditioner 0.134 kW/kW of cooling.

(b) ratio of the power for part (b) to the power for part (a) is: 0.535/0.134 = 3.99

(a) The minimum theoretical power required by the air conditioner can be calculated using the formula:
Power = Q/Δt
Where Q is the heat transfer rate (in kW) and Δt is the temperature difference between the room and outside.
The heat transfer rate can be determined using the formula:
Q = m*Cp*ΔT

Where m is the mass flow rate of air (in kg/s), Cp is the specific heat capacity of air (in kJ/kg·K), and ΔT is the temperature difference between the room and outside.

Assuming a typical value of 400 m^3/h for the air flow rate and using the values for Cp and density of air at room temperature, we can calculate the mass flow rate of air as:
m = (400/3600)*1.2 = 0.1333 kg/s

Using the values given in the problem, we have:
ΔT = 28 - 20 = 8°C
Cp = 1.005 kJ/kg·K

Substituting these values in the above formula, we get:
Q = 0.1333*1.005*8 = 1.07 kW

Finally, substituting the value of Q and Δt in the formula for power, we get:
Power = 1.07/8 = 0.134 kW/kW
Therefore, the minimum theoretical power required by the air conditioner is 0.134 kW/kW of cooling.

(b) In this case, the temperature difference between the hot and cold reservoirs of the air conditioner is 32 - 16 = 16°C. Using the Carnot efficiency formula, we can calculate the theoretical maximum COP (coefficient of performance) as:

COP = TH/(TH - TC) = 32/16 = 2

The COP is defined as the ratio of the heat transferred from the cold reservoir to the work input to the system. Therefore, the minimum theoretical power required by the air conditioner can be calculated as:
Power = Q/COP = Q/2

Using the same value of Q as in part (a), we get:
Power = 1.07/2 = 0.535 kW

The ratio of the power for part (b) to the power for part (a) is:
0.535/0.134 = 3.99

Therefore, the power required by the air conditioner to achieve the required rates of heat transfer with practical sized units is almost 4 times the theoretical minimum power required at the same COP.

Know more about the specific heat capacity

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