An air conditioner operating between $\mathbf{93}\mathbf{°}\mathbf{F}$and$\mathbf{70}\mathbf{°}\mathbf{F}$is rated at$\mathbf{4000}\mathbf{Btu}\mathbf{/}\mathbf{h}$cooling capacity. Its coefficient of performance is$\mathbf{27}\mathbf{\%}$of that of a Carnot refrigerator operating between the same two temperatures. What horsepower is required of the air conditioner motor?

Temperatures at which the air conditioner operates are ${\mathrm{T}}_{\mathrm{L}}=70°\mathrm{F}=294\mathrm{K}$and${\mathrm{T}}_{\mathrm{H}}=93°\mathrm{F}=307\mathrm{K}$ .

The cooling capacity of the air conditioner,$\frac{{\mathrm{Q}}_{\mathrm{L}}}{\mathrm{t}}=4000\mathrm{Btu}/\mathrm{h}$

Coefficient of performance of Carnot refrigerator,$\mathrm{K}=0.27{\mathrm{K}}_{\mathrm{c}}$

The coefficient of performance of the Carnot refrigerator is given. Using this, we can find the coefficient of performance of the air conditioner. Once we get that, using the relation between work done, coefficient of performance and heat absorbed we find the horsepower of the motor.

Formulae:

Coefficient of performance of Carnot refrigerator,

${\mathrm{K}}_{\mathrm{c}}=\frac{{\mathrm{T}}_{\mathrm{L}}}{{\mathrm{T}}_{\mathrm{H}}-{\mathrm{T}}_{\mathrm{L}}}$ (1)

The work done per cycle of the Carnot engine,

$\mathrm{W}=\frac{\mathrm{Q}}{\mathrm{K}}$ (2)

From the given value of the coefficient of performance and using equation (1), we can get the coefficient of performance of the air conditioner as given:

$$\begin{gathered} \mathrm{K}=0.27\times \left(\frac{{\mathrm{T}}_{\mathrm{L}}}{{\mathrm{T}}_{\mathrm{H}}-{\mathrm{T}}_{\mathrm{L}}}\right) \\ =0.27\times \left(\frac{294\mathrm{K}}{307\mathrm{K}-294\mathrm{K}}\right) \\ =0.27\times 23 \\ =6.21 \end{gathered}$$

Dividing t into both sides of equation (2), we can get the power of the air conditioner as given:

$$\begin{gathered} \frac{\mathrm{W}}{\mathrm{t}}=\frac{{\mathrm{Q}}_{\mathrm{L}}/\mathrm{t}}{\mathrm{K}} \\ =\frac{40000\mathrm{Btu}/\mathrm{h}}{6.21} \\ =643\mathrm{Btu}/\mathrm{h} \\ =643\mathrm{Btu}/\mathrm{h}\times \left(\frac{0.0003929\mathrm{hp}}{1\mathrm{Btu}/\mathrm{h}}\right) \\ =0.25\mathrm{hp} \end{gathered}$$

Hence, the value of the required power is 0.25 hp