91Ó°ÊÓ

The rate of heat entering an air conditioned building on a hot day is proportional to the difference in temperature between inside and outside.

Let us write the expression of input work for air conditioner

$W=Q_{\text{hot}}-Q_{\text{cold}}….\text{(1)}$

Here, $Q_{\text{hot}}$is heat transferred to outside and $Q_{\text{cold}}$is heat leakage from outside into the air conditioned room.

Write the expression of $Q_{\text{cold}}$

$Q_{\text{cold}}=A\left(T_{\mathrm{h}}-T_{\mathrm{c}}\right)$

Here, $A$is a constant, $\left(T_{\mathrm{h}}-T_{\mathrm{c}}\right)$is temperature difference between the outside and inside of the air conditioned room.

Write the expression of $Q_{\text{hot}}$from the second law of thermodynamics

$Q_{\text{hot}}=Q_{\text{cold}}\left(\frac{T_{\mathrm{h}}}{T_{\mathrm{c}}}\right)$

Let us substitute $Q_{\text{cold}}\left(\frac{T_{\mathrm{h}}}{T_{\mathrm{c}}}\right)$for $Q_{\text{hot}}$and $A\left(T_{\mathrm{h}}-T_{\mathrm{c}}\right.)$ for $Q_{\text{cold}}$in equation (1)

$W=\left(A\left(T_{\mathrm{h}}-T_{\mathrm{c}}\right)\right)\left(\frac{T_{\mathrm{h}}}{T_{\mathrm{c}}}\right)-\left(A\left(T_{\mathrm{h}}-T_{\mathrm{c}}\right)\right)$

Simplify the above expression

$W=\frac{A}{T_{\mathrm{c}}}{\left(T_{\mathrm{h}}-T_{\mathrm{c}}\right)}^2…..\left(2\right)$

Assume the inside temperature is $295\mathrm{K}$and the outside temperature is $303\mathrm{K}$.

Substitute $303\mathrm{K}$for $T_{\mathrm{h}}$and $295\mathrm{K}$for $T_{\mathrm{c}}$in equation (2)

$W_1=\frac{A}{(295\mathrm{K})}(303\mathrm{K}-295\mathrm{K}{)}^2$

Assume now that the inside temperature is further lowered by $1\mathrm{K}$. Substitute $303\mathrm{K}$for $T_{\mathrm{h}}$and $294\mathrm{K}$for $T_{\mathrm{c}}$in equation (2)

$W_2=\frac{A}{(294\mathrm{K})}(303\mathrm{K}-294\mathrm{K}{)}^2$

The relatively extra work required to decrease the temperature by $1\mathrm{K}$is

$\frac{W_2}{W_1}=\frac{(295\mathrm{K})(303\mathrm{K}-294\mathrm{K}{)}^2}{(294\mathrm{K})(303\mathrm{K}-295\mathrm{K}{)}^2}$

$=1.27$

Thus, it is verified that the cost of air conditioning is approximately proportional to the square of temperature difference and the input work has to be increased by $27\%$ lowering the temperature by only $1\mathrm{K}$ which can prove to be an economical issue sometimes.