Physics: Principles with Applications (7th Edition)

Published by Pearson
ISBN 10: 0-32162-592-7
ISBN 13: 978-0-32162-592-2

Chapter 15 - The Laws of Thermodynamics - General Problems - Page 442: 76

Answer

$61\;\rm W$

Work Step by Step

The 4800 W that enters the room, when the window is not covered, is the rate of cooling heat. In other words, $\left(\dfrac{Q_L}{t}\right)_1=4800\;\rm W$ for the first case. And when we cover the window, the cooling rate became,$\left(\dfrac{Q_L}{t}\right)_2=500\;\rm W$. Now we need to find the power done by the air conditioner in each case and then find the difference between them. $$e=1-\dfrac{T_L}{T_H}=\dfrac{W}{Q_H}=\dfrac{W}{W+Q_L}$$ Thus, $$1-\dfrac{T_L}{T_H}=\dfrac{W}{W+Q_L}$$ Solving for $W$; $$\left[1-\dfrac{T_L}{T_H}\right]W+\left[1-\dfrac{T_L}{T_H}\right]Q_L=W$$ $$W\left(1-\dfrac{T_L}{T_H}-1\right)=-\left[1-\dfrac{T_L}{T_H}\right]Q_L$$ $$W\left( -\dfrac{T_L}{T_H} \right)=-\left[1-\dfrac{T_L}{T_H}\right]Q_L$$ $$W=\dfrac{-\left[1-\dfrac{T_L}{T_H}\right]Q_L}{\left( -\dfrac{T_L}{T_H} \right)}=\dfrac{ \left[1-\dfrac{T_L}{T_H}\right]T_HQ_L}{T_L}$$ $$W=\dfrac{-\left[1-\dfrac{T_L}{T_H}\right]Q_L}{\left( -\dfrac{T_L}{T_H} \right)}=Q_L\cdot\dfrac{T_H-T_L}{T_L}$$ Divide both sides by $t$; $$\dfrac{W}{t} =\dfrac{Q_L}{t}\cdot\dfrac{T_H-T_L}{T_L}$$ Therefore, $$P =\dfrac{Q_L}{t}\cdot\dfrac{T_H-T_L}{T_L}$$ Thus, $$\Delta P=P_1-P_2=\left(\dfrac{Q_{L}}{t}\right)_1\dfrac{T_H-T_L}{T_L}-\left(\dfrac{Q_{L}}{t}\right)_2\dfrac{T_H-T_L}{T_L}$$ $$\Delta P =\dfrac{T_H-T_L}{T_L}\left[\left(\dfrac{Q_{L}}{t}\right)_1-\left(\dfrac{Q_{L}}{t}\right)_2\right]$$ Plugging the known; $$ \Delta P=\dfrac{(32+273)-(21+273)}{21+273}\left[ 4800-500\right]$$ $$ \Delta P=\color{red}{\bf161}\;\rm W$$ So, the power saved is about 161 W.
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