A heat pump is run by a reversible heat engine operating between reservoirs at 800°C and 50°C. The heat pump working on Carnot cycle picks up 15 kW heat from reservoir at 10°C and delivers it to a reservoir at 50°C. The reversible engine also runs a machine that needs 25 kW. Determine the heat rece

Question

A heat pump is run by a reversible heat engine operating between reservoirs at 800°C and 50°C. The heat pump working on Carnot cycle picks up 15 kW heat from reservoir at 10°C and delivers it to a reservoir at 50°C. The reversible engine also runs a machine that needs 25 kW. Determine the heat received from highest temperature reservoir and heat rejected to reservoir at 50°C.

Accepted Answer

[latex] ext{Schematic arrangement for the problem is given in figure.}\\begin{aligned} ext{For heat engine,}\qquad \qquad \qquad & \\eta_{HE} & = \frac{W_{HE}}{Q_{1}} = 1 - \frac{323}{1173}\\Rightarrow \qquad \quad\qquad \qquad \qquad \frac{W_{HE}}{Q_{1}} & = 0.7246\ ext{For heat pump,} \qquad \qquad \qquad& \ W_{HP}& = Q _{4} - Q_{3} = Q _{4} - 15\ ext{COP} & = \frac{T_{4}}{T_{4} - T_{3}}=\frac{Q_{4}}{Q_{4} - Q_{3}}\\Rightarrow \qquad \qquad \qquad \frac{323}{(323 - 283)} & =\frac{Q_{4}}{Q_{4} - 15} \\Rightarrow \quad\qquad\qquad \qquad \qquad \quad Q _{4} & =17.12 kW \\Rightarrow \quad\qquad\qquad \qquad \qquad W_{H P} & =17.12 - 15=2.12 kW\ ext{Since,} \qquad\qquad \qquad \qquad W_{H E} & = W_{HP} + 25\\Rightarrow \qquad\qquad \qquad \qquad \quad W _{HE} & =27.12 kW\\eta_{H E} & =0.7246 = \frac{W_{HE}}{Q_{1}}\\Rightarrow \quad\qquad\qquad \qquad \qquad \quad Q_{1} & = 37.427 kW\\Rightarrow \quad\qquad\qquad \qquad \qquad \quad Q_{2} & = Q_{1} - W_{HE}\& = 37.427 – 27.12\Q_{2} & = 10.307 kW\end{aligned}\ ext{Hence heat rejected to reservoir at 50°C}\\begin{aligned} \qquad \qquad \qquad\qquad \qquad \qquad& =Q_{2} + Q_{4}\\Rightarrow \qquad \qquad \qquad & = 10.307 + 17.12\ &= extbf{27.427 kW}\end{aligned}\ ext{Heat received from highest temperature reservoir = 37.427kW}[/latex]

Question 4.18: A heat pump is run by a reversible heat engine operating bet...

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