Explanation Write the expression for characteristics gas constant. R = R ¯ M (I) Here, the universal gas constant is R ¯ and the molecular mass is M . Write the expression for final mass. m 2 = p 2 V 2 R T 2 (II) Here, the final pressure is P 2 , the volume is V , the characteristics gas constant is R and the temperature is T 2 . Write the expression for inlet mass. d m cv d t = m ˙ i . (III) Write the energy balance equation for tank. d U c v d t = Q ˙ c v − W ˙ c v + m ˙ i h i (IV) Since the tank is insulated so the heat transfer is 0, so Q ˙ cv = 0 . Substitute 0 for Q ˙ cv and d m cv d t for m i in Equation (IV). d U cv d t = 0 − W ˙ cv + h i ( d m cv d t ) d U cv = − W ˙ c v d t + h i ( d m cv ) . (V) Write the expression for change in specific internal energy. d U cv = m 2 u 2 − m 1 u 1 Here, mass at final state is m 1 , specific internal energy at final state is u 2 , mass at initial state is m 1 and the specific internal energy at initial state is u 1 . Write the expression for change in mass. d m cv = m 2 − m 1 Substitute m 2 u 2 − m 1 u 1 for d U cv and m 2 − m 1 for d m cv in Equation (V). m 2 u 2 − m 1 u 1 = − W ˙ cv d t + h i ( m 2 − m 1 ) (VI) Also m 1 = 0 . Substitute 0 for m 1 in Equation (VI). m 2 u 2 − 0 × u 1 = − W ˙ cv d t + h i ( m 2 − 0 ) m 2 u 2 = − W ˙ cv d t + h i m 2 W ˙ cv d t = h i m 2 − m 2 u 2 W ˙ cv = m 2 ( h i − u 2 ) d t

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Chapter 4.12, Problem 119P

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