Explanation Refer the table A-2 (c), “Ideal gas specific heats of various common gases”. The general empirical correlation is c ¯ P = a + b T + c T 2 + d T 3 . Write the formula for enthalpy change in molar basis at ideal gas state ( h ¯ 2 − h ¯ 1 ) ideal . ( h ¯ 2 − h ¯ 1 ) ideal = ∫ T 1 T 2 c P d T = ∫ T 1 T 2 ( a + b T + c T 2 + d T 3 ) d T = [ a T + b 2 T 2 + c 3 T 3 + d 4 T 4 ] T 1 T 2 = a ( T 2 − T 1 ) + b 2 ( T 2 2 − T 1 2 ) + c 3 ( T 2 3 − T 1 3 ) + d 4 ( T 2 4 − T 1 4 ) (I) Here, specific heat capacity at constant pressure is c P , initial temperature is T 1 , final temperature is T 2 , the empirical constants are a , b , c , and d . Write the formula for work input to the compressor ( w in ) . w in = ( h ¯ 2 − h ¯ 1 ) ideal M (II) Here, molar mass of propane is M . Write the formula for entropy change in molar basis at ideal gas state ( s ¯ 2 − s ¯ 1 ) ideal . ( s ¯ 2 − s ¯ 1 ) ideal = ∫ T 1 T 2 c P T d T − R u ln ( P 2 P 1 ) = = ∫ T 1 T 2 ( a T + b + c T + d T 2 ) d T − R u ln ( P 2 P 1 ) = [ a ln ( T ) + b T + c 2 T 2 + d 3 T 3 ] T 1 T 2 − R u ln ( P 2 P 1 ) = a ln ( T 2 T 1 ) + b ( T 2 − T 1 ) + c 2 ( T 2 2 − T 1 2 ) + d 3 ( T 2 3 − T 1 3 ) − R u ln ( P 2 P 1 ) (III) Here, universal gas constant is R u , initial pressure is P 2 , and final pressure is P 2 . Write the formula for entropy change in mass basis ( s 2 − s 1 ) ideal . ( s 2 − s 1 ) ideal = ( s ¯ 2 − s ¯ 1 ) ideal M . (IV) Write the formula for reversible work input on the compressor. w rev,in = w in − T 0 ( s 2 − s 1 ) ideal . (V) Here, the surrounding temperature is T 0 . Write the formula for second law of efficiency ( η II ) . η II = w rev,in w in . (VI) Refer table A-1, “Molar mass, gas constant and critical properties table”. The molar mass ( M ) propane is 44.097 kg/kmol . Refer the table A-2 (c), “Ideal gas specific heats of various common gases”. Obtain the empirical constants as follows. a = − 4.04 b = 30.48 × 10 − 2 c = − 15.72 × 10 − 5 d = 31.74 × 10 − 9 The universal gas constant ( R u ) is 8.314 kJ/kmol ⋅ K . Conclusion: Convert the inlet temperature from degree Celsius to Kelvin. T 1 = 100 ° C + 273 = 373 K Substitute − 4.04 for a , 773 K for T 2 , 373 K for T 1 , 30.48 × 10 − 2 for b , − 15

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Chapter 12.6, Problem 97RP

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