Explanation Show the T-s diagram for ideal gas refrigeration system as in Figure (1). Express temperature at state 2 s . T 2 s = T 1 [ P 2 P 1 ] ( k − 1 ) / k (I) Here, temperature at state 1 is T 1 , pressure at state 1 and 2 is P 1 and P 2 respectively, and heat capacity ratio is k . Express temperature at state 4 s . T 4 s = T 3 [ P 4 P 3 ] ( k − 1 ) / k (II) Here, temperature at state 3 is T 3 , pressure at state 3 and 4 is P 3 and P 4 respectively. Express temperature at state 6 s . T 6 s = T 5 [ P 6 P 5 ] ( k − 1 ) / k (III) Here, temperature at state 5 is T 5 , pressure at state 5 and 6 is P 5 and P 6 respectively. Express temperature at state 2. T 2 = T 1 + ( T 2 s − T 1 ) η C (IV) Here, isentropic efficiency of compressor is η C . Express temperature at state 4. T 4 = T 3 + ( T 4 s − T 3 ) η C (V) Express temperature at state 6. T 6 = T 5 − η T ( T 5 − T 6 s ) (VI) Here, isentropic efficiency of turbine is η T . Express the coefficient of performance of the ideal gas refrigeration cycle. COP R = q L w comp,in − w turb,out = h 1 − h 6 ( h 2 − h 1 ) + ( h 4 − h 3 ) − ( h 5 − h 6 ) = T 1 − T 6 ( T 2 − T 1 ) + ( T 4 − T 3 ) − ( T 5 − T 6 ) (VII) Here, heat lost is q L , power input to compressor is w comp,in , power rejected out from turbine is w turb,out , specific enthalpy at state 1, 2, 3 4, 5 and 6 is h 1 , h 2 , h 3 , h 4 , h 5 and h 6 respectively, temperature at state 1,2 ,3, 4, 5 and 6 is T 1 , T 2 , T 3 , T 4 , T 5 and T 6 respectively. Express the mass flow rate of the air. m ˙ = Q ˙ refrig c p ( T 1 − T 6 ) (VIII) Here, rate of refrigeration rate is Q ˙ refrig and specific heat at constant pressure of air is c p . Conclusion: Perform unit conversion of temperature at state 1 3, and 5 from ° C to K . T 1 = − 24 °C = ( − 24 + 273 ) K = 249 K T 3 = 5 °C = ( 5 + 273 ) K = 278 K T 5 = 5 °C = ( 5 + 273 ) K = 278 K Refer Table A-2, “ideal gas specific heats of various common gas”, and write the properties of air. c p = 1.005 kJ / kg ⋅ K k = 1.4 Substitute 249 K for T 1 , 3 for P 2 / P 1 and 1.4 for k in Equation (I). T 2 s = ( 249 K ) ( 3 ) ( 1.4 − 1 ) / 1

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Chapter 11.10, Problem 78P

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The coefficient of performance of the system and the rate at which air must be circulated through the system.

Solution Summary: The author shows the T-s diagram for ideal gas refrigeration system as in Figure (1).

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