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Question 6.245E: A cylinder with a linear spring-loaded piston contains carbo...

A cylinder with a linear spring-loaded piston contains carbon dioxide gas at 300 lbf/in.2lbf / in .^{2} with a volume of 2 ft3ft ^{3}. The device is of aluminum and has a mass of 8 lbm. Everything (Al and gas) is initially at 400 F. By heat transfer the whole system cools to the ambient temperature of 77 F, at which point the gas pressure is 220 lbf/in.2lbf / in .^{2}. Find the total entropy generation for the process.

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CO2:m=P1V1/RT1=300×2×144/(35.10×860)=2.862  lbmCO _{2}: \quad m = P _{1} V _{1} / RT _{1}=300 \times 2 \times 144 /(35.10 \times 860)=2.862   lbm

 

V2=V1(P1/P2)(T2/T1)=2(300/220)(537/860)=1.703  ft3V _{2}= V _{1}\left( P _{1} / P _{2}\right)\left( T _{2} / T _{1}\right)=2(300 / 220)(537 / 860)=1.703   ft ^{3}

 

1W2 CO2=∫PdV=0.5(P1+P2)(V2−V1)=[(300+220)/2](1.703−2)144778=−14.29  Btu\begin{aligned}{ }_{1} W _{2  CO _{2}} =\int PdV &=0.5\left( P _{1}+ P _{2}\right)\left( V _{2}- V _{1}\right) \\=&[(300+220) / 2](1.703-2) \frac{144}{778}=-14.29   Btu\end{aligned}

 

1Q2 CO2=mCV0(T2−T1)+1W2=0.156×2.862(77−400)−14.29=−158.5  Btu{ }_{1} Q _{2  CO _{2}} = mC _{ V 0}\left( T _{2}- T _{1}\right)+{ }_{1} W _{2}=0.156 \times 2.862(77-400)-14.29=-158.5   Btu

 

1Q2 A1=mC(T2−T1)=8×0.21(77−400)=−542.6  Btu{ }_{1} Q _{2  A1 }= mC \left( T _{2}- T _{1}\right)=8 \times 0.21(77-400)=-542.6   Btu

 

System:  CO2+AlCO _{2}+ Al

Entropy Eq.:      mCO2(s2−s1)CO2+mAL(s2−s1)AL=1Q2/T0+Sgen tot m _{ CO _{2}}\left( s _{2}- s _{1}\right)_{ CO _{2}}+ m _{ AL }\left( s _{2}- s _{1}\right)_{ AL }={ }_{1} Q _{2} / T _{0}+ S _{ gen \text { tot }}

 

1Q2=−542.6−158.5=−701.14  Btu{ }_{1} Q _{2}=-542.6-158.5=-701.14   Btu

 

mCO2(s2−s1)CO2+mAL(s2−s1)AL=2.862[0.201ln⁡(537/860)−(35.10/778)ln⁡(220/300)]+8×0.21ln⁡(537/860)=−0.23086−0.79117=−1.022  Btu/R\begin{array}{l}m _{ CO _{2}}\left( s _{2}- s _{1}\right)_{ CO _{2}}+ m _{ AL }\left( s _{2}- s _{1}\right)_{ AL }= \\2.862[0.201 \ln (537 / 860)-(35.10 / 778) \ln (220 / 300)] \\+8 \times 0.21 \ln (537 / 860)=-0.23086-0.79117=-1.022   Btu / R\end{array}

 

Sgen tot =mCO2(s2−s1)CO2+mAL(s2−s1)AL−1Q2/T0=−1.022+701.14537=+0.2837 Btu/R\begin{aligned}S _{\text {gen tot }} &= m _{ CO _{2}}\left( s _{2}- s _{1}\right)_{ CO _{2}}+ m _{ AL }\left( s _{2}- s _{1}\right)_{ AL }-{}_{1} Q _{2} / T _{0} \\&=-1.022+\frac{701.14}{537}=+ 0 . 2 8 3 7  Btu / R\end{aligned}

 

 

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