Question 7.236E: A rigid 35 ft^3 tank contains water initially at 250 F, with...

A rigid 35 ft ^{3} tank contains water initially at 250 F, with 50 % liquid and 50% vapor, by volume. A pressure-relief valve on the top of the tank is set to 150 lbf / in .^{2} (the tank pressure cannot exceed 150 lbf / in .^{2} – water will be discharged instead). Heat is now transferred to the tank from a 400 F heat source until the tank contains saturated vapor at 150 lbf / in .^{2}. Calculate the heat transfer to the tank and show that this process does not violate the second law.

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C.V. Tank and walls out to the source. Neglect storage in walls. There is flow out and no boundary or shaft work.

Continuity Eq.4.15:        m _{2}- m _{1}=- m _{ e }

Energy Eq.4.16:             m _{2} u _{2}- m _{1} u _{1}=- m _{ e } h _{ e }+{ }_{1} Q _{2}

Entropy Eq.7.12:          m _{2} s _{2}- m _{1} s _{1}=- m _{ e }s_{e}+\int dQ / T +{ }_{1} S _{2  gen }

State 1: 250 F, Table F.7.1                v _{ f1 }=0.017, \quad v _{ g 1}=13.8247   ft ^{3} / lbm

\begin{array}{l}m _{ LIQ }= V _{ LIQ } / v _{ f1 }=0.5 \times 35 / 0.017=1029.4   lbm \\m _{ VAP }= V _{ VAP } / v _{ g 1}=0.5 \times 35 / 13.8247=1.266   lbm \\m =1030.67   lbm \\x = m _{ VAP } /\left( m _{ LIQ }+ m _{ VAP }\right)=0.001228 \\u = u _{ f1 }+ x  u _{ fg 1}=218.48+0.001228 \times 869.41=219.55 \\s = s _{ f1 }+ x  s _{ fg 1}=0.3677+0.001228 \times 1.3324=0.36934\end{array}

 

state 2:

\begin{array}{l}v _{2}= v _{ g }=3.2214   ft ^{3} / lbm , \quad u _{2}=1110.31, h _{2}=1193.77   Btu / lbm \\s _{2}=1.576  Btu / lbm – R \quad m _{2}= V / v _{2}=10.865   lbm\end{array}

 

From the energy equation we get

\begin{aligned}{ }_{1} Q _{2} &= m _{2} u _{2}- m _{1} u _{1}+ m _{ e } h _{ e } \\&=10.865 \times 1110.31-1030.67 \times 219.55+1019.8 \times 1193.77 \\&=1003187   Btu\end{aligned}

 

\begin{array}{l}{ }_{1} S _{2  \text { gen }} = m _{2} s _{2}- m _{1} s _{1}- m _{ e } s _{ e }-{ }_{1} Q _{2} / T _{\text {source }} \\\quad=10.865 \times 1.576-1030.67 \times 0.36934+1019.8 \times 1.57-1003187 / 860 \\\quad= 7 7 . 2   B t u / R\end{array}

……………………………………..

Eq.4.15 : 1=\frac{\dot{m}_{1}}{\dot{m}_{3}}+\frac{\dot{m}_{2}}{\dot{m}_{3}}

Eq.4.16 : 0=\frac{\dot{m}_{1}}{\dot{m}_{3}} h_{1}+\frac{\dot{m}_{2}}{\dot{m}_{3}} h_{2}-h_{3}+\dot{Q} / \dot{m}_{3}

Eq.7.12 : \left(m_{2} s_{2}-m_{1} s_{1}\right)_{ c . v .}=\sum m_{i} s_{i}-\sum m_{e} s_{e}+\int_{0}^{t} \sum_{ c.s. } \frac{\dot{Q}_{ c.v. }}{T} d t+{ }_{1} S_{2 gen }

 

F.7.1
F.7.1'

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