15 Due 11/7

1. A solar-heated house uses a box of dry sand of $3\times 4\times 0.5$ m to store solar heat. If during the night the sand cools from 35$\POW9,{\circ}$C to 20$\POW9,{\circ}$C, then how much heat is released to the 18$\POW9,{\circ}$C house? What is the entropy generated in the total system by this process?

2. Three kg of liquid lead initially at 500$\POW9,{\circ}$C is poured into a form. It then cools at constant pressure down to room temperature of 20$\POW9,{\circ}$C as heat is transferred to the room. The melting point of lead is 327$\POW9,{\circ}$C and the enthalpy change $h_{if}$ between saturated solid and saturated liquid lead at 100 kPa is 24.6 kJ/kg. The specific heats are in Tables A.3 and A.4. Calculate the net entropy generated by this process in the complete system including the room.

3. Carbon dioxide at 400K and 30 kPa in an insulated cylinder is compressed to 625 kPa in a reversible process. Calculate the specific work and final temperature using
   1. the exact data from table A.8.
   2. the specific heat value from Table A.5. Do not use the polytropic formula to find $T$. Use the expression for the entropy change instead.
   Compare the results.

4. Helium is reversibly and isothermally compressed from 100 kPa and 20$\POW9,{\circ}$C to 600 kPa, and then it is expanded reversibly adiabatically back to 100 kPa.
   1. Show this two-step process as a fat curve in both the $Ts$ and $Pv$ diagrams. Label the states as 1, 2, and 3.
   2. Show why the first process is polytropic with $n=1$ and the second with $n=1.667$.
   3. Use the polytropic formulae to get the final temperature $T_3$. It should be 143.06 K.
   4. Use the first law and heat formulae to get the specific work in the first step from 1 to 2.
   5. Compute the same work directly from the appropriate polytropic work formula and compare results.
   6. Use the first law and heat formulae to get the specific work in the second step from 2 to 3.