ENGR 2010 Thermodynamics I: Homework Set 3 Solutions

P, kPa 3 300 . . 4 1. A system is expanded from state 1 to state 4 in the multistep process plotted at right. Compute 200 2. the work done by the system during each separate process (1–2, 2–3, 3–4) and the total work for the 100 1. entire process.

0.10 0.20 0.30 V, m3

Work = area under the process curve on a P − V diagram.

3 2. A piston cylinder of initial volume V1=0.2 m contains m = 1 kg of water at an initial temperature ◦ T1 = 120 C. The piston floats and maintains a constant pressure. Heat is added to the system and 3 the system expands. The process ends at V2 = 1.5 m . Find the final temperature of the system and the work done by the water. 3 ◦ The initial state has v1 = 0.2/1 = 0.2m /kg and T1 = 120 C. This is a saturated mixture, with 3 P1 = 198.5 kPa. The final volume is v2 = 1.5m /kg, and P1 = P2. The final state is superheated, with ◦ T2 = 374 C. The work done for the constant pressure process is ∫ 2 W1−2 = P dV = P (V2 − V1) = 198.5 · (1.5 − 0.2) = 258 kJ 1

3 3. A system contains air, initially at P1 = 100 kPa, V1 = 0.1 m , and T1 = 300 K. The system is now compressed in an isothermal process. The final volume is one half the initial volume. Find the work done on the system. Assume ideal gas behavior. For the ideal gas, ∫ ∫ ∫ 2 2 mRT 2 dV W − = P dV = dV = mRT 1 2 V V 1 ∫ 1 ( ) 1 ( ) 2 dV V2 1 = P1V1 = P1V1 ln = (100)(0.1) ln = −6.93 kJ 1 V V1 2 The negative work signifies that work was done on the system, i.e., the system was compressed.

3 4. A system contains air, initially at P1 = 200 kPa, V1 = 0.1 m , and T1 = 400 K. The system is now allowed to expand, and during the process the pressure and volume are related by P v1.4 = constant. At the final state a total of W1−2 = 10 kJ has been done by the system. Find the final temperature and volume. Assume ideal gas behavior. n n n For a general polytropic process, PV = C = P1V1 = P2V2 where C is a constant, and

∫ ∫ 2 2 ( ) dV C 1−n 1−n W − = P dV = C = V − V 1 2 n − 2 1 1 1 V 1 n 1 = (P V − P V ) 1 − n 2 2 1 1 and for the ideal gas PV = mRT , so

mR (1 − n)W1−2 (−.4)(10)(400) W − = (T − T ) → T = T + = 400 + = 320 K 1 2 1 − n 2 1 2 1 mR (200)(0.1) Also, for the ideal gas and the polytropic process,

( ) ( ) − ( ) P T V V n V T 1/(n 1) 400 1/0.4 1 = 1 2 = 2 → 2 = 1 = = 1.75 P2 T2V1 V1 V1 T2 320

3 and V2 = 1.75V1 = 0.174 m . It is important to get the sign of the work correct; in this problem work is positive.

5. A gas is contained in a spherical balloon. The balloon has an initial diameter of D1 = 0.05 m, and the initial pressure of the gas is P1 = 120 kPa. Gas is now added to the balloon, during which the pressure increases proportionally with diameter, i.e., P D = P1 D1 At the end of the process the diameter of the balloon has doubled. Find the work done by the gas in the balloon during the process. ∼ 3 Since V D , then ( ) P V 1/3 = P1 V1 This is therefore a polytropic process with exponent n = −1/3. The work can be computed from the formula in the previous problem.