Thermodynamics, work and cycle homework.

1. Four kilograms of a certain gas is contained within a piston-cylinder assembly. The gas undergoes a process for which the pressure-volume relationship is pV1.5 = constant. The initial pressure is 3 bar, the initial volume is 0.1 m3, and the final volume is 0.2 m3. The change in specific internal energy of the gas in the process is u2 – u1 = -4.6 kJ/kg. There are no significant changes in kinetic or potential energy. Determine the net heat transfer for the process, in kJ.

2. A gas expands from an initial state where p1 = 500 kPa and V1 = 0.1 m3 to a final state where p2 = 100 kPa. The relationship between pressure and volume during the process is pV = constant. Sketch the process on a p-V diagram and determine the work, in kJ.

3. Five moles of nitrogen gas at stp expands isobarically to twice its volume. Determine:(i) Work done by this gas(ii) The change in internal energy(iii) The heat absorbed during this process

(Cv = 20.7 J mol-1 K-1, relative molecule mass of nitrogen is 28)

4. One kilomole of and ideal gas undergoes an isothermal expansion such that its volume changes if V1 to V2. Show that the work done is given by

W = 1000 RT log e (V 2V 1

)

5. Air contained within a vertical piston-cylinder assembly is shown in Fig. P1.43. On its top, the 10-kg piston is attached to a spring and exposed to an atmospheric pressure of 1 bar. Initially, the bottom of the piston is at x = 0, and the spring exerts a negligible force on the piston. The valve is opened and air enters the cylinder from the supply line, causing the volume of the air within the cylinder to increase by 3.9 x 10-4 m3. The force exerted by the spring as the air expands within the cylinder varies linearly with x according to Fspring = kx where k = 10,000 N/m. The piston face area is 7.8 x 10-3 m2. Ignoring friction between the piston and the cylinder wall, determine the pressure of the air within the cylinder, in bar, when the piston is in its initial position. Repeat when the piston is in its final position. The local acceleration of gravity is 9.81 m/s2.

6. The following table gives date, in kJ, for a system under-going a thermodynamic cycle consisting of four processes in series. For the cycle, kinetic and potential energy effects can be neglected. Determine the missing table entries.

Process ΔU Q W1-2 600 -6002-3 -13003-4 700 04-1 500 700

7. A gas undergoes three processes in series that complete a cycle:

Process 1-2: compression from p1 = 10 lbf/in2, V1 = 4 ft3, to p2 = 50 lbf/in2 during which the pressure-volume relationship is pv = constant

Process 2-3: constant volume from p3 to p1

Process 3-1: constant pressure

Sketch the cycle on a p-V diagram and determine the net work for the cycle, in Btu.

8. A gas is contained in a closed rigid tank. An electric resistor in the tank transfer energy to the gas at a constant rate of 1000 W. Heat transfer between the gas and the surroundings occurs at a rate of Q̇=−50t , where Q̇ is in watts and t is time, in minutes.

(a) Plot the time rate of change of energy of the gas for 0≤ t ≤20min, in watts.(b) Determine the net change in energy of the gas after 20 minutes, in kJ.(c) If electricity is valued at $0.08 per kW-hour, what is the cost of the electrical input to the

resistor for 20 minutes of operation?