Work and Heat

Abstract

This chapter deals with two quantities that affect the thermal energy stored in a system. Work and heat represent the transfer of energy to or from a system, but they are not in any way stored in the system. They represent energy in transition and must carefully be defined to quantify their effect on the thermal energy stored in a system. Once they are quantified, they can be related to the conservation of energy principle known as the first law of thermodynamics.

Problems

Problem 4.1

Energy is added to a piston-cylinder arrangement, and the piston is withdrawn in such a way that the quantity PV = C = constant. The initial pressure and volume are 400 kPa and 2 m³, respectively. If the final pressure is 200 kPa, calculate the work done by the gas on the piston.

Problem 4.2

Calculate the work done due to a volume change using Eq. 4–8 and knowing the pressure as function of volume, using ideal-gas relationshipPV = nRT. Assume temperature is constant (i.e., isothermal condition).

Problem 4.3

Consider a gas enclosed in a piston-cylinder assembly as the system. The gas is initially at a pressure of 500 kPa and occupies a volume of 0.2 m³. The gas is taken to the final state where P₂ = 100 kPa by the following two different processes. Calculate the work done by the gas in each case.

• (a) The volume of the gas is inversely proportional to the pressure.

• (b) The process follows the path PV^γ = constant, where γ = 1.4.

Problem 4.4

Determine the horsepower required to overcome the wind drag on a modern car traveling 90 km/h if the drag coefficient C_D = 0.2. The drag force F_D is given by ${F}_D=\frac{1}{2}\rho {V}^2{AC}_D$, where A is cross-sectional area (projected area) of the car and V is the velocity. The density ρ of air is 1.23 kg/m³. Use A = 2.3 m².

Problem 4.5

A scooter of mass 90 kg is moving at a speed of 60 km/h. Estimate the kinetic energy of the scooter.

Problem 4.6

An aircraft with a mass of 25,000 kg flies at a speed of 1000 km/h at an altitude of 10 km. Calculate the kinetic and potential energy of the aircraft.

Problem 4.7

A hollow sphere of mass m and volume V is immersed in a liquid of density ρ. If the sphere is raised through a distance of h in the liquid by an external agent, determine the work done by the external agent. Is there any energy transfer as work between the sphere and the liquid? What happens to the energy of the fluid? Use Fig. 4.26 to solve this problem.

Fig. 4.26

Schematic for Problem 4.7

Problem 4.8

A gas is compressed reversibly from the initial state of P₁, V₁ to final state of P₂, V₂. During the compression process, the pressure and volume are related by the equation PV^γ = C (constant). Calculate the work done by the gas.

Problem 4.9

A balloon, which is initially collapsed and flat, is slowly filled with helium from a cylinder, forming the balloon into a sphere of 5 m in diameter. The ambient pressure is 100 kPa. During the filling process, the temperature of helium inside the cylinder remains constant at 300 K. Determine the work done by the cylinder on the balloon system.

Problem 4.10

Three thermodynamic quantities A, B, and C are defined as dA = Pdυ, dB = υdP, and dC = Pdυ − υdP, where Pυ = RT. Which of the quantities can be used as properties?

Problem 4.11

A particular gas obeys the Van der Waals equation of state. Calculate the work done per kmol of the gas if the gas is compressed reversibly at constant temperature from the initial volume υ₁ to the final υ₂. The van der Waals equation is given by:

$$ P=\frac{RT}{\upsilon -b}-\frac{a}{\upsilon^2} $$

Problem 4.12

Consider the system shown in Fig. 4.27. Initially the gas is at 500 kPa and occupies a volume of 0.2 m³. The spring exerts a force, which is proportional to the displacement from its equilibrium position. The ambient is 100 kPa. The gas is heated until the volume is doubled at which point the pressure of the gas is 1 MPa. Calculate the work done by the gas.

Fig. 4.27

Schematic for Problem 4.12

Problem 4.13

A gas is compressed reversibly in a piston-cylinder assembly, from the initial state of 2 bar and 0.2 m³ to a final state of 10 bar and 0.04 m³. The pressure-volume relationship during the compression process is P = a + bV. Use Fig. 4.28 and show the path followed by the gas on a pressure versus volume diagram, and calculate the work done on the gas.

Fig. 4.28

Schematic for Problem 4.13

Problem 4.14

Consider the system shown in Fig. 4.29. Initially the gas is at 200 kPa and occupies a volume of 0.1 m³. The spring exerts a force, which is proportional to the displacement from its equilibrium position. The atmospheric pressure of 100 kPa acts on the other side of the position. The gas is heated until the volume is doubled and the final pressure is 500 kPa. Calculate the work done by the gas.

Fig. 4.29

Schematic for Problem 5.14

Problem 4.15

A spherical balloon of 1 m diameter contains a gas at 150 kPa. The gas inside the balloon is heated until the pressure reaches 450 kPa. During the process of heating, the pressure of the gas inside the balloon is proportional to the cube of the diameter of the balloon. Determine the work done by the gas inside the balloon.