Gas Compression and Expansion How can you calculate the energy used or made available when the volume of a gas is changed? Gas Compression and Expansion page: 1 of 10 Contents Initial Problem Statement 2 Narrative 3-8 Appendices 9-10
Gas Compression and Expansion Initial Problem Statement Many machines compress or expand a gas or fluid as part of their working design. Examples include a simple bicycle pump, a refrigerator and an internal combustion engine. To compress a gas you need to expend energy to reduce its volume. When a gas is allowed to expand energy is released. How can you calculate the energy used or made available when the volume of a gas is changed? Gas Compression and Expansion page: 2 of 10
Narrative 1. Isothermal change An isothermal change of volume of a gas is one where the temperature of the gas remains constant. To achieve this, the change in volume must be slow and there must be good thermal contact with the surroundings. In this case, for an ideal gas, the pressure, p, and volume,, are related through the equation p = k, This is Boyle s law. where k is a constant. The energy transfer, w, involved in changing the volume is given by w= p d Activity 1 k The expression p = k can be re-written as p =. Substitute this into w p = d and find the integral. Discussion 1 How do you determine the constant of integration? Activity 2 One cylinder of an engine has a volume available to an air/fuel mix of 500 cm 3 when the piston is in its lowest position. Convert this volume to m 3. Figure 1 olume available: 500 cm 3 Gas Compression and Expansion page: 3 of 10
The initial pressure at this volume is 101 kilopascals. Find the constant, k, in p = k and the constant of integration in your expression for the energy transfer when the volume is changed. Discussion 2 What are the units of k? Activity 3 The gas is isothermally compressed to a volume of 50 cm 3. Figure 2 Find the energy transfer in joules. Give your answer to 1 d.p. Multimedia Compressed volume: 50 cm 3 The activity Gas Compression and Expansion Interactive is available to demonstrate the change of volume that occurs as a pistol moves in a cylinder. Gas Compression and Expansion page: 4 of 10
2. Isothermal change, definite integral The previous section used an indefinite integral to evaluate the energy change when a gas is compressed or expanded. When changes in values are considered in problems involving integration, e.g. the change in energy when there is a change in volume, the result can be obtained by using limits to give a definite integral. This avoids having to determine the constant of integration. For the previous example where a volume of 5 10-4 m 3 is compressed to a volume of 5 10-5 m 3, the change in energy is written as a definite integral where k = 50.5. w = k 5 510 4 510 Discussion 3 1 d Since the variable in the integral is, the limits, 5 10-5 and 5 10-4 are values of. Which of the limits, 5 10-4 or 5 10-5 is the larger? What does this tell you about what is happening? Activity 4 Find w using the definite integral above. Give your answer to 1 d.p. A graph of how the pressure, p, varies with the volume,, in the isothermal system is shown below Pressure (Pa) 6.00E+06 5.00E+06 4.00E+06 3.00E+06 2.00E+06 1.00E+06 Final volume Initial volume 0.00E+00 0.0E+00 5.0E-05 1.0E-04 1.5E-04 2.0E-04 2.5E-04 3.0E-04 3.5E-04 4.0E-04 4.5E-04 5.0E-04 5.5E-04 6.0E-04 6.5E-04 Figure 3 olume (m 3 ) Discussion 4 What does the definite integral you have evaluated represent on the graph? Gas Compression and Expansion page: 5 of 10
Discussion 5 These calculations assume a slow change in volume so that the gas remains at the same temperature as its surroundings. This is done by transferring energy between the gas and its surroundings. Is this likely to be the case for an internal combustion engine? Gas Compression and Expansion page: 6 of 10
3. Adiabatic change The previous sections have assumed a slow change in volume so that the gas remains at the same temperature as its surroundings. This is done by transferring energy between the gas and its surroundings. In many cases, however, this will not be the case. For example, the compression of the air/fuel mixture in an internal combustion engine is very fast, typically taking only a few tens of milliseconds. Under these circumstances energy does not have sufficient time to leave the system and instead heats the gas, changing its temperature. In a diesel engine this effect is used to ignite the fuel by compression alone. In the ideal case where no energy is lost the relationship between pressure and volumes is given for an ideal gas by p γ = k, where k is a constant (not equal to the constant used in the previous sections). The volume is raised to the power of γ which is a property of the gas being compressed. For air γ may be taken as having a constant value of 1.4. The energy transfer involved in changing the volume, w, is still given by w= p d k The expression p γ = k can be re-written as p = γ, so that k w = γ d 1 = k d γ Activity 5 Find the above integral using γ = 1.4. Activity 6 One cylinder of an engine has a volume available to an air/fuel mix of 500 cm 3 when the piston is in its lowest position. Convert this volume to m 3. The initial pressure at this volume is 101 kilopascals. Find the constant, k, in p γ = k where γ = 1.4, and the constant of integration in your expression for the energy transfer when the volume is changed. Activity 7 The gas is adiabatically compressed to a volume of 50 cm 3. Find the energy transfer in joules. Give your answer to 1 d.p. Gas Compression and Expansion page: 7 of 10
Activity 8 Introduce limits to the integral w= k 1 d and find the energy transfer using a γ definite integral. Discussion 6 Do you think it is possible to compress a gas without a loss of energy? Gas Compression and Expansion page: 8 of 10
Appendix 1 using the interactives Gas Compression and Expansion Interactive The activity Gas Compression and Expansion Interactive is available to demonstrate the change of volume that occurs as a pistol moves in a cylinder. Figure 5 The speed of the animation is controlled using the slider at the bottom of the screen. In particular, if you stop the motion when the piston as at the very bottom of its motion and compare the volume available with that which available when the piston is at the very top of its motion, you will see that there is a volume ratio of 10:1. This is the compression ratio of the engine. Figure 6 Figure 7 Gas Compression and Expansion page: 9 of 10
Appendix 2 mathematical coverage Use calculus to solve engineering problems Be able to interpret area Distinguish between definite and indefinite integrals and interpret a definite integral as an area Gas Compression and Expansion page: 10 of 10