Adiabatic Expansion of an Ideal Gas
Adiabatic expansion of an ideal gas refers to a process in which a gas expands without gaining or losing heat to its surroundings. This results in a decrease in the gas's pressure and temperature. The process is characterized by the equation PV^γ = constant, where P is pressure, V is volume, and γ is the heat capacity ratio.
4 Key excerpts on "Adiabatic Expansion of an Ideal Gas"
eBook - ePub- A. D. Johnson(Author)
- 2017(Publication Date)
- Routledge(Publisher)
...and temperature terms P 1 T 1 γ P 1 γ = P 2 T 2 γ P 2 γ which becomes T 1 γ P 1 γ − 1 = T 2 γ P 2 γ − 1 Therefore (P 1 P 2) γ − 1 = (T 1 T 2) γ or, more conveniently, P 1 P 2 = (T 1 T 2) γ / (γ − 1) (12.15) In practice, an adiabatic process can be nearly achieved if the. process is very rapid and there is no time for heat transfer to take place. An adiabatic process approximates closely to the expansion and compression processes in a petrol engine in a car. Similarly an adiabatic process can take place if the closed system is insulated from the surroundings. Example 12.8 An ideal gas at a pressure of 800 kPa and temperature of 500 K expands adiabatically to a pressure of 200 kPa (Figure 12.6). Calculate the work done for each 1 kg of gas during the process if γ = 1.4 and C y = 0.718 kJ/kg K. Fig. 12.6 Solution From the non-flow energy equation (11.6) q − w = u 2 − u 1 but q = 0, so that w = u 1 − u 2 From equation (12.8) w = C V (T 1 − T 2) In order to find T 2 it is necessary to use equation (12.15) : P 1 P 2 = (T 1 T 2) γ /(γ − 1) This can be rearranged to the. form T 1 T 2 = (P 1 P 2) (γ − 1) / γ = (800 200) 0.4/1.4 = 1.486 Therefore T 2 = T 1 1.486 = 500 1.486 = 336.5 K The work done. is w = C V (T 1 − T 2) = 0.718 × 10 2 (500 − 336.5) = 117.4 × 10 2 J/kg = 117.4 kJ/kg Note: the work done is positive, indicating a work output from the expansion process. Fig. 12.7 Adiabatic process. 12.5.3 Adiabatic index It was shown in section 11.6.2 that the work done during a frictionless expansion, or compression, process is equal to the area under the process curve on a P − V diagram. The area under an adiabatic process curve for an ideal gas is illustrated in Figure 12.7...
eBook - ePubEinstein's Fridge
The Science of Fire, Ice and the Universe
- Paul Sen(Author)
- 2020(Publication Date)
- William Collins(Publisher)
...Now press down on the piston, squeezing the air so it occupies, say, half its original volume. The gas will resist – you will have to exert yourself, putting motive power into the cylinder – but as you push down, the air’s temperature will rise by around 60°C. This kind of compression, because no heat flows into or out of the gas, is termed adiabatic. Adiabatic compression can be reversed. Having squeezed the air, let go and allow it to expand and push back the piston to its original position. As long as the cylinder remains insulated, the gas will cool down to the temperature it started at and it will give back the same amount of motive power you put in to squeeze it. Because no heat flowed into or out of the gas, this is known as adiabatic expansion. The second aspect of air’s behaviour Carnot focused on was what happens when heat from a furnace does flow into a gas. This can both raise its temperature and cause it to expand, pressing on the walls of its container. This expansion is what pushes the piston of a steam engine. Carnot concluded that for maximum efficiency, all the heat flowing into the cylinder should go to expanding the gas and none to raising its temperature. But this seems contradictory – how can you add heat to something without making it hotter? It’s almost impossible to achieve in practice, but theoretically, this is how: A piston is positioned almost at the very bottom of a vertical cylinder. Hot air at the same temperature as an adjacent furnace is squeezed into the small space between piston and cylinder floor. The gas expands and pushes the piston, thus creating motive power. If the cylinder was completely insulated – that is, adiabatic – the gas would cool. But as the cylinder is next to the furnace, heat will flow into it, compensating for any drop in temperature. So, as heat flows into the air, it expands, creating a quantity of motive power while its temperature remains unchanged. This is known as an isothermal expansion...
eBook - ePub- Richard Gentle, Peter Edwards, William Bolton(Authors)
- 2001(Publication Date)
- Newnes(Publisher)
...This would require the addition or extraction of heat energy. For instance, if the piston is moved up the cylinder, the heat energy produced would need to be taken away if the pressure was to remain constant (Figure 2.2.2). Figure 2.2.2 Constant pressure process Constant volume The volume remains constant, i.e. the piston is fixed. Clearly, the only process which can occur is heating or cooling of the gas (Figure 2.2.3). Figure 2.2.3 Constant volume process • It is important to remember that these are the only two processes which are straight lines on the p / V diagram. Adiabatic compression and expansion During an adiabatic process, no heat transfer occurs to or from the gas during the process. This would require the perfect insulation of the cylinder, which is not possible, and it is worth noting that even the insulation itself will absorb some heat energy (Figure 2.2.4). Figure 2.2.4 Adiabatic process The index of expansion, γ, for a reversible adiabatic process is the ratio of the specific heats of the gas, i.e., The equation of the curve is • Note that a reversible adiabatic process is known as an isentropic process, i.e. constant entropy. Entropy is discussed later. Polytropic expansion and compression This is the practical process in which the temperature, pressure and volume of the gas all change. All gas processes in the real world are polytropic – think of the gas expanding and compressing in an engine cylinder (Figure 2.2.5). Figure 2.2.5 Polytropic process The equation of the curve is where ‘ n ’ is the index of polytropic expansion or compression. Isothermal process In this case, which is Boyle’s law, the temperature of the gas remains constant during the process...
eBook - ePubBiomolecular Thermodynamics
From Theory to Application
- Douglas Barrick(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...The decrease in pressure for Adiabatic Expansion of an Ideal Gas is similar to that for isothermal expansion. However, the decrease is sharper than the inverse dependence (Figure 3.12), because the heat flow that is required to maintain constant temperature in the isothermal expansion is prevented by the adiabatic boundary. This sharp dependence suggests a decay of the form p = λ V γ (3.17) where γ is a constant larger than one, and λ is a proportionality constant. In the next section, we will show that γ = C p /C V, where C p and C V are heat capacities at constant pressure and volume, respectively. The fact that C p is larger than C V for all simple systems ensures that γ > 1. For monatomic and diatomic ideal gases, γ = 5/3 and 7/5, respectively. The value of λ can be obtained by rearranging Equation 3.17, and recognizing that for a particular adiabat, the value of λ remains unchanged at all (p, V) points on the curve, including the starting point (p i, V i). That is, λ = p V γ = p i V i γ (3.18) Combining Equations 3.17 and 3.18 gives p = p i V i γ V γ (3.19) Substituting Equation 3.19 into the integral (Equation 3.13) gives the work for reversible adiabatic expansion (and compression), we can substitute Equation 3.17 into the work integral (Equation 3.13) and. solve: w = − ∫ V i V f p i V i γ V γ d V = p i V i γ (γ − 1) V γ − 1 | V i V f = p i V i γ (γ − 1) { 1 V f γ − 1 − 1 V i γ − 1 } (3.20) Equation 3.20 can be used to directly determine the work of reversible adiabatic expansion, either by assuming a specific value of γ based the type of gas (Problem 3.6), or by determining the value of γ by fitting experimental data measurement. (Problem 3.7). Although Equation 3.20 is adequate for calculation of the adiabatic expansion work for the ideal gas, there is a much simpler form that can be derived, which takes advantage of the difference between C V and C p...
