Technology & Engineering
Adiabatic Expansion
Adiabatic expansion refers to the process in which a gas expands without any heat being added or removed from the system. This results in a decrease in temperature and an increase in volume. Adiabatic expansion is commonly used in engineering applications such as gas turbines and refrigeration systems.
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eBook - ePubStatistical Thermodynamics
An Information Theory Approach
- Christopher Aubin(Author)
- 2024(Publication Date)
- Wiley(Publisher)
7 Applications of Thermodynamics In Chapter 6, we worked through the mathematics required to study thermodynamic problems, setting us up with the tools to be able to study physical processes. In this chapter, we will work through various examples, calculating observables that can then allow us to test our theory. Recall we made some pretty bold assumptions (such as the fundamental postulate), and now it is time to see that physical observations are consistent with them. In most of our examples, the volume will be the only external parameter, although we will also see some other examples where different external parameters will come into play. After finishing this chapter, you should be able to understand the Adiabatic Expansion of a system, specifically an ideal gas, learn about simple processes used to cool gases (such as free expansion and the Joule–Thomson process), and understand the basics of heat engines and refrigerators. 7.1 Adiabatic Expansion The first process we will study is a thermally isolated system that is allowed to do mechanical work. Such a process has first been discussed in Section 4.6.2 (shown in Figure 4.10), and now we will consider it in more detail. Because our system is thermally isolated, there can be no heat flow into or out of the system, so (7.1) and thus via the first law, (7.2) We would like to determine how the pressure changes with the volume for an adiabatic process. In order to do this generally, let us assume the adiabatic process is also quasistatic. For a quasistatic process, we can relate the heat flow to the entropy change,, so in this case, (7.3) This way we can determine how the pressure depends upon the volume by considering the pressure as a function of volume and entropy, and we can write (7.4) where the second equality is true for an adiabatic process. Now we just crush the derivative to obtain (7.5) with the specific heat ratio defined in Eq. (6.101). Exercise 7.1 Show that Eq
eBook - PDF- Michael A. Box, Gail P. Box(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
2.3.3 ADIABATIC PROCESSES Many important thermodynamic processes take place under well-defined, restricted circumstances. We may define the following: ∞ An isobaric process is one taking place at constant pressure. ∞ An isothermal process is one taking place at con-stant temperature. ∞ An adiabatic process is one taking place in such a way that there is no exchange of heat between the system and its environment, and no phase changes are occurring. 32 Physics of Radiation and Climate Consider the Adiabatic Expansion of a gas. The First Law of Thermodynamics becomes du dw = -. Since the work done is positive, u must decrease, and so must T. We now use Equation 2.40 – another form of the First Law – and also the ideal gas equation, to obtain dq T c dT T dp T c dT T R dp p p p º = -= -0 a We may now integrate this equation from some initial state, defined by ( p 0 , T 0 ), to a final state defined by ( p, T ), obtaining ln ln T R c p T T p p p 0 0 = And hence, T T p p 0 0 = æ è ç ö ø ÷ k (2.45) where κ = R / c p = 0.286. Equation 2.45 is known as Poisson’s equation. We may derive two other versions of this result if we use the ideal gas equation: T T 0 0 1 = æ è ç ö ø ÷ -r r g (2.46) and p p 0 0 = æ è ç ö ø ÷ r r g (2.47) These are clearly of limited use in the atmosphere as we usually seek to remove density from our equations at the first opportunity . 2.3.3.1 POTENTIAL TEMPERATURE Poisson’s equation is the basis of the important concept of potential temperature. The potential temperature of a parcel of (dry or unsaturated) air, θ , is the temperature that air would have if it was brought adiabatically from its current temperature, T , and pressure, p , to a reference pressure level of 1000 hPa. Hence, q k = æ è ç ö ø ÷ T p 1000 (2.48) By this (physical) definition, we see that the potential temperature of an air parcel is conserved during an adiabatic process.
Available until 4 Dec |Learn more- Dagmar Klostermeier, Markus G. Rudolph(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
16 Chapter 2 . State Functions and the Laws of Thermodynamics From Figure 2.10 , we see that the work performed during a reversible expansion constitutes an upper limit for the work performed by the system: w w re v i rrev ≥ eq. 2.16 For the compression, on the other hand, the work performed under reversible conditions is a lower boundary for the work performed on the system: w w re v i rrev ≤ eq. 2.17 In other words, for a real process that is not entirely reversible, the work w irrev the system performs is most likely smaller than w rev , whereas the work we have to perform on a system is most likely larger. This relation does not only apply to state changes of gases but to all processes: in general, we can have a system perform more work on the way from state 1 to state 2 if changes are reversible. We will come back to this in Section 2.4.2 . 2.2.4.4 Adiabatic Expansion and Compression So far, we have considered volume changes of an ideal gas under isothermal conditions, where the temperature of the gas remains constant because the gas is coupled to a large heat reservoir, such that heat released by the gas is taken up by the reservoir, and heat taken up by the gas is removed from the reservoir. We can also change the state of an ideal gas without allowing heat exchange with the surroundings. A change in state without heat exchange is called an adiabatic process . In this case, the temperature of the system will change. Upon expansion, the system cannot take up heat from the surroundings, and the temperature will decrease. Conversely, upon compression, no heat can be released, and the temperature of the system will increase. Adiabatic compression of air masses crossing a mountain range is the reason for warm winds in the valleys ( Box 2.3 ). BOX 2.3: ADIABATIC COMPRESSION OF AIR MASSES GENERATES DRY, WARM WINDS . When winds move air towards high mountains, the air masses ascend to higher levels. They thereby reach regions with lower air pressure, and expand.
eBook - PDF- Lucien Borel, Daniel Favrat, Dinh Lan Nguyen, Magdi Batato(Authors)
- 2012(Publication Date)
- EPFL PRESS(Publisher)
Chapter 8 Thermodynamic processes and diagrams 8.A Adiabatic Expansion of an ideal gas Description A mass of air M is expanded from an initial state 1 to a final state 2 in the power system represented in Figure 8.1. Hypotheses • The cylinder is thermally insulated. • The changes of the kinetic and potential energies are negligible. • The process is done without dissipation. • The isobaric specific heat of air is given by a polynomial of the type: c p = 5 X i=1 a i T i-1 where a i are constants. • The air can be considered as an ideal gas. Data • Initial thermodynamic state: V 1 = 0.25 · 10 -3 m 3 P 1 = 100 bar ˆ T 1 = 500 ◦ C • Final volume: V 2 = 2.5 · 10 -3 m 3 • Specific constant of air: r = 0.2882 kJ/(K kg) • Constants of the polynomial of the specific heat: c p in kJ/(K kg) T in K a 1 a 2 a 3 a 4 a 5 1 057.69 -462.92 · 10 -3 1 182.59 · 10 -6 -835.11 · 10 -9 198.8 · 10 -12 232 Adiabatic Expansion of an ideal gas Questions • Calculate the mass of air. • Determine the final thermodynamic state (P 2 , T 2 ). • Calculate the work supplied by the expansion. Figure 8.1 Solution Mass of air The equation of state of ideal gases (5.72) gives the mass of air: M = V 1 P 1 rT 1 = 11.2 · 10 -3 kg Thermodynamic processes and diagrams 233 Final thermodynamic state Considering (2.2) and the first and third hypotheses, the process 1-2 is isen- tropic.
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