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Unformatted text preview: AA 311 Lecture 10: Elementary Thermodynamics Reading:  Chapters 4.5. High-speed flow of gas is also a high-energy flow. The kinetic energy of the fluid elements in a high- speed flow is large, and must be taken into account. When high-speed flow is slowed down, the reduction in kinetic energy may result in a substantial increase in temperature and density. Hence high-speed flows, compressibility, and vast energy changes are all related. To study high-speed aerodynamics and compressible flows, one first needs to examine some of the fundamentals of the energy changes in a gas, and the consequent changes in pressure and temperature. First, recall the first law of thermodynamics. We will study a system of gas of unit mass that has a boundary, A , takes up some volume, v , and is able to exchange energy with its surroundings. Such a system is depicted in Figure 1. The internal energy of this system will be denoted as e . An incremental Figure 1: System of unit mass of gas. change to the internal energy is denoted de . There are two ways the system’s internal energy can be increased: • Heat is added to the system across the boundary: δq . • Work is done on the system: δw . The first law of thermodynamics states that an increase in a system’s internal energy can come from either or both of two sources only: 1) heat added to the system; 2) work done on the system: de = δq + δw. (1) Consider the system shown in Figure 2. Suppose work is done on the system, by effect of which an elemental surface area dA is pushed in an incremental distance s . By definition, work is force times distance, hence Δ W = pdAs. The total work done on the system is the sum of all the work done over the boundary, or in terms of calculus δw = Z A psdA. Suppose that pressure, p , is constant, then δw = p Z A sdA. 1 Figure 2: System of unit mass of gas. The integral above is geometrically the change in volume of unit mass of gas inside the system (see Figure 2, right), hence Z A sdA =- dv. Using this relation, we can write the work done on the system in terms of pressure and specific volume: δw =- pdv. (2) Substituting equation 2 into (1), we get an alternative form of the first law δq = de + pdv....
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- Fall '09
- Thermodynamics, isentropic flow