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Unformatted text preview: Chapter 6 First law analysis for a control volume Read BS, Chapter 6 Problems in previous chapters have focused on systems. These systems always were com- posed of the same matter. However, for a wide variety of engineering devices, for example • flow in pipes, • jet engines, • heat exchangers, • gas turbines, • pumps, • furnaces, or • air conditioners, a constant flow of new fluid continuously enters and exits the device. In fact, once the fluid has left the device, we often are not very concerned with that fluid, as far as the performance of the device is concerned. Of course, we might care about the pollution emitted by the device and the long term fate of expelled particles. Pollution dispersion, in contrast to pollution- creation, is more a problem of fluid mechanics than thermodynamics. Analysis of control volumes is slightly more complicated than for systems, and the equa- tions we will ultimately use are slightly more complex. Unfortunately, the underlying mathe- matics and physics which lead to the development of our simplified control volume equations are highly challenging! Worse still, most beginning thermodynamics texts do not expose the student to all of the many nuances required for the simplification. In this chapter, we will summarize the key results and refer the student to an appendix for a more rigorous development. We will introduce no new axioms in this chapter. We shall simply formulate our mass and energy conservation axioms for a control volume configuration. A sketch of a generic apparatus for control volume analysis is given in Fig. 6.1. 137 138 CHAPTER 6. FIRST LAW ANALYSIS FOR A CONTROL VOLUME Q cv W cv . . m i (h i + v i 2 /2 + gz i ) . m e (h e + v e 2 /2 + gz e ) . m e (h e + v e 2 /2 + gz e ) . m e (h e + v e 2 /2 + gz e ) . Figure 6.1: Sketch of generic configuration for control volume analysis. 6.1 Detailed derivations of control volume equations This section will give a summary of the necessary mathematical operations necessary to cast the conservation of mass and energy principles in a traditional control volume formulation. The analysis presented has been amalgamated from a variety of sources. Most directly, it is a specialization of course notes for AME 60635, Intermediate Fluid Mechanics. 1 Basic mathematical foundations are covered well by Kaplan. 2 A detailed and readable description, which has a stronger emphasis on fluid mechanics, is given in the undergraduate text of Whitaker. 3 A rigorous treatment of the development of all equations presented here is included in the graduate text of Aris. 4 Popular mechanical engineering undergraduate fluids texts have closely related expositions. 5 6 However, despite their detail, these texts have some minor flaws! The treatment given by BS is not as detailed. This section will use a notation generally consistent with BS and show in detail how to arrive at its results....
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This note was uploaded on 03/02/2012 for the course THERMO 20231 taught by Professor Powers during the Spring '10 term at Notre Dame.
- Spring '10