We thus can get h 2 parenleftbigg 2870 5 kj kg

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. We thus can get h 2 = parenleftbigg 2870 . 5 kJ kg parenrightbigg + 1 2 parenleftbigg parenleftBig 700 m s parenrightBig 2 parenleftBig 70 m s parenrightBig 2 parenrightbigg kJ kg 1000 m 2 s = 3113 . 05 kJ kg . (9.26) Now we know two properties, h 2 and s 2 . To find the final state, we have to double interpolate the superheated steam tables. Doing so, we find T 2 = 324 . 1 C, P 2 = 542 kPa. (9.27) See Fig. 9.4 for a diagram of the process. Note the temperature rises in this process. The kinetic s v T P 1 2 T = 3 2 4 . 1 ˚ C T = 2 0 0 ˚ C 1 2 P=200 kPa P = 542 kPa Figure 9.4: Steam diffuser schematic. energy is being converted to thermal energy. 9.2 Bernoulli’s principle Let us consider our thermodynamics in appropriate limit to develop the well known Bernoulli principle : a useful equation in thermal science, often used beyond its realm of validity, relating pressure, fluid velocity, density, and fluid height, valid only in the limit in which mechanical energy is conserved. CC BY-NC-ND. 2011, J. M. Powers.
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270 CHAPTER 9. SECOND LAW ANALYSIS FOR A CONTROL VOLUME Figure 9.5: Daniel Bernoulli (1700-1782), Dutch-born mathematician and physicist; image from history/Biographies/Bernoulli Daniel.html . The principle was first elucidated, though not without considerable turmoil within his prolific family, by Daniel Bernoulli, 1 depicted in Fig. 9.5. For the Bernoulli principle to be formally valid requires some very restrictive assumptions. We shall make them here in the context of thermodynamics. The same assumptions allow one to equivalently obtain the principle from an analysis of the linear momentum equation developed in fluid mechanics courses. In such a fluids development, we would need to make several additional, but roughly equivalent, assumptions. This equivalence is obtained because we shall develop the equation in the limit that mechanical energy is not dissipated. For our analysis here, we will make the following assumptions: the flow is steady, all processes are fully reversible, there is one inlet and exit, and there is contact with one thermal reservoir in which thermal energy is transferred reversibly. Though we will not study it, there is another important version of the Bernoulli principle for unsteady flows. 1 D. Bernoulli, 1738, Hydrodynamica, sive de Viribus et Motibus Fluidorum Commentarii , J. H. Deckeri, Strasbourg. CC BY-NC-ND. 2011, J. M. Powers.
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9.2. BERNOULLI’S PRINCIPLE 271 Our second law becomes in the limits we study, dS cv dt bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright =0 = ˙ Q cv T + ˙ ms 1 ˙ ms 2 + ˙ σ cv bracehtipupleftbracehtipdownrightbracehtipdownleftbracehtipupright =0 , (9.28) 0 = ˙ Q cv T + ˙ m ( s 1 s 2 ) , (9.29) ˙ m ( s 2 s 1 ) = ˙ Q cv T , (9.30) ˙ mT ( s 2 s 1 ) = ˙ Q cv . (9.31) Now let us non-rigorously generalize this somewhat and allow for differential heat transfer at a variety of temperatures so as to get ˙ m integraldisplay 2 1 Tds = ˙ Q cv . (9.32) In a more formal analysis from continuum mechanics, this step is much cleaner, but would require significant development. What we really wanted was a simplification for
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