Chapter7_B - Chapter 7 - Section B - Non-Numerical...

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Chapter 7 - Section B - Non-Numerical Solutions 7.2 ( a ) Apply the general equation given in the footnote on page 266 to the particular derivative of interest here: µ T P S =− µ T S P µ S P T The two partial derivatives on the right are found from Eqs. (6.17) and (6.16); thus, µ T P S = T C P µ V T P For gases, this derivative is positive. It applies to reversible adiabatic expansions and compressions in turbines and compressors. ( b ) Application of the same general relation (page 266) yields: µ T V U µ T U V µ U V T The two partial derivatives on the right are found from Eqs. (2.16) and (6.31); thus, µ T V U = 1 C V · P T µ P T V ¸ For gases, this may be positive or negative, depending on conditions. Note that it is zero for an ideal gas. It applies directly to the Joule expansion , an adiabatic expansion of gas confined in a portion of a container to fill the entire container. 7.3 The equation giving the thermodynamic sound speed appears in the middle of page 257. As written, it implicitly requires that V represent specific volume. This is easily confirmed by a dimensional analysis. If V is to be molar volume, then the right side must be divided by molar mass: c 2 V 2 M µ P V S ( A ) Applying the equation given in the footnote on page 266 to the derivative yields: µ P V S µ P S V µ S V P This can also be written: µ P V S ·µ P T V µ T S V ¸· µ S T P µ T V P ¸ ·µ T S V µ S T P µ P T V µ T V P ¸ Division of Eq. (6.17) by Eq. (6.30) shows that the first product in square brackets on the far right is the ratio C P / C V . Reference again to the equation of the footnote on page 266 shows that the second product in square brackets on the far right is (∂ P /∂ V ) T , which is given by Eq. (3.3). Therefore, µ P V S = C P C V µ P V T = C P C V µ 1 κ V 662
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Substitute into Eq. ( A ): c 2 = VC P M C V κ or c = ± P M C V κ ( a ) For an ideal gas, V = RT / P and κ = 1 / P . Therefore, c ig = ± M C P C V ( b ) For an incompressible liquid, V is constant, and κ = 0, leading to the result: c =∞ . This of course leads to the conclusion that the sound speed in liquids is much greater than in gases. 7.6 As P 2 decreases from an initial value of P 2 = P 1 , both u 2 and ˙ m steadily increase until the critical- pressure ratio is reached. At this value of P 2 , u 2 equals the speed of sound in the gas, and further reduction in P 2 does not affect u 2 or ˙ m . 7.7 The mass-flow rate ˙ m is of course constant throughout the nozzle from entrance to exit. The velocity u rises monotonically from nozzle entrance ( P / P 1 = 1 ) to nozzle exit as P and P / P 1 decrease.
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Chapter7_B - Chapter 7 - Section B - Non-Numerical...

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