Department of Mechanical Engineering ME 322 Mechanical Engineering Thermodynamics. Calculation of Entropy Changes. Lecture 19
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1 Department of Mecanical Engineering ME Mecanical Engineering ermodynamics Calculation of Entropy Canges Lecture 9
2 e Gibbs Equations How are entropy alues calculated? Clausius found tat, dq dq m re re ds s s Wile tis is correct, it is not ery easily done! We need to deelop an alternate metod. Consider a closed simple compressible system undergoing a reersible process wit negligible kinetic and potential canges. For tis system, e First Law dq dw du re e Second Law dqre ds
3 e Gibbs Equations Since te system is a simple compressible system, dw dv Substituting te Second Law into te First Law and writing te work term as aboe gies, dq dw du re ds dv du Diiding troug by te mass of te system and rearranging gies, ds du d e First Gibbs Equation
4 e Gibbs Equations A second Gibbs equation can be deried tat inoles entalpy, u du d d d d d Substituting tis expression into te first Gibbs equation and rearranging gies, ds d d d d ds d d e Second Gibbs Equation 4
5 Significance of te Gibbs Equations Rearranging te Gibbs Equations allows us to determine entropy canges for a substance... ds du d ds du d du s s d u u ds d d d d ds d s s d ese equations are muc easier to use to find entropy canges. ese equations can now be used wit any of te fluid property models we ae preiously discussed. 5
6 e Real Fluid Model ds du d ds d d u du du u ds d s s d d d ds d s s d ese equations must be integrated between te two states. e calculus is complex and beyond te scope of ME e good news is tat te results are tabulated in property tables or tey can be determined from software suc as EES 6
7 Real Fluid Model Reference States ds d d d d table 0 table table 0 0 ds d s s d How can te reference state be arbitrary? table table d stable s d s BOC Now consider te difference between two table alues... table (Bunc Of Calculus) table, table, BOC BOC 0 0 s s s s table, table, 0 0 s,,, BOC table table stable table, e datum state is arbitrary! 7
8 Real Fluid Model Reference States Example: R at two states as sown below... 00F 40 psia 00F 40 psia u s ft /lbm 0.75 Btu/lbm 4.8 Btu/lbm Btu/lbm-R u s ft /lbm 7.5 Btu/lbm 5.84 Btu/lbm Btu/lbm-R 0.06 ft /lbm u 6.40 Btu/lbm 9.0 Btu/lbm s Btu/lbm-R 00F 40 psia ables 00F 40 psia EES u s ft /lbm 0.0 Btu/lbm 0.0 Btu/lbm Btu/lbm-R E ES u s ft /lbm Btu/lbm 0.65 Btu/lbm Btu/lbm-R u s ft /lbm Btu/lbm 0.78 Btu/lbm Btu/lbm-R 0.07 ft /lbm u 6.50 Btu/lbm 9. Btu/lbm s Btu/lbm-R 8
9 Incompressible Substance Model For te incompressible substance, recall tat en, constant and du cd u du d s s d c u Furtermore, if te eat capacity can be assumed constant, d s s c cln Notice tat te entropy of an incompressible substance is only a function of temperature. 9
10 e Ideal Gas Model For te ideal gas, recall tat R, du c d, and d c d en from te first Gibbs equation, u du d R d d s s d c c R ln u Furtermore, if te eat capacity can be assumed constant, s s c ln R ln Notice tat te entropy of an ideal gas is a function of bot temperature and specific olume (or pressure). p 0
11 e Ideal Gas Model For te ideal gas, recall tat R, du c d, and d c d en from te second Gibbs equation, d d R d d s s d cp cp Rln Furtermore, if te eat capacity can be assumed constant, s s cp ln Rln Notice tat te entropy of an ideal gas is a function of bot temperature and pressure. p
12 Variable Heat Capacities d d s s c R s s c R ln p ln Options to include te effects of ariable eat capacity. Integrate te ideal gas eat capacity equation found in able C.4. c c c R p p. Use a constant specific eat ealuated at te aerage temperature of te process.. Let someone else do te calculus for you by using software (a.k.a... EES!).
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