ES 202 Fluid and Thermal Systems Lecture 20 Isentropic Processes 1 28 2003 Road Map of Lecture 20 Comments on Quiz 3 saturated liquid x 0 and saturated vapor x 1 good job on interpolation problem weak on Compressed Liquid Approximation quality is undefined in compressed liquid region constant pressure and temperature curves on phase diagrams shape and direction Supplement to Lecture 19 Property variation in an ideal gas variable specific heats Gibbs equation graphical interpretation newly defined variables Isentropic processes When is constant entropy a good assumption entropy change for an ideal gas with constant specific heats special case entropy change for an ideal gas with variable specific heats general case Examples Lecture 20 ES 202 Fluid Thermal Systems 2 1 Interpretation of CLA on Phase Diagrams P v diagram T v diagram exact state con s Co ns ta nt p re ss ur el in e exact state tant t em p er atur e line approximate state approximate state CLA stands for Compressed Liquid Approximation Lecture 20 ES 202 Fluid Thermal Systems 3 Supplement to Lecture 19 Critical point properties Substance Temperature Pressure Air 132 5 K 3 77 MPa Water 647 3 K 22 09 MPa extracted from Table A 1 in Cengel Turner Relationship between cv and cp Which one has a higher value What is the reason for the difference Apply your reasoning to the Ideal Gas Model and Incompressible Substance Model Lecture 20 ES 202 Fluid Thermal Systems 4 2 Property Variation in an Ideal Gas Recall the Gibbs equation relationship between changes in properties Tds du Pdv Tds dh vdP or For an ideal gas with finite temperature change du c T dT dT dv s ds c T T R v u v dh c T dT dT dP s c T T R P h or v p p How to evaluate the integrals graphical interpretation cv or cp would be a straight horizontal line for constant specific heats u or h T Lecture 20 ES 202 Fluid Thermal Systems 5 An Easy Way Out For variable specific heats due to the frequent usage of the integrals its value from a common reference point is tabulated e g Table A17 in Cengel Turner A new variable is defined T dT T 0 which is the temperature dependent part in entropy change s0 c p T By making use of this newly defined variable the entropy difference between any two states can be easily expressed as P s2 s1 s20 s10 R log 2 13 P4 14 424 contribution due to pressure change Lecture 20 ES 202 Fluid Thermal Systems 6 3 Constant Property Processes constant pressure isobaric piston cylinder model constant volume rigid system boundaries constant temperature sufficient time for heat transfer with environment constant entropy reversible adiabatic They serve as good models for complex problems When is constant entropy a good assumption process time scale short compared with heat transfer time scale i e heat transfer rate much slower than other processes in the problem for example rapid compression expansion process Lecture 20 ES 202 Fluid Thermal Systems 7 Isentropic Processes Recall the Gibbs equation for an ideal gas ds cv T dT dv R T v or ds c p T dT dP R T P For an isentropic proces ds 0 you only need to know one more thermodynamic property to fix the state the Gibbs equation can be reduced to cv T dT dv R T v or c p T dT dP R T P The relationship between temperature pressure and specific volume can be obtained by direct integration of the above equations Lecture 20 ES 202 Fluid Thermal Systems 8 4 Isentropic Processes II For the special case of constant specific heats direct integration yields k 1 T2 v1 T v 1 2 Tv k 1 constant T2 P2 T P 1 1 k 1 k P2 v1 P v 1 2 TP 1 k k constant k Pv k constant where k cp cv 1 The results confirm the previous claim that if you know one more thermodynamic property temperature pressure or specific volume you know everything else Lecture 20 ES 202 Fluid Thermal Systems 9 Isentropic Processes III For the general case of variable specific heats we can recall the newly defined variable s0 T dT s0 c p T T 0 which further defines two new variables useful for isentropic analysis Relative pressure s0 P2 Pr 2 Pr exp R P1 Pr1 Relative specific volume vr Lecture 20 T Pr ES 202 Fluid Thermal Systems v2 vr 2 v1 vr1 10 5