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11 Pages

### engr210lecture7

Course: ENGR 210, Winter 2011
School: Drexel
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Word Count: 854

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210 Introduction ENGR to Thermodynamics Course announcements First term exam: TOMORROW, 8 am in Main Auditorium Closed Book, Closed Notes Only pencils, erasers, calculator at workplace No internet, wireless or telephony capable devices in your workspace Exam Details 50 minutes of work You will need your recitation section number You asked for a formula sheet I will provide it; you only bring pencils,...

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210 Introduction ENGR to Thermodynamics Course announcements First term exam: TOMORROW, 8 am in Main Auditorium Closed Book, Closed Notes Only pencils, erasers, calculator at workplace No internet, wireless or telephony capable devices in your workspace Exam Details 50 minutes of work You will need your recitation section number You asked for a formula sheet I will provide it; you only bring pencils, eraser, and calculator to exam Lecture 7- Winter 201125 ENGR 210 Week 4 Recitation Problems covered this week Chap 3: 24E, 60, 66 Xtra Problems Chap 3: 26, 27, 29E, 42, 64 Lecture 7- Winter 201125 Ideal gas equation of state Equation of state - equation relating P, T, v Early results Boyle: P v-1 Charles / Gay-Lussac: P = R (T / v) at low pressures Ideal gas law P v = R T R = gas constant = Ru / M Ru = universal gas constant = 8.31447 kJ / (kmolK); 1.98588 BTU/(lbmol-R) M = molar mass = kg/kmol, lbm/lbmol kmol = 6.023 x 1026 molecules lbmol is not a specific number of things Lecture 7- Winter 201125 Different Forms of Ideal Gas Law PV = mRT PV = NRuT P = absolute pressure usually in MPa or kPa V = molar specific volume n m3/kmol = V/n T = absolute temperature in K or R Ru = 8.314 kJ/(kmolK) N = number of moles , , Pv = RT Pv = RuT Universal Gas Constant Ru = MR 8.3145 kJ/(kmolK) 8.3145 kPam3/(kmolK) 1.986 Btu/(lbmolR) 1545.4 ftlbf/(lbmolR) 10.73 psiaft3/(lbmolR) Lecture 7- Winter 201125 Alternative Origin of Ideal Gas Law The ideal gas equation of state can be derived from basic principles if one assumes Intermolecular forces are small Volume occupied by the particles is small. Lecture 7- Winter 201125 Example 1 Determine the particular gas constant for air and hydrogen. R= Ru M kJ kmol K = 0.2870 kJ kg kg K 28.97 kmol 8.314 Rair = kJ kmol K = 4.124 kJ kg kg K 2.016 kmol 8.314 Rhydrogen = Lecture 7- Winter 201125 Table A-1: Ideal Gas Information Lecture 7- Winter 201125 Ideal Gas Rules of Thumb The ideal gas equation of state is appropriate when: the pressure is small compared to the critical pressure (P< 0.25* Pcrit) or the temperature is larger than twice the critical temperature and the pressure is less than 10 times the critical pressure. Critical point data are also in Table A-1. Lecture 7- Winter 201125 Manipulating the Ideal Gas Law By writing the ideal gas equation twice and simplifying, the properties of an ideal gas at two different states 1 and 2 are related by R = P1V1 / (m1T1) = P2V2 /(m2T2) R = P1v1 / T1 = P2v2 /T2 . If, in addition, we have m1 = m2, then we may write PV1 P2V2 1 = T1 T2 Lecture 7- Winter 201125 Example 2 An ideal gas having an initial temperature of 25C undergoes the two processes described below. final Determine the temperature of the gas. Process 1-2: The volume is held constant while the pressure doubles. Process 2-3: The pressure is held constant while the volume is reduced to one-third of the original volume. T2 P 3 2 T3 T1 Ideal Gas 1 V Lecture 7- Winter 201125 Example 2 (cont) Process 1-3: m1 = m3 so that PV1 P3V3 1 = T1 T3 but V3 = V1/3 and P3 = P2 = 2P1 , so T3 = T1 P3 V3 P V1 1 T3 = T1 2 P V1 / 3 2 1 = T1 P V1 3 1 T3 = 2 (25 + 273) K = 198.7 K = 74.3C 3 Lecture 7- Winter 201125 Superheated H2O vapor = ideal gas? In general NO !!!!!! Use tables for properties All the complex interactions are included in tabulated (empirical) values Can water vapor EVER be treated as an ideal gas? YES but under certain conditions!!! How close is the empirical value to the ideal gas value? Lecture 7- Winter 201125 2-13 When can Steam be treated as an IG Lecture 7- Winter 201125 What about working fluids without tables? Need a measure of deviation from ideal gas behavior One example is Compressibility factor: Z Z= P v vact ual = R T vide al Also, find that Z = function of a scaled temperature and pressure called reduced temperature and reduced pressure Lecture 7- Winter 201125 Reduced Pressure and Temperature Define PR and TR to be values normalized to critical pressure and temperature, respectively PR = P / Pcr TR = T / Tcr (absolute temperature) Universal relationship Principle of corresponding states Generalized compressibility chart PR 1 (actually < 0.25) - behaves like ideal gas TR 2 - behaves like ideal gas Largest deviations from ideal gas behavior near critical point Lecture 7- Winter 201125 2-14 Generalized Compressibility: A-15 Various gases Lecture 7- Winter 201125 Example 3 Find specific volume, v, for superheated water vapor at 10 MPa and 400C using ideal gas law generalized compressibility chart steam tables (accurate value) Lecture 7- Winter 201125 Example 3 (cont) Ideal Gas T v = Ru MP v= 8.314kJ /(kmol K )(400 + 273.15) K (18kg / kmol )(10000kPa) v = 0.0311 m3/kg Lecture 7- Winter 201125 Example 3 (cont) Z factor P = P / P ;T = T / T R crit R crit P = 10MPa / 22.06MPa = 0.4533 T = 400+273.15 / 647.1 = 1.0403 R R From Appendix A-15, Z = 0.85 v actual =Z v ideal v = 0.0264 m3/kg Lecture 7- Winter 201125 Example 3 (cont) fini Table A-6, v = 0.026436 m3/kg Ideal Gas, v = 0.0311 m3/kg Z, v = 0.0264 m3/kg Ideal gas is off by about 20% Z is pretty close Lecture 7- Winter 201125 Next Prepare for the Exercise for Credit Work problems to understand concepts Review assumptions and physical laws invoked to solve each problem Get a good nights sleep Read Chapter 4 Lecture 7- Winter 201125
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