Thermodynamics I Simple Equation of State and Z, Lecture notes of Thermodynamics

Download Thermodynamics I Simple Equation of State and Z and more Lecture notes Thermodynamics in PDF only on Docsity! Handout HW 4. Due next Wednesday. • Questions on HW due today?• Finish specific heat• Phase diagrams• Revisit: the "compressed liquid approximation" and how to best approximate the enthalpy• Incompressible substance model: solids and liquids• Generalize compressibility and equations of state• How to determine if a substance can be modeled as an ideal gas○ Z, pr, Tr○ Today: • ME 291 L7 9-21-16 Wednesday, September 21, 2016 10:33 AM ME 291 - F16 Page 1 yw) = hgG) + % Or) (4-puti) | Tr compressi Le SunbtHonce Mode et solid or Diquid Tncomp => denelty , p or Spee, ( 3 pe dency pvr spac wk chanets Yn Prevlr | Ahn = Aur w Lp = Seat + vA [ah= eAT + cap] if ty 15 cost  Values of Cp ¢ Cy (2) TABLE D.1_Temperature-Dependent Molar Specific Heats of Gases at Zero Pressure — SI Units mat bT+eP+aP {Tin K, Zp in ks/\kmal > K)) Temperature Error % _ Substance Formula a 6 ¢ d Range.K Max. Avg Nitrogen N 28.90 —O.1571X10? 0.8081 X10 | —2.873X 10% 273-1800 0.59 ‘Oxygen O: 25.48 1.520 10-2 0.7155 10-8 131210 273-1800 1.19 Air 28.11 0.1967 x 10? 0.4802 X 10" = 1.966 10- 273-1800 0.72 Hydrogen H, 29.11 -0.1916X 10 0.4003 X 10 —0.8704X 10-* 273-1800 1.01 Carbon monoxide CO 28.16 0.1675 X 10 0.5372 10-8 2x10 273-1800 0.89 Carbon dioxide CO, 22.26 5.981 10-2 =3,501X 10-8 -7.469X 10 273-1800 0.67 Water vapor H,0 32.24 0.1923 x 10 1.05510 =3.595X 10% 273-1800 0.53 Nitric oxide NO 29.34 — —0.09395 x 107 0.9747 X 10-5 —4.187 X 10° 273-1500 0.97 Nitrous oxide N,0 24.11 5.8632X 10 — =3.562X 10-7 10.58 10 273-1500 0.59. Nitrogen dioxide NO, 29 SMSX102 — —3,52X 10% 787X 10% 273-1300 0.46 Ammonia NH 27568 «2.5630 10? 0.99072 X 10-7 —6.6909X 10- 273-1500 0.91 Sulfur Ss 2721 2218x102 =1.628X10-$ «3.986 X10 273-1800 0.99 Sulfur dioxide SO; 25.78 S.198X 102 —3812X 10 8.612X 10% 273-1800 0.45 Sulfur trioxide $0, 16.40 14.58 10% = 11.20X 10-8 32.42 X 10% 273-1300 0.29 Acetylene CH, 218 9.2143 X10 = -6.527X 1031821 10% 273-1500 1.46 Benzene CH, 36.22 48475X 102 -31.57X10 =—77.62X 10% 273-1500 0.34 Methanol 19.0 9.152X102 = 1.22X10-$ 8.039 X10 273-1000 0.18 Ethanol 19.9 20.9610 = 10.38X 10-3 -20.05X 10% 273-1500 0.40 Hydrogen chloride 30.33 —0.7620 107 1327X 10-3 -4.338X 10% 273-1500 0.22 ‘Methane 19.89 5024x102 1.269X10-$ = 11.01 X 10% 273-1500 1.33 Ethane 6.900 17.27X 10? 6.406 X 10-7 7.285 X 10% 273-1500 0.83, Propane — 4.04 3048x1072 =15.72X 10-8 31.74 K 10 273-1500 0.40 n-Butane 3.96 B71SX 102 -18.34X 10S 3500 X 10 273-1500 0.54 i-Butane CH = -7.913 41.60 102 = 23.01 10-3 49.91 X10 273-1500 0.25 mPentane GH 6.774 4543x102 246X103 42.29 X 10% 273-1500 0.56 mHexane CH 6.938 5522x102 = 28.65X 10S 57.69X 10% 273-1500 0.72 Ethylene GH, 3.95 15.64X 107 — —8.344X 10-* 1767X 10% 273-1500 0.54 Propylene Hs 3.15 2B83X 107? -121SX 10S 2462X 10% __ 273-1500 0.73 0.17 Data from B. G. Kyle, Chemical and Process Thermodynamics, Prentice-Hall, Englewood Clif. N., 1984 (used with permission). Table B-10 Specific heats of selected liquids Table B-11 Specific heats of selected solids (P = 1 atm) Substance State ¢p. Btu/(Ibm-R) Substance T,°C_— cp, cal(g*K) Substance T, Cp call(g-K) Water 1 atm, 32°F 1.007 Ice 0.168 Lead 0.00001 Latm, 77°F 0.98 0.262 0.0073 atm, 212°F 1.007 0.392 0.0283 Ammonia sat, 0°F 1.08 0.468 coz eae ie 0.500 0.0320 Freon 12 sat, —40°F o211 Aurainam 0.0059 ons sal. OF 0217 0.076 Iron 20 0.107 sat, 120°F 0.244 tae Silver 20 0.0588 Benzene | atm, 60°F 043 x Sodium 2» 0.295 { atm, 150°F 0.46 Tungsten 2 0.034 Light oil 1 atm, 60°F 03 liao aoe ie uals Graphite 20 0.7 — Wood 20 oz Rubber 20 oss Mica 20 021 1000 Based on values from Handbook of Chemistry and Physics, American Rubber Company. ME 291 - F16 Page 5 Phase diagrams Wednesday, September 21, 2016 11:02 AM ME 291 - F16 Page 6 Phase Diagrams Substance that Expands upon Freezing, such as H2O Pe = 22.09 MPa (3204 Ibifin.2) 2 E & 1.014 bar (14.7 Ibffin.?) E 2 (a) Specific volume Critical Liquid point Melting ‘Vaporization, g Condensation p 2 2| sola z vi Sublimation “Triple point Y#POF é Temperature Saleen a {e) ME 291 - F16 Page 7  ME 291 Thermodynamics I Some Simple Equations of State Ideal Gas Relation — Physical model: molecules are small compared to the distance between molecules (they have negligible volume), and there are no intermolecular forces... all collisions are clastic. Applics to gases only and valid when P > Per and T > 1.4 Ter (general guideline). PY=nRT- or = PV=mRT or Pv=RT n= number of moles of the gas; m= mass of the gas; R =universal gas constant; R=specitic gas constant= R/M M= molecular weight of gas Internal energy- ideal gas: u =u(T) (function of temperature only) ay = c, -(= = since u(T) only — du=c,dT — |u,-uy =fdu =|c,dT lar), af : Enthalpy- ideal gas: h=ut+?v=u(1)+R1 (function of temperature only) oh c, -(2 -4 since h(1) only dh-c,dl’ > |h,-h,=[dh=[e,d7 or), also, Gi HU pie aR dt dt The van der Waals Equation — accounts for some intermolecular forces, so valid for gases at high pressure, a RT, QTR = > b= a= v-b w 8P,, 64P. The Redlich-Kwong Equation of State — valid for gases above the critical temperature. RT Ong p= - V— Bre vv Dg IT where dpe = 0.4275R°12* / P,, and Dey = 0.0867KT,, / P, ME 291 - F16 Page 10  Benedict-Webb-Rubin Equation of State — used primarily for hydrocarbons over a large range of temperatures and pressures. pafl, [ase — Ay -— a3) + (bRT - af 1 y T KY Here A,,.By,C,,a,b,¢,@, and y are constants. These constants are given in Table C.5 of Fundamentals of Engineering Thermodynamics, 2! ed., by Howell and Buckius. The Virial Equation — general form of an equation of state; basic form derived from statistical thermodynamics. B, C, etc. are functions the temperature. RT RIB RTC = + + v v v P Principle of Corresponding States Idea — all gascous substances obey the same general equation of state when expressed in terms of their critical properties. Z=Z(Pq.Tp) Z is called the compressibility factor Prand Tp are the reduced pressure and reduced iemperature, respectively. > sce compressibility charts Determination of Z is an excellent way to ascertain if a gas can be considered ideal (if Z = 1) , and how to modify the ideal gas equation if it is not. Incompressible Liquids and Solids v= constant > ¢, =¢, =c du =c,dT =cdT and u,—u, =!) cd? The enthalpy is #=u+ Py.so dh = du + Pdv + vdP Since dv = 0,dh = caT + vdP and hy —h, =P cdT +P, — P) =u, —u, +V(P,-P) ME 291 - F16 Page 11 The "Mole" Wednesday, September 21, 2016 10:38 AM ME 291 - F16 Page 12 Z Charts Wednesday, September 21, 2016 10:38 AM ME 291 - F16 Page 15 Z= Po/RT Z= Po/RT 0 05 10 15 20 25 30 35 inydrocarbons | 40 45 50 55 60 65 10 Reduced preaure. P* Figure 8-9. Demonstration of the principle of corresponding states FIG. B-14a Generalized compressibility chart—low-pressure rat nge. Data from L. C. Nelson and E. F. Obert, Gen- eralized Compressibility Charts, Chem. En., vol. 61, P. 203, 1964. Note: v/v, = P.v/RT, ME 291 - F16 Page 16 70 T/T, = 15.0)  TABLE ¢.5_ Empirical Constants for the Benedict-Webb-Rubin Equation Ao Nem /kg?_lbf-fe*/ibm* By om /kg #/lbom q N-m*-K2/kg? Tot fe“? /lm? 731.195, 430.550 302.865 249,391 3918.49 2307.33 2498.75 2.65735 X10 4.25667 x 10 | 0.889635 x 10” 1.98649 X 10 3.18205 X 10 | 1.69071 x 10" 2.08914 x 10 2.01509 10" x107 2.51642 Xx 10" © 2.20855 X 10> 2.65194 x 10” X10 2.55256 X10" 2.06958 X 107 2.98871 x 10” 2.14127 10% 3.43000 X 1077 | 2.98168 x 10” 2.22006 X 10> 3.55620 X 10 | 3.40357 x 10” 316% 10° 17426 X YO" 3.48283 X 10-2 4.13424 X 107 497242 1057.02 - | 2.06498 10% 3.30778 x.10-7, | 4.53487 “1.98756 X 10 — 3.18377 X 10 | 4.79543, a 6 € Gas Formuta | Nem?/kg? _ Ibf-ft”/Ibm* m*/kaq? ft /om Nem? K2/kg? bf ft?-"R?/Ibm? ‘Methane cH, 1.21466 104.270 1.31523 X 10° 3.37476 X 10° 0.62577 X 10° 1.74047 X 107 Ethylene CH, L919 102.256 1,09451 X 10% 2.80842 Xx 107 0.97139 x 10° 2.70175 X 10° Ethane . 3.16097 X 107? 1.28892 1.23191 X 107% 18582 X 10-* 3.04271 x 107 1.28545 x 10-3 _ 3.29835 X 107 EB C10t < 1.63610 X 10° 1.87887 X 10° TABLE C.5 Empirical Constants for the Benedict-Webb-Rubin Equation (Continued) @ 7 Gas Formula m*/kg? ft?/Ibm? mt /kg? ft*/Ibm? ‘Methane CH, 30.1853 X10 12.4068 X 10+ | 23.3469 x10~* 5.99061 X 10° Ethylene CH, 8.08173 X 10% 3.32175 X10 | 11.7469 X10 — 3.01415 X 10 Ethane GH, 8.97220 X 10 3.68775 X 10+ 13.0701 XI 3.35367 X 107 5.16963 X 10° 5.62184 X 10% ” po] 4.33983 a For SI units, R is Pa-m?/(kg-K) or J/(kg-K), P in Pa, » in m?/kg, and Tin K 2.12482 X10 | 9.41616 X 10 231069 x 10+ | 10.0799 x 10~ 1.86472 x 10~* 538K For USCS units, R is in ft Ibf/(Ibm-"R), Pin Ib(/A?, vin f?/Ibm, and T in “R. ‘Si data, a corrected, from Ernest Cravalho and Joseph L. Smith, Engineering Thermodynamics, Pitman, Marshfield, Mass, 1981 (used with permission). ME 291 - F16 Page 17 2.41610 X 107 2.58641 X 107 2.28573 X 107 4.39605 X 107 4.55052 X 10° 5.22574 X10"