ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR Progress Report No. 12 DATA AND EQUATIONS FOR SOME THERMODYNAMIC PROPERTIES "FREON-12" DICHLORODIFLUOROMETHANE G, E. GRYKA R... G. RIEMUS J. J. MARTIN Project Supervisor Project 1777 E. I. DU PONT DE NEMOURS AND COMPANY WILMINGT ON, DELAWARE May 1955

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN TABLE OF CONTENTS page OBJECTIVE i i INTRODUCTION 1 RESULTS 1 1. CRITICAL PROPERTIES AND NORMAL BOILING POINT 1 2. SATURATED LIQUID DENSITY (dL) 2 3. SATURATED LIQUID HEAT CAPACITY (Cs) 3 4. GAS HEAT CAPACITIES 3 5. PVT RELATIONSHIPS OF THE GAS 4 6. CHANGES OF ENTHALPY AND ENTROPY USING THE IDEAL GAS HEAT CAPACITY AND THE- MARTIN-HOU EQUATION OF STATE WITH C5 TERM 13 7. VAPOR PRESSURE 17 8. THE THERMODYNAMIC CONSISTENCY OF THE VAPORPRESSURE EQUATION AND THE LIQUID AND GAS HEAT CAPAC ITIES 19 9. THE VARIATION OF HEAT CAPACITY (Cp) WITH PRESSURE 23 10. CONVERSION FACTORS 26 BIBLIOGRAPHY 28._ _. _ii_ _,_ _._._ _._-ii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN OBJECTIVE The objective of this project is to determine the thermodynamic properties of "Freon" refrigerants. iii

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN INTRODUCTION The purpose of this report is to present the information necessary for the calculation of a table of thermodynamic properties (pressure, temperature, volume, enthalpy, and entropy) for the saturated liquid, saturated vapor, and superheated gas of "Freon-12" dichlorodifluoromethane. Some new findings and a complete summary of published data, including the vapor pressure, the density and heat capacity of the saturated liquid, the PVT behavior and heat capacity of the gas, the normal boiling point, and the critical temperature, pressure and volume are presented. The thermodynamic consistency of the vapor pressure equation, equation of state, and the liquid and vapor heat capacities for this compound are also discussed. The variation of heat capacity (Cp) of the gas with pressure is evaluated from the equation of state. RESULTS 1. CRITICAL PROPERTIES AND NORMAL BOILING POINT A. Critical temperature was determined at Jackson Laboratory and reported by Whitney.12 This temperature is 233.60F or 693.3~R, where absolute zero is taken as -459.7 ~R. B. The critical pressure was calculated by inserting the critical temperature into the vapor pressure equation and found to be 596.9 psia. C. The critical density (345.84 lb/cu ft) was determined by a rectilinear diameter of temperature versus saturated liquid or gas density. D. The normal boiling point was calculated as -21,920F or 438.080R, by solving the recommended vapor pressure equation at a pressure of 14.696 psia. ____ ___ ___ ___ ___ ____ ___ ___ ___ ___1 ___,_,__....

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 2. SATURATED LIQUID DENSITY (dL) Density of the saturated liquid is presented in two alternate forms of equations. The first was reported by Eiseman;4 the second is consistent with our previous report on "Freon" refrigerants and is recommended for use in the determination of tables of thermodynamic properties. The observed data were obtained from Jackson Laboratory.6'9 I. dL = 34.84 + 18o0048 (1-T/693.3) + 74.1153 (1-T/693.3)~0'3643 II. dL = 34.84 + 0.0269600 (693.3-T) + 0.834921 (693.3-T)1/2 + 6.02683 (693.3-T)1/3 - 6.55549 x 10-6 (693.3-T)2 dL = lb/cu ft TABLE I COMPARISON OF' EXPERIMENTAL AND CALCULATED SATURATED LIQUID DENSIT ES Te-mp De~iation Deviation FTemp obs. d calc.(I) D dL calc.(II) D -187.44 108.263 107.511 -.70 1017.332 -o.87 -100.68 100.403 100.252 -.15 100.211 -.19 - 36.o04 94.2 35 94.278 +.05 94.276 +o 04 - 19.12 92.531 92.608 +.08 92.609 +.08 - 11.02 91.700 91.790 +.10 91.793 +.10 - 0.04 90.589 90.660 +.0o8 90o662 +.08 + 1.94 90.452 90.454 +.00 90.458 +o01 + 14.90 88.985 89.082 +.11 89.082 +ll + 32.00 87. 062 87.211 205 +.16 +t 56.84 84.4'409 84.348 07 84 324 -o10 + 78.08 81.662 81.738 +.09 81.694 +.04 + 95.18 79.440 79.503 +.0o8 79.446 +.01 + 95.36 79.421 79.479 +.08 79.423 +.00 +116.42 76.537 76.520 -.02 76.442 -.12 +133.70 7 3.877 73.877 0 73.777 -.14 +195.98 61.086 61.296 +.34 61.162 +.12 +210.02 56.809 56.923 +.20 56.805 -.01 +224.06* 50.816* 50.499 -o 62* 50.452 -.72* +233.6 34.84 34.84 0 34084 0 Average deviation 0.16 0.15 Average deviation (excluding 1st and 2nd to last points) 0.09 0.08 *This point lies out of line on a smooth curve graph............ 2...

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 3. SATURATED LIQUID HEAT CAPACITY (Ce) Heat capacities for liquid along the saturation line (Cs) were represented by an equation given in a confidential report by J. G. Aston. 1 The equation fits his datawithin 0.5% deviation over the range of temperatures from 216~R to 455~Ro Cs = 23.68? - 0.02584T + 0.0001417T2 moale K T = OK Converting this equation to engineering units, Btu Cs = 0.195708 - 1.1864 x 10-4T + 3.6142 x 10-7T2 lb ~R T = ~R. 4. GAS HEAT CAPACITIES 8 Masi measured ideal gas heat capacities for "Freon-12"<. A fourconstant equation was used to represent his data over the temperature range shown in Table II. The constants were evaluated by requiring the equation to reproduce the observed values at four temperatures. The equation for Co differs from Cp by R equal to 1.98589 Btu v P lb mole ~R or 0.0164225 Btu A. Constant-pressure heat capacity for the ideal gas: C = 0.0245028 + 3.32442 x 10-4T 2.412299 x 10-7T2 + 6.719186 x 10-l"T3 Btu lb OR B. Constant-volume heat capacity for the ideal gas: C~ = 0.0080803 + 3.32442 x 10'4T - 2.412299 x 10'7T2 + 6.719186 x 10'11T3 Btu lb ~R Here T is in ~R. The degree sign (-) on the heat capacities indicates zero pressure or ideal gas state......~~~~~~~~~~~

ENGINEERING RESEARCH INSTITUTE. UNIVERSITY OF MICHIGAN TABLE II IDEAL GAS HEAT CAPACITY (C~) FOR "FREON-12"t "Tp.",,'" C~,-os........ Temp Co obs. C0 calc. Deviation Experimental ~K..cal/mole ~K ca/mole K Investigator 200 14.043* 14.043 0 8 273.16 16.654 16.653 0 8 300 17.456* 17.456 0 8 400 19.831* 19.831 0 8 500 21.469 21.452 -.08 8 600 22.605* 22.605 0 8 700 23.411 235.573.70 8 * Equation fitted through this value. 5. PVT RELATIONSHIPS OCE TIE GAS The Martin-Hou7 equation of state was used to correlate the PVT data, The following values were used to evaluate the constants in the equation: R = 10.73 psia-ft3/lb mole-~R, Mol.wt. = 120.924 lb/lb mole, PC = 596.9 psia, Tc - 693 3~R, TB = 1570 ~R, T' = 554~R, 1 = 3.23, m = 6.16, Vc = 0.0287026406 ft3/lb, Zc = 0.278456844, b = 0.0065093886 ft3/lb, d2P " - 0 at 2.5 times the critical density. dT2,, ~4,,

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Surmmary of the Formulae and Procedures for Evaluating the Arbitrary Constants in the Martin-Hou Equation of State with a C5 Term Equation of state: T T =RT A2+B2T+C2e 5.4 A3+B3T+C3e T 75 A4 P 3- $- ++ V-b (V-b)2 (V-b)3 (V-b)4 B5T+C5e-5' 475r (V-b)s Formulae (in order of Evaluation): b = Vc - V-c where Z = PcVc 15Zc -RTc f2(Tc) = 9 P(Vc-b)2 - 3.8 RTC(Vc-b) f3(Tc) = 5.4 RTC (2V-b)2 - 17 Pc(Vc-b)3 f4(Tc) = 12 Pc(Vc -b)4 - 3.4 RTc (V-b )3 f5(T ) = o.8 RT (V -b)4 - 3 Pc(Vc-b) Lf2(Tc) +bRT' + 2(l-Zc)] (TB-Tc) + [f2(Tc) + bRTB](Tc-T' ) C2 RT' [(T ) (e 5' 475 -e5 -475) - (TC-Tc) (e T 5+47-5. — 547 ) TB B2 =-f, f(Tc)_- bRTB - C2(e -5e47_ -e5475) TB-T A2 = f2(Tc) - B2Tc - C2e-s.475 The above formulae are derived by the same procedure as in the original Martin-Hou equation of state7 which did not have the C5 term. 5 - - - -

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN In order to derive expressions for the two remaining curvature term constants, the following facts are utilized:, d2p d1. = 0 at critical density dT'? d2P 2. = 0 at 2.5 x critical density dT2 d p = C( -t C2 + 5 c = 0 at V andVc dT2 (V-b)2 (V-b)3 (V-b)5 2.5 Therefore, eliminating C5 and solving for C3 in terms of C2, C2 [(VL-b) - (5-b)3] C3 = [(5b)_ (Vc-b) ] C5 = -C2 (V -b)3 C3 (v-b)2 or: -C2 —b) 3- _Cs 2 A4 = f4(T ) fs(Tc) - C5e-5' 475 TC B5 B3 = m(Vc-b)3 - R(Vc-b)2 - B2(Vc-b) - (Vb)2 A3 = f3(Tc) - B3Te - C3e-5.475 The equation of state thus has the following form and constants: T T T p = RT + A2+B2T+C2e- +T + A+BT+Ce+ T+Ce-k psia V-b (V-b)2 (V-b)3 (V-b)4 (B-b)5 where R = 0.088734 psi-ft3/lb- ~R, A2 = -3.4097271

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN B2 = 1.59434848 x 10 3 T = ~R C2 = -56.7627671 V = ft3/lb A3 = 0.06023944654 B3 = -1.879618431 x 10-5 C3 = 1.31139908 A4 = -5.48737007 x 10-4 B5 = 3.46883400 x lO-9 C5 = -2.54390678 x lO-5 b = 0.0065093886 k = 5.475 Tc = 693.3 k/Tc = 7.8970143 x 10`3 A comparison of the experimentally determined PVT data with the equation of state is given in Table III. TABLE III1 COMPARISON OF EQUATION OF STATE WITH EXPERIMENTALLY DETERMINED PVT DATA Density Temp. P abs. P calc. Deviation EKterimental lb/ft 3 R, psia, psia t _ I.i'vestigator 61.633 673.79 737.4 1315.07 43.9 this report 684.16 944.0 1582.40 40.34 693.84 1109.4 1846.54 39.92 Beyond ran.ge cf 710.06 1424".4 2317.90 38.55 validit y of 721.16 1630.2 2659.31 38.70 equation 731.60 1827.9 2993.05 38.93 738.75 1958.8 3228.17 39.32 52.62 698.99 762.1 757.81 -.57 this report 714.00 947.6 943.21 -..47 731.26 1154.3 1170.25 1.36 746.08 1353.1 1375.64 1.64 762.11 1551.3 1607.36 3.49 777.25 1748.2 1834.32 4.69 791.69 193.5.1 2057.27 5.94 50.53 696.22 68o.0 672.12 -1.17 this report 716. 34 911 i7 889.76 -2.47 733.98 1123.2 1093.47 -2.72 754.10 1357.0 1338.48 -1.38 772.30 1560.8 1570.13.59 789.69 1767.2 1799.13 1.77 804.00 1942.0 1992. 51 2.53 7.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN TABLE III, (Continued) Density Temp P abs. P calc. Deviation Experimental lb/ft3' oR psia a psia Investigator 49.47 693.57 645.3 629.77 -2.47 this report 703.72 760.6 730.55 -4.11 725.92 1004,7 963.95 -4.23 743.34 1199.2 1158.03 -3.55 761.39 1410.0 1367.84 -3.08 779.63 1626.9 1587.68 -2.47 799.30 1861.1 1832.35 -1.57 45.17 719.27 846.8 822.20 -2.97 this report 739.63 1035.0 1003.67 -3.12 760.84 1230.0 1199.42 -2.55 779.77 1424.7 1379.04 -3.31 800.17 1636.3 1577o05 -3.76 818.40 1798.4 1757.34 -2.34 836.09 1980.0 1934.88 -2.33 44.71 687.86 571.7 556.90 -2.66 this report 714.34 789.0 775.57 -1.73 737.40 1000.3 975.96 -2.49 758.86 1199.5 1169.34 -2.58 780.15 1400.0 1366.71 -2.44 802.24 1612.1 1576.41 -2.26 824.18 1827.7 1788.85 -2.17 43.76 685.11 546.2 535.31 -2.03 this report 710.31 745.0 735.72 -1.26 732.76 937.8 922.42 -1.67 756.17 1145.9 1123.89 -1.96 778.28 1341.3 1319.50 -1.65 799.02 1535.2 1506.93 -1.88 821.46 1746.4 1713.35 -1.93 840.94 1929.9 1895.13 -1.83 39.43 714.88 745.9 749.73.51 this report 741.61 947.1 942.38 -.50 767.15 1136.5 1129.48 -.53 793.09 1331.4 1321.99 -.71 818.76 1532.0 1514.50 -1.16 845.65 1741.0 1717.89 -1.35 868.24 1917.6 1889.89 -1.47............__ __ __ __ __ __ __ 8,__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN TABLE III, (Continued) Density Temp P abs. P caic. Deviation Experimental lb/ft3 OR psia psia % Investigator 34.72 698.08 613.8 626.24 1.99 this report 732.64 829.7 838.27 1.02 766.30 1042.0 1044.67.26 796.45 1226.0 1229.49.28 820.17 1371.1 1374.85.27 34.64 706.97 684 680.58 -.50 2 767.13 1027 1048.47 2.05 807.81 1284 1296.99 1.00 835.06 1453 1463.39.71 846.47 1522 1533.05.72 34.38 694.89 600.3 606.56 1.03 this report 699.08 638.3 632.00 -1.00 730.40 610.1 821.95 1.44 770.47 1058.5 1064.49.56 797.89 1225.9 1230.23.35 32.46 670.09 467.3 463.81 -.75 this report 672.73 480.3 479.01 -.27 676.75 500.3 502.11.36 697.55 608.2 621.10 2.08 734.80 823.2 832.16 1.08 769.97 1022.7 1029.56.67 806.15 1226.5 1231.20.38 847.10 1450.5 1458.12.52 32.41 697.0 612.1 617.93.94 this report 709.9 684.7 691.17.94 738.2 841.2 850.85 1.13 781.9 1085.6 1095.23.88 818.7 1288.7 1299.46.83 852.1 1477.1 1483.92.46 31.73 695.4 616.5 608.49 -1.32 this report 721.5 750.2 752.84.35 736.1 829.3 832.97.44 756.9 942.4 946.96.48 772.7 1026.5 1032.29.56 794.7 1146.9 1151.20.37 818.7 1280.4 1280. 31 -.01

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN TABLE III, (Continued) Density Temp P abs. P calco Deviation Experimental lb/ft3 ~R;psia psia pi Investigator 20.27 709.8 630 626.57 -.55 this report 737~2 720 711.59 -1.18 757o 8 781 774.17 -.88 775.6 833 827o46 -.67 794.1 885 882.18 -.o 32 14.06 676.91 470.3 476.37 1.27 this report 692.87 512.3 508.26 -.79 750.97 612.5 619.92 1.20 779.58 681.1 672.93 -1.21 805.4 716.3 719.96 o51 11.936 663.46 416 421.56 1.32 2 704.09 477 487.18 2.09 745.45 535 551.24 2.95 786.53 60o 612.90 2.10 828.29 657 674.12 2.54 869.83 715 733.96 2 58 10.75 682.43 424.0 428.26 ~ 99 this report 745.62 511.1 514.72 o70 814.12 6o 0.5 604.08.43 843.48 639.9 641.47.24 6.537 624.22 264.1 263.78 -.13 this report 627.88 266.1 266.71.23 659.14 298.2 291.21 -2.40 696.65 312.4 319.64 2.27 735~ 75 344.7 348.42 1.07 5.293 608.9 219.8 218.29 -.69 3 635.0 236.7 234.42 -.97 659.8 251.2 249.36 -.74 680.0 262.8 261.29 -.58 699.8 273.9 272.81 -.40 5.187 600.14 211.6 209.78 -.87 2 628.13 227.3 226.81 -.22 655.0 243.3 242.68 -.26 683.07 256.5 258.87.92 707.89 273.3 272.93 -.14 734.87 288.9 287.97 -.32 766.82 304 305.53.o50 10

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE III (Continued) Density Temp. P abs. P calc. Deviation Experimental lb/ft3 ~R psia psia Investigator 4.756 612.7 204.8 204.08 - o35 3 614.6 205.7 205,12 -.28 641.6 220.2 219,67 - ~24 669.7 235.2 234.44 -.32 688.8 245.3 244.31 -.41 4.528 595.7 188.5 187.81 - o 37 3 625.1 203.7 203.01 -.34 639.8 210.5 210.45 -.02 654.8 218.4 217.94 -.21 678.0 229.5 229.35 -.07 700 3 240.6 240,16 -.18 718.7 249.8 248.97 -.33 3.770 590.7 164.8 161.15 -2.26 3 671.7 197.6 194.21 -1.75 3.337 588.9 149.2 145,56 -2.50 3 671.7 177.7 174.89 -1.61 3.170 570.86 132.8 133.33.40 this report 609.76 145.8 146.73.63 663.49 162.8 164.57 1.08 717.42 180.3 181.93.90 764,04 195.3 196.66.69 3.00 558.25 122.5 123.34.68 this report 572.52 127.2 128.06.67 622.06 142.0 143.97 1,37 673550 158.1 159.88 1.11 715.98 169.9 172,72 1.63 766.92 185.4 187,86 1.31 2.891 566.45 121.8 122.32.43 this report 577.69 124.9 125.86 o76 616.28 136.4 137.72.96 658.20 148.5 150.24 1.16 735.20 170.5 172.56 1.19 11

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN TABLE III (Continued) Density Temp. P abs. P calc. Deviation Experimental lb/ft3 ~R psia psia Investigator 2.821 587.1 129.2 126.17 -2,40 3 669.9 153.6 150.37 -2.15 2.361 587.1 110.3 108o37 -1.78 3 669.9 130.2 128.16 -1.59 1.804 545.7 79.5 77.79 -2.20 3 552.9 80.9 79.13 -2.24 588.9 87.5 85.72 -2.0o8 669.9 102.2 100.08 -2.12 Michelsl~ has recently measured PVT properties of "Freon-12." The results of his work are compared below with the values calculated from the equation of state used in this report. COMPARISON OF EQUATION OF STATE WITH EXPERIMENTAL DATA OF MICHELS FOR "FREON-12", DICHLORODIFLUOROMETHANE Volume Temp PMichels Peqn of state Deviation ft3/lb R r..440581 581.67 103.83586 103.433 -.39 626.67 114.07666 113.866 -.28 671.67 124.137059 124.040 -.08 716.67 134.0720377 133.895 -.13 761.67 143.910541 143.914 +.002 692.4636 128.737944 128.677 -.05.342716 581.67 128.690974 128.080 -.47 626.67 142.2988822 141.985 -.22 671.67 155.6004512 155.466 -. 09 716.67 168.6998611 168.650 -.03 761.67 181.6405203 181.627 -.007 692.4636 161.65902128 161.589 -.04.280690 581.67 151.3684367 150,526 -.56 626.67 168.5452009 168.095 -.27 671 67 185.2343589 185.039 -.11 716.67 201.6055128 201.544 -.03 761.67 217.7782926 217.741 -.02 692.4636 192.8179995 192.711 -. 06 12

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MILHIKOAN Volume Temp P P Deviation ft 3/lb OR Michels eqn of state %.236055 626.67 194.0389929 193.440 -.31 671.67 214.4510717 214.182 -.13 716.67 234.4075881 234. 310 -.04 761.67 254.0615965 254.007 -.02 692.4636 223.7045714 223.546 -.07.1983965 626.67 221.951149 221.167 - 35 671.67 247.045396 246.682 - 15 716.67 271.471206 271.336 -.05 761.67 295.477100 295. 388 -.03 692.4636 258. 391689 258.163 -.09.1715460 626.67 246.720504 245.780 - 38 671.67 276.696448 276.224 -.17 716.67 305. 733321 305.530 -.07 761.67 334.130931 334.040 -.02 692.4636 290.180419 289.884 -.10.1463545 626.67 274.631964 274.694 +.02 671.67 311.173274 312.066 +.29 716.67 346. 364106 34 7.882 +.44 761.67 380.701089 382.606 +.50 692.4636 327.544548 328.777 +.3__ 8 Average Deviation 17 6. CHANGES OF ENTHALPY AND ENTROPY USING THE IDEAL GAS HEAT CAPACITY AND THE MARTIN-HOU EQUATION OF STATE WITH C5 TERM A. Equation of state: KT -KT P = RT/(V-b) + (A2+B2T+C2e )/(V-b)2 + (A3+B3T+C3e )/(V-b) (1) + A4/(v-b)4 + (B5T+Ce-KT )/(V-b) where K = 5.475 =k Tc Tc 13

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Derivative with respect to temperature: (dP/dT) = R/(,-b) + B2/(V-b)2 + B3/(V-b)3 + Bs/(V-b)5 (2) - [C2/(V-b)2 + C3/(V-b)3 + Cs/(V-b)5] Ke-KT B. Change of enthalpy: dH = CpdTp + [V-T(d/dT)p]dPT. (3) But _dPT = d.PV)T - PdvT (4) and -(dV/dT)pdPT = (dP/dT)Vd-V T (5) Putting 4 and 5 into 3, dH =CpdTp + d(PV)T - PdVT + T(dP/dT)VdVT (6) Integrating between T~ at O pressure and any given T and P or V, where H~ is the enthalpy at To and 0 pressure. H-H~ = j) Co dT + [(PV)T]- V+ [T(dP/dT)V-P]dVT. (7) T~p V~ V=o Substituting 1 and 2 into 7, H _H = C9 &C1o +,()T1 dV VV= V=b) T(V-b) -T Voo V=oo ~4 -- A3+ (1+V.~) C,3e _ A: (vlb)3 (V-b)'T v=a, (v-b)5 LVT 14

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Integrating; V 1 + i+KT) C2e A3+(1+K..)C3e,H-H~ = C0 dT + [(PV)T] + + V=c0 L(V-b) 2 (V-b)2 A4 (1+KT)C5e KT1 (9) 53(V-b 4(V_.b)4 jV V=oo or T -KI -KIt - C dT + ( PV) -RT + A2+(1+KT)C2e + A3+ 1+KT C3e -KT (10),AA + (1+KP)Cq5eK + 3(v-b)3 4(V-b)4 AH - H1 = (2 - H). (H 11 ) Substituting 10 into 11, AH CpdT + (=V)TV - RT2 + A2+C~I(pL)C2e p ~ T2 (v2-b) A3+(1+KT2)C3e T2 A4 (l+KT2)C5e KT2 + 2(V2b)2+ j + (12) 2(V2-b)2 3(V2-b) 4(v2-b) T - _ A+(1 +K1)C3e Al4 (1+KTi)C5e 2(V0-b)2 3(V-b)3 4(Vb) +F +KT(- (3)T =CdT - R(T2-T1) + _ ~ + A3+(1+KIP)C3e (L (V-b) 2(V-b)2 (V -b)3 4(V-b) V, () 15 -T 1~~~~~~~~.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Since C = Cv + R, T2 T, KT 3I-KI -KKT AH C=, C~ dT + + A2+(1+KT)C2e + A3+(1+KT)C3e v (V-b) 2(V-b)2 (14) + A4 + (i+KT)CseT V2,,T2 (14) 3(v~b)3 4(V-b)4 VT C. Change of entropy: With S as a function of V and T: dS = (d/dT)vdT + (dp/dv)TdV (15) Since CV = T(ds/dT)V and (d/dV)T = (dP/dT)V -~~~~~ -~~~ ~ ~ (16) dS= CdT/TV + (dP/dT)vdVT Integrating between T* at a low pressure P* where volume is V*, and any given T and P or V where S* is the entropy in this ideal gas state, and using 2, S-S* = T C~ T + V + B 2-CKe /TS /V BV BC 2Ke-KT <0 C~ dT + 1 ln (v-b) T* T V(Vl b) (V-b)2 B3+-C3Ke- B5-C5Ke K (V-b)3 (-b) 5 (V-T or S-S_ = in ) (V-b) - (1 ) *C- (V-b) 2(_V-b)2 - (.b) ++ -S + Ke (Vb) v-4 b) 2(V-b)2 (V-b)4 and AS = (S2-S*) - (S-s*) (19) 16

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Putting 18 into 19, S T2 dR CrAS CV + R in (V-b) - B-h BT 4(V-b) ]*-b 40 T* (V2(V-b)2 (V-b)4 SinceTV* is - - - V-b) 2(V-Vb)2 + 4(V-b) V v VTV* T* (20) R (V-b) - B2 B3 B5 -in V_ ) (V-b) 2(V-b) 4(V-b) C2 C3 C5 KTV + + + -.v-b)"- Ke 2(V-b)2 _4(v-b) Since V* is very large, e ~ rtdT + in (V-b) B2 B3 B5 e SnC0 V- - (v-b) 2(V-b)2 4 (v-b)4 -T1 -- - 2 T + n+ Ke- (V-b) + ( v-b) (-b) 7. VAPOR PRESSURE Three different vapor pressure equations are given below. The first is a four-constant equation of the form log P = A +BC log T + DT T reported in a private communication.4 In an effort to obtain a better representation, the data of several investigators were fitted by the method of least squares to a five-constant equation of the form log P A+ + B T T2 Subsequent calculations, which were-done to check the thermodynamic consistency of the vapor pressure equation with the liquid and vapor heat the equation causes relatively poor checks at low temperatures. 17

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN A four-constant equation of the same form as the first was fitted to the data. This equation reproduced the data very well and also provided the best check with the heat capacities. It is recommended for use in the computation of tables of thermodynamic properties of "Freon-12." (1) Previously reported Equation (4) 15446.029 log P = 40.1941 -3446 029 12.59294 log T + 0.00480348 T T (2) Five-constant equation log P = 3.712389 + 1.0390177 x lo0-2 T - 1.31524 x 10o-1T2 863.6109 197644.4 T T2 (3) Equation developed in this report (recommended for tables) log P = 39.883817 - 3436.6322 _ 12.471522 log T + 0.0047304424 T T P = psia T = OR TABLE IV VAPOR PRESSURE SUMMARY AND COMPARISON WITH DATA Temp....... RTPP. 1De. Pc (2) De. (3)Dev Dev. Experimental.... P~Ros Pcalc(l) calc (2) cal TR m robs calc ai c( Invest igator 310.73 0.1622.16224 -.02 0.1618.25.16219.01 this report 350.96. 0200' 1.02044 -.02 1.0275 -.71 1.0202*.00his report 401.70 5.6889 5.679.16 5.689 -.01 5.6780*.18 5 455.70 21.894 21.90 -.03 21.83.29 21.891.01 5 491.70 44.76 44.787 -. 6 44.60.36 44.760*. 0 11 545.70 107.9 108.2 -.28 107.92 -.02 108.04 -.13 5 569.78 151.8 151.556.16 151.4.26 151.27 35 13 622.88 290.4 291.013 -.21 290.0 -.17 290.08.11 13 635.70 331.3 335.5 -1.25 335.1 -1.13 334.26 -.88 5 688.21 573.1 572.029.19 569.3.67 568.95.73 this report 693.3 critical 600.25 -- 596.9 -- 596.9* -- -- * Equation fitted through this value Subsequent to the work of this report, Michels~0 presented some vapor-pressure data. The agreement of the recommended vapor-pressure equation is indeed good, as shown in Table IVA. 18

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE IVA Temp Pob s(atm) Pcal 3 Dev. ac 1() Dev. ~R _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _........._ _ _._ _ _. _ __. 545.602 7.328 7.3410 +0.177% 7.35510 +0.313% 580.867 11.873 11.9055 +0.274% 11.9304 +0.481% _ _ _ _ _ _ _. I _ I 8. THE THERMODYNAMIC CONSISTENCY OF THE VAPOR-PRESSURE EQUATION AND THE LIQUID AND GAS HEAT CAPACITIES Consider an infinitesimal cycle as shown in the diagram here. The enthalpy changes along the five steps permit comparison of the satu- Constant rated-liquid heat capacity and the | P+dP g Temp ideal gas heat capacity through the /x"Constant vapor-pressure equation. If these 1 - Vol. heat capacities are reliable, the cycle permits checking of the vaporpressure equation in the following H manner: From 1 to 2, 1) dH = (AHvap)T; from 2 to 3, 2) dH = CVdT + V dT; from 3 to 4, _ [(c +V d 1 d 5) H = T - + VdP = T- + VV 1dV from 4 to 5, 4) dH = ( -AHvap )T+dT; 19

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN from 5 to 1, 5) dH = - C IdT - vQ () dT Summing Equations 1 through 5 and solving for CV, 6) CV = Cls + v P + d(AHrap V (dP d dT _,dT)V - -z ) Note: Subscript s refers to saturation conditions. Superscript I refers to the liquid state. Using, (AH)vap = T(V-V' ) ) i d (AH T= (V _.) s d(AH)\T__" [(ds (d S dPp dT + d dV dT \d dT \dT/s sd/ SdV T 9) (dTP) =. + )T dV T 20

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Substitute 7 into 6: 10) CV = Cs + Vd) + s-V L [(-T) + T( S \dT/s ( dT \T -(/sJ I [dTIS dTV dVT dTs Substitute 8 into 10: I + vdP dP + 2( p ~ + 11) CV = Cs + - - s (d ) 2'dT sLs dT dT2 s TI dP d dVe V + d T 1\dP dT — ) V- T-) ()V T Simplifying, 12) C CI + (VV' )dT2/ T(dT/s\ T s [ (dT; (dT)s dT VI V = 1 d ddV! 1 d(d5 ]13) =_ - dT d*2 dT Thus equations 9, 12, and 13 determine Cv as a function of previously given equations and their derivatives. CV is related to Co (the ideal gas heat capacity) by the following expression: KT 14) CV = C + c T e F-h + (.185052) V V \T 21LV-b (b) 21

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The conversion factor (.185052) is used when CV and Cv are expressed in Btu/lb ~R. The vapor pressure equation, the equation of state, and the liquid and vapor heat capacities are checked for consistency by comparing the calculated ideal gas heat capacity (Ci) with those reported by Masi.8 Table V presents these comparisons by use of the three vapor-pressure equations. The presence of the second derivative of the vapor-pressure equation in the check makes the test quite sensitive. Each of the three vapor-pressure equations fit the vapor-pressure data well; the main influence on the cyclical check appears to be the form of the vapor-pressure equation. At higher temperatures the comparisons are similar for the three equations. However, at low temperatures the second equation predicted large deviations, apparently because of the presence of the 1/T2 term, which becomes very important at low temperatures. This was the major factor in the decision to use the third vapor-pressure equation, which was recommended in Section 7. TABLE V (1) log P = 40.1941 - 344,6029 - 12.59294 log T + 0.00480348 T (2) log P = 3.712388 + 0.001039018 T - 1.31524 x 10-11T2 863.610928. 197644.385 T T2 (3) log P = 39.883817 - 3436.6322 _ 12.471522 log T + 0.0047304424 T T co = o.oo807939 + 3.32442 x 104T - 2 412299 x 10 7T2 + 6.719186 x 10 "T3 |mp |Cvo (Exp.) CV calc(l) Dev CV calc(2) Dev. CV calc(3) Dev. ~RP Btu/lb OR Btu/lb ~R Btu/lb oR % Btu/lb OR % 540.12792.13578 5.79.137 6.63.1354 5.52 450.115031.11806 2.56.1227 6.25.11730 1.93 400.10683.10492 -1.82.1046 -2.13 - 360.09970.09437 -5.65.0835 -16.2.09440 -5.61 310.09005 - -.0430 -52.2.07916 -13.76 22......

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 9. THE VARIATION OF HEAT CAPACITY (Cp) WITH PRESSURE The variation of heat capacity with pressure can be calculated from the equation of state by means of the relationship Thisprovides a check on the equation of state. Masi8 measured the heat capacity of "Freon-12" at low pressures. Since the second derivative in the above expression is difficult to evaluate, it has for convenience been evaluated here in the limit as pressure approaches zero. This also corresponds to the low pressures of the observed values. To obtain the variation of Cp with pressure, begin with dP LTdT2 p Now dV (dP dP dT p \Td TI and ddV ~dT dT2 dT dV \dT/p Therefore, dP" (d P dP d2 (dai1 ld2V] \- /T dT2 TV d dT vdT2p T (dP d2 \a T V T (dV )T d_ T V dP P\ dP \d(VT TT2 V KVT 2dP,3 \dJT 23

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN For the equation of state, p RT F2(T) F3(T) F(T) V-b (V-b) (V- )4 (V-b) so that V v-b +(V-b)2 (Vb)3 (v-b) (v-b)5 (p = + t(T + F T) + (d2P - F2'(T) F3'(T) F "(T F5 (T) (V-b)2 + (-b)3 + (Vb)4 (-b )5 V d -dP. RT 2F2(T) 3FS(T) 4F4(T) 5Fs(T) (v-b)2 (- ( VVb)3 -b)4 (V-b) (V-b) T d2p - 2RT2T 6F2(T) +12FS(T) 20Fs(T) 30F_ (T) (V-b (V-b)4 (V b)b ) (-b ((V-b) V_4TI (V-b )2 (v-b)3 (v-b)4. (V-b) ) (v-b)6 Substitution of these derivatives where required gives RT 2F2(T) ]2 [F2 "(T) F31(T) 7 (cdi2VA - (v..b )2 (V-b)3 + (V-b) 2 +(V-b 3 \ci.T2/ Lv~b2 + (b) \d2JP [ RT 2F2(T) +, F * t (V- b) (V-b) b 2+ F2'(T) [ ~ T.' [ 6~ +-b (bb);) (V-bb j" RT 2F2(T) 3 (Vb )2 (V-b)3 R +F~2 (T) 2 RT lb 6F2(T) + ] + - [Vih)2 + + The denominator of the above expression is of the order -6 in (V-b) as P+O and V+oo............_____ ~24...

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Therefore, in the numerator, consider terms of order -5. -2(R)(RT)(R) + (2RT)(R2) = -?R3T + 2R3T = 0 Considering terms of order -6 in the numerator, (RT)2F2t(T)-2jR(RT)[2F2T(T)] + R[2F2(T)IR + [F2'(T)] (RT)R + R2[6F2(T)] + 2[F2t(T)] R(2RT) which simplifies to R2T2F2"(T) + 2R2F2(T)-2R2TF2'(T) In the limit as P+O and V-o, all terms of order -7, -8, etc. in (V-b) in both the numerator and denominator approach zero as compared with the terms of order -6. Therefore, lim (d-2V = R2T2F2" (T) + 2R2F2(T) - 2R2T F2'(T) P+O~0 1~ J -'___________________F_ 2 wo \/P0 R3T3 -_ t + 72+F2(T) _2F2'(T) RT RT3 RT2 1 im (d lim T2 _ 2F2 (T F"(T ) F 2Fs(T) P+o dPT Po \dT /pj RT R RT2 - KT F2(T) = A2 + B2T + C2e c ra C,, T1 K )2 T-F-T -m 2[B2-T C2e"1 ()2 C2eC2e Te P+O P /T RT R 2 A2+B2T+C2e-tck RT2 (T2R~ T T 2- K 2 Ce RTT TR RT2, RT___ ___ ___ ___ ___ ___ ___ ___ __ 25

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Therefore, at low pressure ( -_ ____ K P2 P TR \TcT TC2 T2 RT2 The change of the ideal gas heat capacity (Cp) with pressure, calculated from the equation of state is compared in Table VI with the values reported by Masi? TABLE VI VARIATION OF Cp WITH PRESSURE Temp TeRp calc. obs. Dev. 0C "OR 4 -30 437.7 o.8544 0.633 +35 0 491.7 0.5496 o.430 +27.8 45 572.7 0.2965 0.218 +36 90 653.7 0.1691 0.137 +23.4 The above values are reported in the units of cal mole.K atm The deviations which are of the order of 30% may seem large. However, since this is a very sensitive test of the second derivatives of the equation of state, it is not considered too extreme. 10. CONVERSION FACTORS Since there has been some confusion as to the most acceptable values of certain constants required in thermodynamic calculations, the following values have been taken from publications of the National Bureau of Standards. It is noted that R = 10.7315 is slightly higher than that used in the equation of state but is well within the experimental data................. 26

FNGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Gas law constant R = 10.7315 psia ft3/~R lb mole C based on chemical scale of atomic weights R = 1.98719 thermochemical cal/OK gm mole R = 1.98589 Btu/OR lb mole 1 Btu = 778.156 ft lb 1 int. steam cal = 1.000654 thermochemical cal 1 atm = 14.696 psia Btu psia-ft3 lb oR = 0.185052 lb OR 27.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN BIBLIOGRAPHY 1. Aston, J. G., Pennsylvania State College, Final report to E, I. du Pont de Nemours and Company, October, 1950. 2. Benning, A. F., and McHarness, R. C., Jackson Laboratory Report 30-89, No. 11, SN-15954. 3. Buffington, R. M., and Gilkey, W. K., Ind. Eng. Chem., 23, 254, (1931). 4. Eiseman, B. J., Jr., Personal Communication, September, 1955. 5. Gilkey, W. K., Gerard, F. W., and Bixler, M. E., Ind. Eng. Chem. 23, 364, (1931). 6. Markwood, W. H., E. I. du Pont de Nemours and Company, Jackson Laboratory Report 30-89-12, No. 1, SN-17635. 7. Martin, JO J.,and Hou, Y. C., Amer. Inst. Chem. Eng. J., Vol. 1, No. 2 (1955). 8. Masi, J. F., J. Amer. Chem. Soc., 74, 4741, (1952). 9. McHarness, R.C., E. I. du Pont de Nemours and Company, Jackson Laboratory Report 4386-83-86. 10. Michels, A., Personal Communication with D. E. Kvalnes, Jackson Laboratory March, 1955. 11. Penningtqn, W.A., and Reed, W.H., "The Evolution of a New Refrigerant", Modern Refrigeration, May, June, July (1950). 12. Whitney, J. H., E. I. du Pont de Nemours and Company, Jackson Laboratory Notebook 4713-54. 13. Whitney, J. H., E. I. du Pont de Nemours and Company, Jackson Laboratory Notebook 4713-64-65-83-92. 28