MT3230 Supplemental Data 4.2 Spring, 2018 Dr. Sam Miller COMPUTING THE LATENT HEAT OF VAPORIZATION OF WATER AS A FUNCTION OF TEMPERATURE Abstract The latent heat of vaporization parameterizes the amount of energy per unit mass that water must either gain to evaporate or lose to condense. It is known to vary with temperature, and appears in many of the important equations in atmospheric thermodynamics. This research evaluates four different methods for computing latent heat at varying temperatures in the range between freezing and boiling, and compares the results of each to published values. The first method tests a 2 nd -order polynomial fit from Miller (2017) and finds a large, positive bias and an error of about three and a half percent. The second method uses a linear expression shown in Rogers and Yau (1989) with a constant slope, yielding a bias about 17 times smaller and an error about 16 times smaller than the first method. The third method develops and uses temperature-dependent functions for the specific heats of liquid water and water vapor at constant pressure (c and cpv) to compute a temperature-varying slope, but shows no significant improvement over the second method. The fourth method further modifies the Rogers and Yau (1989) relationship, fitting a 2 nd -order function to the data, using temperature-dependent functions for c and cpv as terms in the polynomial, and achieves a very small bias, and an error effectively equal to zero. Introduction. According to the AMS Glossary of Meteorology, the latent heat of vaporization (lv) of water is defined as is the specific enthalpy of the water vapor minus the specific enthalpy of the liquid water (Glickman, 2000). As a parameter, lv states the amount of energy per unit mass that water must gain to evaporate, or lose to condense, and is it known to vary inversely with temperature. It is also a term in many of the important equations of atmospheric thermodynamics. For example, the Classius- Clapeyron Equation states that the saturation (or equilibrium) vapor pressure at a selected temperature is given by: e s = e 0 exp [ l v R v ( 1 T 0 1 T )] (1) where es is the saturation vapor pressure [Pa], e0 is the vapor pressure at the triple point of water (611.12 Pa), lv is the latent heat of vaporization (equal to 2.5008 x 10 6 J kg -1 at 0 C, decreasing by about 10 percent as temperature increases to 100 C), Rv is the individual gas constant for vapor (461.39 J kg -1 K -1 ), T0 is the temperature at the triple point (273.16 K), and T is the temperature [K] (Miller, 2015; Miller, 2017). Miller (2017) showed that the method used to calculate lv has an important influence on the outcome of the calculations. In that publication, the context was computing the boiling temperature of MT3230 Atmospheric Thermodynamics 1
water, but the principle is more broadly applicable. The purpose and motivation of the present research is to evaluate four different methods (including two new approaches) for computing latent heat at varying temperatures in the range between 0 and 100 C, compare the outcomes to known values, and determine the biases and percentage errors associated with each. For the purposes of this research, the values of latent heat reported by The Engineering Toolbox (2017) are considered correct. Methods and Results. Method 1: Miller (2017). The first method uses a 2 nd -order polynomial fit published by Miller (2017), originally based on latent heat data in Rogers and Yau (1989): l v = l v0 + l v1 T + l v2 T 2 (2) where lv is the latent heat of vaporization ; lv0 is the zeroth-order coefficient (2.5007 x 10 6 ), lv1 is the first-order coefficient (-1173.7723), lv2 is the second-order coefficient (1.1315), and T is the temperature. When compared to published values of lv in Rogers and Yau (1989), which only covered the range from 0 to 50 C, this function yields an R 2 value of 0.9992, a mean bias of 0.0309 x 10 6, and a mean error of 1.28 %. Mean bias is defined as the computed value published value, and mean error by: (Computed value) (Published value) Error = abs [ ] x 100 % (Published value) (3) Table 1 shows the computed values of lv using (2) compared to those published in The Engineering Toolbox (2017), along with the applicable biases and error percentages. The specific temperatures chosen correspond to the data in The Engineering Toolbox (2017), and the temperature range (roughly between the freezing and boiling temperatures of water at sea level) chosen corresponds to the general range of temperatures applicable to operational meteorologists. The mean bias in the range from 17.51 C and 102.32 C is 79,920, which indicates that the polynomial approximation yields values higher than those reported in The Engineering Toolbox (2017), and is somewhat higher than the bias reported in the narrower temperature range used in Miller (2017). The bias varies between 20,300 (at 17.51 C) and 141,690 (at 102.32 C), for a total range of 121,390. The mean error is about 3.45 percent, again somewhat higher than the mean error reported by Miller (2017), because of the wider range of temperatures used here. The error varies as a direct function of temperature it is low at the lower end of the temperature range, and increases as the temperature increases. MT3230 Atmospheric Thermodynamics 2
Table 1: Computed values of lv using Miller (2017), compared to published values, along with bias and error, as a function of temperature. Temperature Computed Value Engineering Toolbox Bias Error [%] 17.51 2.48049 2.46019 20 300 0.83 24.10 2.47307 2.44465 28 420 1.16 28.98 2.46763 2.43310 34 530 1.42 32.90 2.46331 2.42382 39 490 1.63 36.18 2.45971 2.41601 43 700 1.81 39.02 2.45662 2.40924 47 380 1.97 41.53 2.45390 2.40325 50 650 2.11 43.79 2.45147 2.39785 53 620 2.24 45.83 2.44928 2.39294 56 340 2.35 60.09 2.43425 2.35840 75 850 3.22 69.13 2.42496 2.33613 88 830 3.80 75.89 2.41814 2.31923 98 910 4.26 81.35 2.41270 2.30542 107 280 4.65 85.95 2.40817 2.29364 114 530 4.99 89.96 2.40426 2.28330 120 960 5.30 93.51 2.40083 2.27405 126 780 5.58 96.71 2.39777 2.26565 132 120 5.83 99.63 2.39499 2.25792 137 070 6.07 102.32 2.39245 2.25076 141 690 6.30 Method 2: Rogers and Yau (1989). The second method uses an expression shown in Rogers and Yau (1989), that describes the latent heat of vaporization at a given temperature by: l v = l v0 (c c pv )(T T 0 ) (4) where lv0 is the value of lv at 0 C (2.5008 x 10 6 J kg -1 ), c is the specific heat of liquid water at constant pressure, taken as a constant equal to 4187 [J kg -1 K -1 ] (an estimate of its average value between 0 and 100 C), cpv is the specific heat of water vapor at constant pressure, taken as a constant equal to 1870 [J kg -1 K - 1 ] (an estimate of its average value), T is the in situ temperature [K], and T0 is the reference temperature, equal to 273.16 [K]. This linear equation can be rewritten as: l v = l v0 l v1 T (5) where lv1 is a constant equal to (c cpv), or 2317 [J kg -1 K -1 ], and T is now the temperature in C. Table 2 shows the computed values of lv using (5) compared to those published in The Engineering Toolbox (2017), along with the applicable biases and error percentages. The mean bias in the range from 17.51 C and 102.32 C is 4831, which indicates that the linear approximation yields values higher than those reported in The Engineering Toolbox (2017), but is 17 times smaller than the mean bias associated with Method 1. The bias varies between 39 (at 17.51 C) and 12,965 (at 102.32 C), for a total range of 12,926, which is about 10 times smaller than the range found for Method 1. The mean MT3230 Atmospheric Thermodynamics 3
error is about 0.21 percent, which is about 16 times smaller than the error associated with Method 1, but shares a characteristic of Method 1 error in that it is directly proportional to temperature. The fact that this simple, linear method yields much better results than the polynomial described in Method 1 indicates that some more fundamental error may be involved, since the lv values computed using Method 1 yielded much better values of the boiling temperature in Miller (2017) than the values of lv computed using Method 2. This is discussed further in Summary and Conclusions (below). Table 2: Computed values of lv using Rogers and Tau (1989), compared to published values, along with bias and error, as a function of temperature. Temperature Computed Value Engineering Toolbox Bias Error [%] 17.51 2.46023 2.46019 39 0.00 24.10 2.44496 2.44465 310 0.01 28.98 2.43365 2.43310 553 0.02 32.90 2.42457 2.42382 751 0.03 36.18 2.41697 2.41601 961 0.04 39.02 2.41039 2.40924 1 151 0.05 41.53 2.40457 2.40325 1 325 0.06 43.79 2.39934 2.39785 1 489 0.06 45.83 2.39461 2.39294 1 672 0.07 60.09 2.36157 2.35840 3 172 0.13 69.13 2.34063 2.33613 4 496 0.19 75.89 2.32496 2.31923 5 733 0.25 81.35 2.31231 2.30542 6 892 0.30 85.95 2.30165 2.29364 8 014 0.35 89.96 2.29236 2.28330 9 063 0.40 93.51 2.28414 2.27405 10 087 0.44 96.71 2.27672 2.26565 11 073 0.49 99.63 2.26996 2.25792 12 037 0.53 102.32 2.26372 2.25076 12 965 0.58 Method 3: Modified, linear Rogers and Yau (1989). The third method begins with the expression shown in (4), and rewrites it with the temperature (T) in C: l v = l v0 (c c pv )T (6) where lv0 (the intercept) is the same as it is in (4) and (5). The main difference between (6) and (4) is that (6) recognizes that both c and cpv are not constants, but are themselves separate and quite different functions of temperature, so the first step in this method is developing those functions. The theoretical basis for the variation in c as a function of temperature is discussed in Dougherty and Howard (1998), and in Chaplin (2016), where it is shown that c reaches a minimum at about 36 C when the ambient pressure is 1000 hpa. In the present work, Lide (2006) is used for data describing c as a function of temperature (reproduced in Table 3). A plot of the data, along with a curve fit, is shown in MT3230 Atmospheric Thermodynamics 4
Fig. 1. The shape of the function requires a 5 th -order polynomial fit (7), which yields an R 2 value of 0.99973: c = a 0 + a 1 T + a 2 T 2 + a 3 T 3 + a 4 T 4 + a 5 T 5 (7) where c is the specific heat of liquid water at constant pressure [J kg -1 K -1 ], and T is the temperature. The constants associated with each term (an) are shown in Table 4. The largest of the coefficients corresponds to the zeroth-order term (a0), and is almost exactly equal to the value of c at 0 C. Table 3: Specific heat of pure liquid water at constant pressure as a function of temperature, at pressure of 1000 hpa (Lide, 2006). Temperature c [J kg -1 K -1 ] 0 4217.6 10 4192.1 20 4181.8 30 4178.4 40 4178.5 50 4180.6 60 4184.3 70 4189.5 80 4196.3 90 4205.0 100 4215.9 Fig. 1: Specific heat of pure liquid water at constant pressure as a function of temperature, at pressure of 1000 hpa, with 5 th -order polynomial curve fit. (Base data from Lide, 2006.) MT3230 Atmospheric Thermodynamics 5
Table 4: Constants associated with 5 th -order polynomial fit for c as a function of T. Constant Value a0 4217.5 a1-3.4918 a2 0.1162 a3-0.001861 a4 1.5179 x 10-5 a5-4.7115x10-8 The specific heat of water vapor, at constant pressure (cpv) and at 0 C, can be derived from a theoretical model described in Miller (2015). For a gas in general, c p = ( f + 2 2 ) R M (8) where f is the number of degrees of freedom for the gas, M is the molar mass of the gas, and R* is the Universal Gas Constant. Water is a tri-atomic gas, and therefore has 6 degrees of freedom, which includes translational and rotational degrees of freedom (Miller, 2015). The molar mass of water vapor is equal to 18.02 [kg kmol -1 ], and the Universal Gas Constant from the U.S. Standard Atmosphere is equal to 8314.32 [J kmol -1 K -1 ] (Miller, 2015). Substituting these into (8) results in: c pv = ( 6 + 2 2 ) 8314.32 18.02 = 1845.58 J kg K (9) As the temperature increases, vibrational degrees of freedom are added, and cpv increases. Rogers and Mayhew (1988) is used for data describing cpv as a function of temperature (reproduced in Table 5). The original temperature references in Rogers and Mayhew (1988) are on the absolute scale, and are converted to Celsius to make them parallel to the rest of this work. A plot of the data, along with a curve fit, is shown in Fig. 2. The shape of the function can be accommodated with a 2 nd -order polynomial fit (10), which yields an R 2 value of 0.9998: c pv = b 0 + b 1 T + b 2 T 2 (10) where cpv is the specific heat at constant pressure of water vapor [J kg -1 K -1 ], T is the temperature, b0 is equal to 1858.6, b1 is equal to 0.1766, and b2 is equal to 0.001314. The largest of the coefficients corresponds to the zeroth-order term (b0), and is close to the value of cpv at 0 C. MT3230 Atmospheric Thermodynamics 6
Table 5: Specific heat at constant pressure of water vapor as a function of temperature (Rogers and Mayhew, 1988). Temperatures originally on the Absolute scale were converted to Celsius to make them parallel to the rest of this work. Temperature cpv [J kg -1 K -1 ] -23.16 1855 1.84 1859 26.84 1864 51.84 1871 76.84 1880 101.84 1890 Fig. 2: Specific heat at constant pressure of water vapor as a function of temperature, with 2 nd -order polynomial curve fit. (Base data from Rogers and Mayhew, 1988.) Table 6 shows the computed values of lv, using (7) and (10) to compute the variable terms of (6), compared to those published in The Engineering Toolbox (2017), along with the applicable biases and error percentages. The mean bias in the range from 17.51 C and 102.32 C is 4771, which indicates that the linear approximation yields values higher than those reported in The Engineering Toolbox (2017), is slightly smaller than the mean bias associated with Method 2, and 17 times smaller than the mean bias associated with Method 1. The bias varies from -37 (at 17.51 C) to 11,835 (at 102.32 C), for a total range of 11,872, which is narrower than the range of biases noted in Method 2. The mean error is about 0.21 percent, which is about 16 times smaller than the error associated with Method 1, and the same as the error associated with Method 2. The distribution of the error across the range of temperatures is different than the distribution associated with Method 2: It is somewhat higher in the middle range of temperatures than those associated with Method 2, and lower in the higher range of temperatures than those associated with Method 2. MT3230 Atmospheric Thermodynamics 7
Table 6: Computed values of lv using modified, linear Rogers and Tau (1989), compared to published values, along with bias and error, as a function of temperature. Temperature Computed Value Engineering Toolbox Bias Error [%] 17.51 2.46015 2.46019-37 0.00 24.10 2.44498 2.44465 334 0.01 28.98 2.43375 2.43310 651 0.03 32.90 2.42472 2.42382 903 0.04 36.18 2.41717 2.41601 1 156 0.05 39.02 2.41062 2.40924 1 379 0.06 41.53 2.40483 2.40325 1 581 0.07 43.79 2.39962 2.39785 1 768 0.07 45.83 2.39491 2.39294 1 971 0.08 60.09 2.36196 2.35840 3 558 0.15 69.13 2.34098 2.33613 4 853 0.21 75.89 2.32522 2.31923 5 986 0.26 81.35 2.31241 2.30542 6 990 0.30 85.95 2.30157 2.29364 7 925 0.35 89.96 2.29207 2.28330 8 767 0.38 93.51 2.28363 2.27405 9 578 0.42 96.71 2.27510 2.26565 10 348 0.46 99.63 2.26902 2.25792 11 104 0.49 102.32 2.262594 2.25076 11 835 0.53 Method 4: Modified, non-linear Rogers and Yau (1989). The fourth method begins with the expression shown in (6): l v = l v0 (c c pv )T then subtracts lv0 from both sides, multiplies by -1, and divides by (c - cpv): l v0 l v c c pv = T (11) which implies a direct, linear relationship. Using lv0 = 2.5008 x 10 6 [Jkg -1 K -1 ], the values of lv (as a function of T) given in The Engineering Toolbox (2017), and temperature-dependent values of c and cpv computed using (7) and (10), respectively, the data shown in Table 7 were computed. MT3230 Atmospheric Thermodynamics 8
Table 7: Comparison of LHS of (11) to temperature. Temperature LHS of (11) 17.51 17.49 24.10 24.24 28.98 29.26 32.90 33.29 36.18 36.68 39.02 39.62 41.53 42.21 43.79 44.56 45.83 46.68 60.09 61.63 69.13 71.23 75.89 78.48 81.35 84.37 85.95 89.37 89.96 93.74 93.51 97.63 96.71 101.16 99.63 104.40 102.32 107.40 In this case, bias is defined as the left-hand side of (11) (LHS) temperature (T), and error by: (LHS) (T) Error = abs [ ] x 100 % (T) (12) where T in the denominator is converted to Kelvins. Using these definitions and the results shown in Table 7, the mean bias for this calculation is about 2.1 C, and the mean error is about 0.58 percent. To further reduce bias and error, the relation shown in (11) can be generalized to: l v0 l v c c pv = f(t) (13) where f(t) is an n th -order polynomial to be determined. Experimentation shows that a 2 nd -order polynomial (14) in T ( C) can be fitted to the RHS of (13) with an R 2 value of 1.00000: l v0 l v c c pv = d 0 + d 1 T + d 2 T 2 (14) where d0 is equal to 0.02909, d1 is equal to 0.99201, and d2 is equal to 0.00055534. The coefficient associated with the first-order term (d1) would be equal to unity if (11) were correct. Rearranging (14), we arrive at (15): MT3230 Atmospheric Thermodynamics 9
l v = l v0 d 0 (c c pv ) d 1 (c c pv )T d 2 (c c pv )T 2 (15) Table 8 shows the computed values of lv using (15), compared to those published in The Engineering Toolbox (2017), along with the applicable biases and error percentages. The mean bias in the range from 17.51 C and 102.32 C is -0.45, which indicates that, unlike the previous three methods, the polynomial approximation yields slightly lower values than those reported in The Engineering Toolbox (2017). More importantly, the bias is at least four orders of magnitude smaller than those associated with all three previous methods. The bias varies from -175 (at 17.51 C) to 135 (102.32 C), for a total range of 310, which is about 40 times narrower than the range of biases noted in any of the previous methods discussed. The mean error is about zero percent. Table 8: Computed values of lv using modified, 2 nd -order Rogers and Tau (1989), compared to published values, along with bias and error, as a function of temperature. Temperature Computed Value Engineering Toolbox Bias Error [%] 17.51 2.46001 2.46019-175 0.01 24.10 2.44462 2.44465-35 0.00 28.98 2.43314 2.43310 40 0.00 32.90 2.42387 2.42382 54 0.00 36.18 2.41609 2.41601 76 0.00 39.02 2.40932 2.40924 78 0.00 41.53 2.40332 2.40325 67 0.00 43.79 2.39790 2.39785 49 0.00 45.83 2.39299 2.39294 55 0.00 60.09 2.35837 2.35840-33 0.00 69.13 2.33606 2.33613-73 0.00 75.89 2.31915 2.31923-79 0.00 81.35 2.30534 2.30542-83 0.00 85.95 2.29358 2.29364-60 0.00 89.96 2.28324 2.28330-60 0.00 93.51 2.27402 2.27405-33 0.00 96.71 2.26565 2.26565 3 0.00 99.63 2.25798 2.25792 65 0.00 102.32 2.25089 2.25076 135 0.01 Summary and Conclusions. The latent heat of vaporization (lv) parameterizes the amount of energy per unit mass that water must either gain to evaporate or lose to condense. It is known to vary with temperature, and appears in many of the important equations in atmospheric thermodynamics. Miller (2017) showed that the method used to compute lv has an important impact on the outcomes of other calculations, such as the boiling temperature of water. The purpose and motivation of this research is to evaluate four different methods (including two new approaches) for computing latent heat at varying temperatures in the range between 0 and 100 C, compare the outcomes to known values, and determine the biases and percentage errors associated with each. The values of latent heat reported by The Engineering Toolbox (2017) are considered correct. Results are summarized in Table 9. MT3230 Atmospheric Thermodynamics 10
Table 9: Comparison of bias and error between 17.21 C and 102.32 C for methods test to compute lv. Bias and error are as defined above. Method Mean Bias Low Bias High Bias Range of Bias Mean Error [%] 1. Miller (2017) 79 920 20 300 141 690 121 390 3.45 2. Rogers and Yau (1989) Constant slope 3. Rogers and Yau (1989) Variable slope 4. Rogers and Yau (1989) 2 nd -order function 4 831 39 12 965 12 926 0.21 4 771-37 11 835 11 872 0.21-0.45-175 135 310 0.00 The first method tested used a 2 nd -order polynomial fit published by Miller (2017), originally based on latent heat data in Rogers and Yau (1989). For the range of temperatures examined in the present work, the Miller (2017) function yielded a mean bias of 79,920, varying from 20,300 at 17.51 C to 141,690 at 102.32 C, for a total range of 121,390. The mean error was about 3.45 percent in the applicable temperature range. The second method used a linear expression shown in Rogers and Yau (1989), that describes the latent heat of vaporization at as a function of an intercept (the value at 0 C), a constant slope taken as the difference between representative values of the specific heat of liquid water at constant pressure (c) and the specific heat of water vapor at constant pressure (cpv), and the temperature. This method yielded a mean bias of 4831 (about 17 times smaller than the bias of Method 1), varying between 39 at the low temperature end to 12,965 at the high temperature end, for a total range of 12,926. The mean error was about 0.21 percent, which is about 16 times smaller than the error associated with Method 1. The third method was similar to the second, but developed and used temperature-dependent functions for c and cpv, instead of fixed values, to compute the slope of the linear function. This method yielded a mean bias of 4771, nearly identical to Method 2. The bias varied between -37 at the low temperature end to 11,835 at the high temperature end, for a total range of 11,872, which was somewhat narrower than Method 2. The mean error was about 0.21 percent, identical to Method 2, but the distribution of these errors through the range of temperatures examined differed than Method 2. The fourth and final method further modified the Rogers and Yau (1989) relationship, fitting a 2 nd -order function to the data, rather than a simple 1 st -order (linear) function. In this sense, this method is similar to the Miller (2017) function used in Method 1. It differs from Miller (2017) in that it uses temperature-dependent functions for c and cpv as terms in the polynomial, and by doing so achieves an R 2 of 1.0000, and a mean bias of only -0.45 (at least four orders of magnitude smaller than all previous methods discussed), which varies between -175 at the low temperature end to 135 at the high temperature end. The total range of the bias is 310, which is about 40 times narrower than all previous methods. The mean error is zero. MT3230 Atmospheric Thermodynamics 11
An interesting contradiction occurs when comparing the results of this paper with the results for boiling temperature in Miller (2017). This paper showed that Method 1, which was developed in Miller (2017), yielded an error in computed latent heat values that was on the order of about three and a half percent in the range between 0 and 100 C, yet the values of lv computed by this method, when used in the Classius-Clapeyron Equation, produced the correct boiling temperature to within a quarter of a degree Celsius in Miller (2017). Method 2 produced much better latent heat values (with errors on the order of only two tenths of a percent), yet, when tested in Miller (2017), its use in the Classius-Clapeyron Equation was shown to produce boiling temperatures that were as much as ten degrees Celsius too warm. This apparent contradiction may be the result of an underlying assumption used when obtaining the closed form of the Classius-Clapeyron Equation, namely, that water vapor can accurately be treated as an ideal gas, and it should be explored further in future work. Acknowledgments. I wish to thank my colleagues Stephen J. Augustyn and R. Bruce Telfeyan for their valuable feedback on this work. References:, 2017: Properties of saturated steam SI Units, The Engineering Toolbox. [Available on-line at http://www.engineeringtoolbox.com/saturated-steam-properties-d_457.html] Dougherty, R.C., and L.N. Howard, 1998: Equilibrium structural model of liquid water: Evidence from heat capacity, spectra, density, and other properties, J. Chem. Phys., 109 (17), 7379-7393; doi: 10.1063/1.477344 Chaplin, M., 2016: Explanation of the Thermodynamic Anomalies of Water (T1-T11). [Available on-line at http://www1.lsbu.ac.uk/water/thermodynamic_anomalies.html] Glickman, T.S., 2000: Glossary of Meteorology, American Meteorological Society, 855 pp. Lide, D.R., 2006: CRC Handbook of Chemistry and Physics, 87 th Ed., CRC Press. Miller, S., 2015: Applied Thermodynamics for Meteorologists, Cambridge University Press, 385 pp. Miller, S., 2017: Methods for computing the boiling temperature of water at varying pressures, Bull. Amer. Meteor. Soc., 98 (7), 1485 1491; doi: 10.1175/BAMS-D-16-0174.1. Rogers, G.F.C., and Y.R. Mayhew, 1988: Thermodynamic and Transport Properties of Fluids, 4 th Ed., Blackwell Publishers, 24 pp. Rogers, R.R., and M.K. Yau, 1989: A Short Course in Cloud Physics, 3 rd Ed., Butterworth-Heinemann, 290 pp. Tsonis, A.A., 2007: An Introduction to Atmospheric Thermodynamics, 2 nd Ed., Cambridge University Press, 187 pp. MT3230 Atmospheric Thermodynamics 12