"""Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, 2017, 2018, 2019 Caleb Bell <Caleb.Andrew.Bell@gmail.com>
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
This module contains functionality for converting between the temperature scales
ITS-90, ITS-76, ITS-68, ITS-48, and ITS-27. These historical temperature scales
can deviate quite a bit from modern temperature measurements! It is important
to convert old measurements of temperature to their modern equivalent.
For reporting bugs, adding feature requests, or submitting pull requests,
please use the `GitHub issue tracker <https://github.com/CalebBell/chemicals/>`_.
.. contents:: :local:
Conversion functions
--------------------
.. autofunction:: chemicals.temperature.T_converter
.. autofunction:: chemicals.temperature.ITS90_68_difference
"""
__all__ = ['T_converter', 'T_scales', 'ITS90_68_difference', 'Ts_68',
'diffs_68', 'Ts_48', 'diffs_48', 'Ts_76', 'diffs_76', 'Ts_27', 'diffs_27']
"""Tabulated values of T68 vs. difference as in [2]_"""
Ts_68 = [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73,
74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88,
89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 110, 120,
130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240,
250, 260, 270, 273.15, 280, 290, 300, 310, 320, 330, 340,
350, 360, 370, 380, 390, 400, 410, 420, 430, 440, 450, 460,
470, 480, 490, 500, 510, 520, 530, 540, 550, 560, 570, 580,
590, 600, 610, 620, 630, 640, 650, 660, 670, 680, 690, 700,
710, 720, 730, 740, 750, 760, 770, 780, 790, 800, 810, 820,
830, 840, 850, 860, 870, 880, 890, 900, 903.89, 910, 920,
930, 940, 950, 960, 970, 980, 990, 1000, 1010, 1020, 1030,
1040, 1050, 1060, 1070, 1080, 1090, 1100, 1110, 1120, 1130,
1140, 1150, 1160, 1170, 1180, 1190, 1200, 1210, 1220, 1230,
1235.08, 1240, 1250, 1260, 1270, 1280, 1290, 1300, 1310,
1320, 1330, 1337.58, 1340, 1350, 1360, 1400, 1500, 1600,
1700, 1800, 1900, 2000, 2100, 2200, 2300, 2400, 2500, 2600,
2700, 2800, 2900, 3000, 3100, 3200, 3300, 3400, 3500, 3600,
3700, 3800, 3900, 4000, 4100, 4200, 4300]
diffs_68 = [-0.006, -0.003, -0.004, -0.006, -0.008, -0.009, -0.009,
-0.008, -0.007, -0.007, -0.006, -0.005, -0.004, -0.004,
-0.005, -0.006, -0.006, -0.007, -0.008, -0.008, -0.008,
-0.007, -0.007, -0.007, -0.006, -0.006, -0.006, -0.006,
-0.006, -0.006, -0.006, -0.007, -0.007, -0.007, -0.006,
-0.006, -0.006, -0.005, -0.005, -0.004, -0.003, -0.002,
-0.001, 0, 0.001, 0.002, 0.003, 0.003, 0.004, 0.004,
0.005, 0.005, 0.006, 0.006, 0.007, 0.007, 0.007, 0.007,
0.007, 0.007, 0.007, 0.008, 0.008, 0.008, 0.008, 0.008,
0.008, 0.008, 0.008, 0.008, 0.008, 0.008, 0.008, 0.008,
0.008, 0.008, 0.008, 0.008, 0.008, 0.008, 0.008, 0.008,
0.008, 0.009, 0.009, 0.009, 0.009, 0.011, 0.013, 0.014,
0.014, 0.014, 0.014, 0.013, 0.012, 0.012, 0.011, 0.01,
0.009, 0.008, 0.007, 0.005, 0.003, 0.001, 0, -0.001,
-0.004, -0.006, -0.009, -0.012, -0.015, -0.017, -0.02,
-0.023, -0.025, -0.027, -0.029, -0.031, -0.033, -0.035,
-0.037, -0.038, -0.039, -0.039, -0.04, -0.04, -0.04,
-0.04, -0.04, -0.04, -0.04, -0.039, -0.039, -0.039,
-0.039, -0.039, -0.039, -0.04, -0.04, -0.041, -0.042,
-0.043, -0.044, -0.046, -0.047, -0.05, -0.052, -0.055,
-0.058, -0.061, -0.064, -0.067, -0.071, -0.074, -0.078,
-0.082, -0.086, -0.089, -0.093, -0.097, -0.1, -0.104,
-0.107, -0.111, -0.114, -0.117, -0.121, -0.124, -0.125,
-0.12, -0.1, -0.09, -0.07, -0.06, -0.04, -0.03, -0.01, 0,
0.01, 0.02, 0.03, 0.04, 0.05, 0.05, 0.05, 0.05, 0.05,
0.04, 0.04, 0.03, 0.02, 0.01, 0, -0.01, -0.03, -0.04,
-0.06, -0.08, -0.09, -0.11, -0.13, -0.14, -0.15, -0.16,
-0.17, -0.19, -0.2, -0.21, -0.22, -0.23, -0.24, -0.25,
-0.25, -0.25, -0.25, -0.26, -0.26, -0.27, -0.31, -0.36,
-0.4, -0.45, -0.5, -0.56, -0.62, -0.68, -0.74, -0.81,
-0.87, -0.95, -1.02, -1.09, -1.17, -1.26, -1.34, -1.43,
-1.52, -1.62, -1.71, -1.81, -1.92, -2.02, -2.13, -2.24,
-2.35, -2.46, -2.58]
Ts_48 = [93.15, 103.15, 113.15, 123.15, 133.15, 143.15, 153.15, 163.15, 173.15,
183.15, 193.15, 203.15, 213.15, 223.15, 233.15, 243.15, 253.15, 263.15,
273.15, 283.15, 293.15, 303.15, 313.15, 323.15, 333.15, 343.15, 353.15,
363.15, 373.15, 383.15, 393.15, 403.15, 413.15, 423.15, 433.15, 443.15,
453.15, 463.15, 473.15, 483.15, 493.15, 503.15, 513.15, 523.15, 533.15,
543.15, 553.15, 563.15, 573.15, 583.15, 593.15, 603.15, 613.15, 623.15,
633.15, 643.15, 653.15, 663.15, 673.15, 683.15, 693.15, 703.15, 713.15,
723.15, 733.15, 743.15, 753.15, 763.15, 773.15, 783.15, 793.15, 803.15,
813.15, 823.15, 833.15, 843.15, 853.15, 863.15, 873.15, 883.15, 893.15,
903.15, 913.15, 923.15, 933.15, 943.15, 953.15, 963.15, 973.15, 983.15,
993.15, 1003.15, 1013.15, 1023.15, 1033.15, 1043.15, 1053.15, 1063.15,
1073.15, 1083.15, 1093.15, 1103.15, 1113.15, 1123.15, 1133.15, 1143.15,
1153.15, 1163.15, 1173.15, 1183.15, 1193.15, 1203.15, 1213.15, 1223.15,
1233.15, 1243.15, 1253.15, 1263.15, 1273.15, 1283.15, 1293.15, 1303.15,
1313.15, 1323.15, 1333.15, 1343.15, 1373.15, 1473.15, 1573.15, 1673.15,
1773.15, 1873.15, 1973.15, 2073.15, 2173.15, 2273.15, 2373.15, 2473.15,
2573.15, 2673.15, 2773.15, 2873.15, 2973.15, 3073.15, 3173.15, 3273.15,
3373.15, 3473.15, 3573.15, 3673.15, 3773.15, 3873.15, 3973.15, 4073.15,
4173.15, 4273.15]
diffs_48 = [0.02, 0.017, 0.007, 0, 0.001, 0.008, 0.017, 0.026, 0.035,
0.041, 0.045, 0.045, 0.042, 0.038, 0.032, 0.024, 0.016,
0.008, 0, -0.006, -0.012, -0.016, -0.02, -0.023, -0.026,
-0.026, -0.027, -0.027, -0.026, -0.024, -0.023, -0.02,
-0.018, -0.016, -0.012, -0.009, -0.005, -0.001, 0.003,
0.007, 0.011, 0.014, 0.018, 0.021, 0.024, 0.028, 0.03,
0.032, 0.034, 0.035, 0.036, 0.036, 0.037, 0.036, 0.035,
0.034, 0.032, 0.03, 0.028, 0.024, 0.022, 0.019, 0.015,
0.012, 0.009, 0.007, 0.004, 0.002, 0, -0.001, -0.002,
-0.001, 0, 0.002, 0.007, 0.011, 0.018, 0.025, 0.035,
0.047, 0.06, 0.075, 0.15, 0.22, 0.3, 0.37, 0.45, 0.52,
0.59, 0.66, 0.73, 0.78, 0.83, 0.88, 0.92, 0.94, 0.97,
0.99, 1.01, 1.02, 1.01, 1, 1, 0.99, 0.98, 0.97, 0.95,
0.95, 0.94, 0.95, 0.95, 0.96, 0.97, 0.98, 0.98, 0.99,
1.01, 1.03, 1.05, 1.07, 1.09, 1.11, 1.13, 1.15, 1.17,
1.19, 1.2, 1.4, 1.5, 1.6, 1.8, 1.9, 2.1, 2.2, 2.3, 2.5,
2.7, 2.9, 3.1, 3.2, 3.4, 3.7, 3.8, 4, 4.2, 4.4, 4.6, 4.8,
5.1, 5.3, 5.5, 5.8, 6, 6.3, 6.6, 6.8]
Ts_76 = [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27]
diffs_76 = [-0.0001, -0.0002, -0.0003, -0.0004, -0.0005, -0.0006,
-0.0007, -0.0008, -0.001, -0.0011, -0.0013, -0.0014,
-0.0016, -0.0018, -0.002, -0.0022, -0.0025, -0.0027,
-0.003, -0.0032, -0.0035, -0.0038, -0.0041]
Ts_27 = [903.15, 913.15, 923.15, 933.15, 943.15, 953.15, 963.15, 973.15, 983.15,
993.15, 1003.15, 1013.15, 1023.15, 1033.15, 1043.15, 1053.15, 1063.15,
1073.15, 1083.15, 1093.15, 1103.15, 1113.15, 1123.15, 1133.15, 1143.15,
1153.15, 1163.15, 1173.15, 1183.15, 1193.15, 1203.15, 1213.15, 1223.15,
1233.15, 1243.15, 1253.15, 1263.15, 1273.15, 1283.15, 1293.15, 1303.15,
1313.15, 1323.15, 1333.15, 1373.15, 1473.15, 1573.15, 1673.15, 1773.15,
1873.15, 1973.15, 2073.15, 2173.15, 2273.15, 2373.15, 2473.15, 2573.15,
2673.15, 2773.15, 2873.15, 2973.15, 3073.15, 3173.15, 3273.15, 3373.15,
3473.15, 3573.15, 3673.15, 3773.15, 3873.15, 3973.15, 4073.15, 4173.15,
4273.15]
diffs_27 = [0.08, 0.19, 0.3, 0.42, 0.52, 0.63, 0.73, 0.83, 0.93, 1.02,
1.09, 1.16, 1.23, 1.29, 1.32, 1.37, 1.4, 1.42, 1.44, 1.44,
1.43, 1.43, 1.42, 1.41, 1.39, 1.36, 1.36, 1.34, 1.33,
1.32, 1.32, 1.31, 1.3, 1.28, 1.27, 1.27, 1.26, 1.25, 1.25,
1.24, 1.22, 1.21, 1.2, 1.18, 1.04, 0.9, 0.35, -0.09,
-0.54, -1.09, -1.64, -2.4, -3.06, -3.92, -4.69, -5.55,
-6.53, -7.6, -8.57, -9.75, -11, -12.2, -13.6, -15.1,
-16.6, -18.3, -19.9, -21.7, -23.7, -25.7, -27.9, -30.1,
-32.4, -35.1]
built_splines_90_68 = False
def build_splines():
global built_splines_90_68
global T68_to_T90, T48_to_T90, T76_to_T90, T27_to_T90
global T90_to_T68, T90_to_T48, T90_to_T76, T90_to_T27
import numpy as np
from scipy.interpolate import UnivariateSpline
Ts_90_from_68 = np.array(Ts_68) + diffs_68
Ts_90_from_48 = np.array(Ts_48) + diffs_48
Ts_90_from_76 = np.array(Ts_76) + diffs_76
Ts_90_from_27 = np.array(Ts_27) + diffs_27
T68_to_T90 = UnivariateSpline(Ts_68, Ts_90_from_68, s=0)
T48_to_T90 = UnivariateSpline(Ts_48, Ts_90_from_48, s=0)
T76_to_T90 = UnivariateSpline(Ts_76, Ts_90_from_76, s=0)
T27_to_T90 = UnivariateSpline(Ts_27, Ts_90_from_27, s=0)
T90_to_T68 = UnivariateSpline(Ts_90_from_68, Ts_68, s=0)
T90_to_T48 = UnivariateSpline(Ts_90_from_48, Ts_48, s=0)
T90_to_T76 = UnivariateSpline(Ts_90_from_76, Ts_76, s=0)
T90_to_T27 = UnivariateSpline(Ts_90_from_27, Ts_27, s=0)
built_splines_90_68 = True
[docs]def ITS90_68_difference(T):
r'''Calculates the difference between ITS-90 and ITS-68 scales using a
series of models listed in [1]_, [2]_, and [3]_.
The temperature difference is given by the following equations:
From 13.8 K to 73.15 K:
.. math::
T_{90} - T_{68} = a_0 + \sum_{i=1}^{12} a_i[(T_{90}/K-40)/40]^i
From 83.8 K to 903.75 K:
.. math::
T_{90} - T_{68} = \sum_{i=1}^8 b_i[(T_{90}/K - 273.15)/630]^i
From 903.75 K to 1337.33 K:
.. math::
T_{90} - T_{68} = \sum_{i=0}^5 c_i[T_{90}/^\circ C]^i
Above 1337.33 K:
.. math::
T_{90} - T_{68} = -1.398\cdot 10^{-7}\left(\frac{T_{90}}{K}\right)^2
Parameters
----------
T : float
Temperature, ITS-90, or approximately ITS-68 [K]
Returns
-------
dT : float
Temperature, difference between ITS-90 and ITS-68 at T [K]
Notes
-----
The conversion is straightforward when T90 is known. Theoretically, the
model should be solved numerically to convert the reverse way. However,
according to [4]_, the difference is under 0.05 mK from 73.15 K to
903.15 K, and under 0.26 mK up to 1337.33 K.
For temperatures under 13.8 K, no conversion is performed.
The first set of coefficients is:
-0.005903, 0.008174, -0.061924, -0.193388, 1.490793, 1.252347, -9.835868,
1.411912, 25.277595, -19.183815, -18.437089, 27.000895, -8.716324.
The second set of coefficients is:
0, -0.148759, -0.267408, 1.08076, 1.269056, -4.089591, -1.871251,
7.438081, -3.536296.
The third set of coefficients is:
7.8687209E1, -4.7135991E-1, 1.0954715E-3, -1.2357884E-6, 6.7736583E-10,
-1.4458081E-13.
These last coefficients use the temperature in degrees Celcius. A slightly
older model used the following coefficients but a different equation over
the same range:
-0.00317, -0.97737, 1.2559, 2.03295, -5.91887, -3.23561, 7.23364,
5.04151.
The model for these coefficients was:
.. math::
T_{90} - T_{68} = c_0 + \sum_{i=1}^7 c_i[(T_{90}/K - 1173.15)/300]^i
For temperatures larger than several thousand K, the differences have no
meaning and grows quadratically.
Examples
--------
>>> ITS90_68_difference(1000.)
0.01231818956580355
References
----------
.. [1] Bedford, R. E., G. Bonnier, H. Maas, and F. Pavese. "Techniques for
Approximating the International Temperature Scale of 1990." Bureau
International Des Poids et Mesures, Sfievres, 1990.
.. [2] Wier, Ron D., and Robert N. Goldberg. "On the Conversion of
Thermodynamic Properties to the Basis of the International Temperature
Scale of 1990." The Journal of Chemical Thermodynamics 28, no. 3
(March 1996): 261-76. doi:10.1006/jcht.1996.0026.
.. [3] Goldberg, Robert N., and R. D. Weir. "Conversion of Temperatures
and Thermodynamic Properties to the Basis of the International
Temperature Scale of 1990 (Technical Report)." Pure and Applied
Chemistry 64, no. 10 (1992): 1545-1562. doi:10.1351/pac199264101545.
.. [4] Code10.info. "Conversions among International Temperature Scales."
Accessed May 22, 2016. http://www.code10.info/index.php%3Foption%3Dcom_content%26view%3Darticle%26id%3D83:conversions-among-international-temperature-scales%26catid%3D60:temperature%26Itemid%3D83.
'''
# Numba requires these coefficients to be in a list not a tuple
ais = [-0.005903, 0.008174, -0.061924, -0.193388, 1.490793, 1.252347,
-9.835868, 1.411912, 25.277595, -19.183815, -18.437089, 27.000895,
-8.716324]
bis = [0, -0.148759, -0.267408, 1.08076, 1.269056, -4.089591, -1.871251,
7.438081, -3.536296]
# cis = (-0.00317, -0.97737, 1.2559, 2.03295, -5.91887, -3.23561, 7.23364,
# 5.04151)
new_cs = [7.8687209E1, -4.7135991E-1, 1.0954715E-3, -1.2357884E-6,
6.7736583E-10, -1.4458081E-13]
dT = 0.0
if T < 13.8:
dT = 0.0
elif T >= 13.8 and T <= 73.15:
for i in range(13):
dT += ais[i]*((T - 40.)/40.)**i
elif T > 73.15 and T < 83.8:
dT = 0.0
elif T >= 83.8 and T <= 903.75:
for i in range(9):
dT += bis[i]*((T - 273.15)/630.)**i
elif T > 903.75 and T <= 1337.33:
# Revised function exists, but does not match the tabulated data
# for i in range(8):
# dT += cis[i]*((T - 1173.15)/300.)**i
for i in range(6):
dT += new_cs[i]*(T-273.15)**i
elif T > 1337.33:
dT = -1.398E-7*T**2
return dT
T_scales = ['ITS-90', 'ITS-68', 'ITS-27', 'ITS-48', 'ITS-76']
[docs]def T_converter(T, current, desired):
r'''Converts the a temperature reading made in any of the scales
'ITS-90', 'ITS-68','ITS-48', 'ITS-76', or 'ITS-27' to any of the other
scales. Not all temperature ranges can be converted to other ranges; for
instance, 'ITS-76' is purely for low temperatures, and 5 K on it has no
conversion to 'ITS-90' or any other scale. Both a conversion to ITS-90 and
to the desired scale must be possible for the conversion to occur.
The conversion uses cubic spline interpolation.
ITS-68 conversion is valid from 14 K to 4300 K.
ITS-48 conversion is valid from 93.15 K to 4273.15 K
ITS-76 conversion is valid from 5 K to 27 K.
ITS-27 is valid from 903.15 K to 4273.15 K.
Parameters
----------
T : float
Temperature, on `current` scale [K]
current : str
String representing the scale T is in, 'ITS-90', 'ITS-68',
'ITS-48', 'ITS-76', or 'ITS-27'.
desired : str
String representing the scale T will be returned in, 'ITS-90',
'ITS-68', 'ITS-48', 'ITS-76', or 'ITS-27'.
Returns
-------
T : float
Temperature, on scale `desired` [K]
Notes
-----
Because the conversion is performed by spline functions, a re-conversion
of a value will not yield exactly the original value. However, it is quite
close.
The use of splines is quite quick (20 micro seconds/calculation). While
just a spline for one-way conversion could be used, a numerical solver
would have to be used to obtain an exact result for the reverse conversion.
This was found to take approximately 1 ms/calculation, depending on the
region.
Examples
--------
>>> T_converter(500, 'ITS-68', 'ITS-48')
499.9470092992346
References
----------
.. [1] Wier, Ron D., and Robert N. Goldberg. "On the Conversion of
Thermodynamic Properties to the Basis of the International Temperature
Scale of 1990." The Journal of Chemical Thermodynamics 28, no. 3
(March 1996): 261-76. doi:10.1006/jcht.1996.0026.
.. [2] Goldberg, Robert N., and R. D. Weir. "Conversion of Temperatures
and Thermodynamic Properties to the Basis of the International
Temperature Scale of 1990 (Technical Report)." Pure and Applied
Chemistry 64, no. 10 (1992): 1545-1562. doi:10.1351/pac199264101545.
'''
def range_check(T, Tmin, Tmax):
if T < Tmin or T > Tmax:
raise Exception('Temperature conversion is outside one or both scales')
if not built_splines_90_68:
build_splines()
try:
if current == 'ITS-90':
pass
elif current == 'ITS-68':
range_check(T, 13.999, 4300.0001)
T = T68_to_T90(T)
elif current == 'ITS-76':
range_check(T, 4.9999, 27.0001)
T = T76_to_T90(T)
elif current == 'ITS-48':
range_check(T, 93.149999, 4273.15001)
T = T48_to_T90(T)
elif current == 'ITS-27':
range_check(T, 903.15, 4273.15)
T = T27_to_T90(T)
else:
raise Exception('Current scale not supported')
# T should be in ITS-90 now
if desired == 'ITS-90':
pass
elif desired == 'ITS-68':
range_check(T, 13.999, 4300.0001)
T = T90_to_T68(T)
elif desired == 'ITS-76':
range_check(T, 4.9999, 27.0001)
T = T90_to_T76(T)
elif desired == 'ITS-48':
range_check(T, 93.149999, 4273.15001)
T = T90_to_T48(T)
elif desired == 'ITS-27':
range_check(T, 903.15, 4273.15)
T = T90_to_T27(T)
else:
raise Exception('Desired scale not supported')
except ValueError:
raise Exception('Temperature could not be converted to desired scale')
return float(T)