def test_sres(print_result=False):
"""Test the residual entropy function to see if it is working correctly."""
if print_result:
print('------ 325 K ------')
print('\t\t\t PC-SAFT\t Reference \tRelative error')
t = 325 # K
p = 101325 # Pa
# all reference values are from PC-SAFT implemented in Aspen Plus
# Toluene ----------
x = np.asarray([1.])
m = np.asarray([2.8149])
s = np.asarray([3.7169])
e = np.asarray([285.69])
pyargs = {'x': x, 'm': m, 's': s, 'e': e}
den = pcsaft_den(t, p, pyargs, phase='liq')
ref = -96.3692 # J mol^-1 K^-1
calc = pcsaft_sres(t, den, pyargs)
assert abs((calc - ref) / ref * 100) < 1e-2
if print_result:
print('Toluene, liquid:\t\t', calc, ref, (calc - ref) / ref * 100,
'J/mol/K')
den = pcsaft_den(t, p, pyargs, phase='vap')
ref = -0.71398 # J mol^-1 K^-1
calc = pcsaft_sres(t, den, pyargs)
assert abs((calc - ref) / ref * 100) < 1e-2
if print_result:
print('Toluene, vapor:\t\t', calc, ref, (calc - ref) / ref * 100,
'J/mol/K')
# Acetic acid ---------
m = np.asarray([1.3403])
s = np.asarray([3.8582])
e = np.asarray([211.59])
volAB = np.asarray([0.075550])
eAB = np.asarray([3044.4])
pyargs = {'x': x, 'm': m, 's': s, 'e': e, 'e_assoc': eAB, 'vol_a': volAB}
den = pcsaft_den(t, p, pyargs, phase='liq')
ref = -98.1127 # J mol^-1 K^-1
calc = pcsaft_sres(t, den, pyargs)
assert abs((calc - ref) / ref * 100) < 1e-2
if print_result:
print('Acetic acid, liquid:\t\t', calc, ref, (calc - ref) / ref * 100,
'J/mol/K')
den = pcsaft_den(t, p, pyargs, phase='vap')
ref = -40.8743 # J mol^-1 K^-1
calc = pcsaft_sres(t, den, pyargs)
assert abs((calc - ref) / ref * 100) < 1e-2
if print_result:
print('Acetic acid, vapor:\t\t', calc, ref, (calc - ref) / ref * 100,
'J/mol/K')
def test_osmoticC(print_result=False):
"""Test the function for calculating osmotic coefficients to see if it is working correctly."""
# NaCl in water
# 0 = Na+, 1 = Cl-, 2 = H2O
x = np.asarray([0.0629838206, 0.0629838206, 0.8740323588])
m = np.asarray([1, 1, 1.2047])
s = np.asarray([2.8232, 2.7599589, 0.])
e = np.asarray([230.00, 170.00, 353.9449])
volAB = np.asarray([0, 0, 0.0451])
eAB = np.asarray([0, 0, 2425.67])
k_ij = np.asarray([[0, 0.317, 0], [0.317, 0, -0.25], [0, -0.25, 0]])
z = np.asarray([1., -1., 0.])
ref = 1.116 # source: R. A. Robinson and R. H. Stokes, Electrolyte Solutions: Second Revised Edition. Dover Publications, 1959.
t = 293.15 # K
s[2] = 2.7927 + 10.11 * np.exp(-0.01775 * t) - 1.417 * np.exp(
-0.01146 * t) # temperature dependent segment diameter for water
k_ij[0, 2] = -0.007981 * t + 2.37999
k_ij[2, 0] = -0.007981 * t + 2.37999
dielc = dielc_water(t)
pyargs = {
'x': x,
'm': m,
's': s,
'e': e,
'e_assoc': eAB,
'vol_a': volAB,
'k_ij': k_ij,
'z': z,
'dielc': dielc
}
rho = pcsaft_den(t, 2339.3, pyargs, phase='liq')
result = pcsaft_osmoticC(t, rho, pyargs)
calc = result[0]
assert abs((calc - ref) / ref * 100) < 2
if print_result:
print('\n########## Test with aqueous NaCl ##########')
print('----- Osmotic coefficient at 293.15 K -----')
print(' Reference:', ref)
print(' PC-SAFT:', calc)
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
import numpy as np
from pcsaft import pcsaft_den
# Toluene
x = np.asarray([1.])
m = np.asarray([2.8149])
s = np.asarray([3.7169])
e = np.asarray([285.69])
pyargs = {'m':m, 's':s, 'e':e}
t = 320 # K
p = 101325 # Pa
den = pcsaft_den(t, p, x, pyargs, phase='liq')
print('Density of toluene at {} K:'.format(t), den, 'mol m^-3')
# Water using default 2B association scheme
x = np.asarray([1.])
m = np.asarray([1.2047])
e = np.asarray([353.95])
volAB = np.asarray([0.0451])
eAB = np.asarray([2425.67])
t = 274
p = 101325
s = np.asarray([2.7927 + 10.11*np.exp(-0.01775*t) - 1.417*np.exp(-0.01146*t)]) # temperature dependent sigma is used for better accuracy
pyargs = {'m':m, 's':s, 'e':e, 'e_assoc':eAB, 'vol_a':volAB}
den = pcsaft_den(t, p, x, pyargs, phase='liq')
print('Density of water at {} K:'.format(t), den, 'mol m^-3')
# Water using 4C association scheme
x = np.asarray([1.])
def test_dadt(print_result=False):
"""Test the function for the temperature derivative of the Helmholtz energy."""
# Toluene
x = np.asarray([1.])
m = np.asarray([2.8149])
s = np.asarray([3.7169])
e = np.asarray([285.69])
pyargs = {'x': x, 'm': m, 's': s, 'e': e}
p = 100000.
t = 330.
rho = pcsaft_den(t, p, pyargs, phase='liq')
dadt_eos = pcsaft_dadt(t, rho, pyargs)
# calculating numerical derivative
der1 = pcsaft_ares(t - 1, rho, pyargs)
der2 = pcsaft_ares(t + 1, rho, pyargs)
dadt_num = (der2 - der1) / 2.
assert abs((dadt_eos - dadt_num) / dadt_num * 100) < 2e-2
if print_result:
print('\n########## Test with toluene ##########')
print(' Numerical derivative:', dadt_num)
print(' PC-SAFT derivative:', dadt_eos)
print(' Relative deviation:',
(dadt_eos - dadt_num) / dadt_num * 100, '%')
# Acetic acid
m = np.asarray([1.3403])
s = np.asarray([3.8582])
e = np.asarray([211.59])
volAB = np.asarray([0.075550])
eAB = np.asarray([3044.4])
pyargs = {'x': x, 'm': m, 's': s, 'e': e, 'e_assoc': eAB, 'vol_a': volAB}
p = 100000.
t = 310.
rho = pcsaft_den(t, p, pyargs, phase='liq')
dadt_eos = pcsaft_dadt(t, rho, pyargs)
# calculating numerical derivative
der1 = pcsaft_ares(t - 1, rho, pyargs)
der2 = pcsaft_ares(t + 1, rho, pyargs)
dadt_num = (der2 - der1) / 2.
assert abs((dadt_eos - dadt_num) / dadt_num * 100) < 2e-2
if print_result:
print('\n########## Test with acetic acid ##########')
print(' Numerical derivative:', dadt_num)
print(' PC-SAFT derivative:', dadt_eos)
print(' Relative deviation:',
(dadt_eos - dadt_num) / dadt_num * 100, '%')
# Water
m = np.asarray([1.2047])
e = np.asarray([353.95])
volAB = np.asarray([0.0451])
eAB = np.asarray([2425.67])
p = 100000.
t = 290.
s = np.asarray(
[2.7927 + 10.11 * np.exp(-0.01775 * t) - 1.417 * np.exp(-0.01146 * t)])
pyargs = {'x': x, 'm': m, 's': s, 'e': e, 'e_assoc': eAB, 'vol_a': volAB}
rho = pcsaft_den(t, p, pyargs, phase='liq')
dadt_eos = pcsaft_dadt(t, rho, pyargs)
# calculating numerical derivative
der1 = pcsaft_ares(t - 1, rho, pyargs)
der2 = pcsaft_ares(t + 1, rho, pyargs)
dadt_num = (der2 - der1) / 2.
assert abs((dadt_eos - dadt_num) / dadt_num * 100) < 2e-2
if print_result:
print('\n########## Test with water ##########')
print(' Numerical derivative:', dadt_num)
print(' PC-SAFT derivative:', dadt_eos)
print(' Relative deviation:',
(dadt_eos - dadt_num) / dadt_num * 100, '%')
# Dimethyl ether
m = np.asarray([2.2634])
s = np.asarray([3.2723])
e = np.asarray([210.29])
dpm = np.asarray([1.3])
dip_num = np.asarray([1.0])
pyargs = {'x': x, 'm': m, 's': s, 'e': e, 'dipm': dpm, 'dip_num': dip_num}
p = 100000.
t = 370.
rho = pcsaft_den(t, p, pyargs, phase='liq')
dadt_eos = pcsaft_dadt(t, rho, pyargs)
# calculating numerical derivative
der1 = pcsaft_ares(t - 1, rho, pyargs)
der2 = pcsaft_ares(t + 1, rho, pyargs)
dadt_num = (der2 - der1) / 2.
assert abs((dadt_eos - dadt_num) / dadt_num * 100) < 2e-2
if print_result:
print('\n########## Test with dimethyl ether ##########')
print(' Numerical derivative:', dadt_num)
print(' PC-SAFT derivative:', dadt_eos)
print(' Relative deviation:',
(dadt_eos - dadt_num) / dadt_num * 100, '%')
# Aqueous NaCl
# 0 = Na+, 1 = Cl-, 2 = H2O
x = np.asarray([0.0907304774758426, 0.0907304774758426, 0.818539045048315])
m = np.asarray([1, 1, 1.2047])
s = np.asarray([2.8232, 2.7599589, 0.])
e = np.asarray([230.00, 170.00, 353.9449])
volAB = np.asarray([0, 0, 0.0451])
eAB = np.asarray([0, 0, 2425.67])
k_ij = np.asarray([[0, 0.317, 0], [0.317, 0, -0.25], [0, -0.25, 0]])
z = np.asarray([1., -1., 0.])
t = 298.15 # K
p = 100000. # Pa
s[2] = 2.7927 + 10.11 * np.exp(-0.01775 * t) - 1.417 * np.exp(
-0.01146 * t) # temperature dependent segment diameter for water
k_ij[0, 2] = -0.007981 * t + 2.37999
k_ij[2, 0] = -0.007981 * t + 2.37999
dielc = dielc_water(t)
pyargs = {
'x': x,
'm': m,
's': s,
'e': e,
'e_assoc': eAB,
'vol_a': volAB,
'k_ij': k_ij,
'z': z,
'dielc': dielc
}
rho = pcsaft_den(t, p, pyargs, phase='liq')
dadt_eos = pcsaft_dadt(t, rho, pyargs)
# calculating numerical derivative
der1 = pcsaft_ares(t - 1, rho, pyargs)
der2 = pcsaft_ares(t + 1, rho, pyargs)
dadt_num = (der2 - der1) / 2.
assert abs((dadt_eos - dadt_num) / dadt_num * 100) < 2e-2
if print_result:
print('\n########## Test with aqueous NaCl ##########')
print(' Numerical derivative:', dadt_num)
print(' PC-SAFT derivative:', dadt_eos)
print(' Relative deviation:',
(dadt_eos - dadt_num) / dadt_num * 100, '%')
def test_density(print_result=False):
"""Test the density function to see if it is working correctly."""
# Toluene
x = np.asarray([1.])
m = np.asarray([2.8149])
s = np.asarray([3.7169])
e = np.asarray([285.69])
pyargs = {'x': x, 'm': m, 's': s, 'e': e}
ref = 9135.590853014008 # source: reference EOS in CoolProp
calc = pcsaft_den(320, 101325, pyargs, phase='liq')
assert abs((calc - ref) / ref * 100) < 2
if print_result:
print('########## Test with toluene ##########')
print('----- Density at 320 K and 101325 Pa -----')
print(' Reference:', ref, 'mol m^-3')
print(' PC-SAFT:', calc, 'mol m^-3')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
# Water
m = np.asarray([1.2047])
e = np.asarray([353.95])
volAB = np.asarray([0.0451])
eAB = np.asarray([2425.67])
ref = 55502.5970532902 # source: IAWPS95 EOS
t = 274
s = np.asarray(
[2.7927 + 10.11 * np.exp(-0.01775 * t) - 1.417 * np.exp(-0.01146 * t)])
pyargs = {'x': x, 'm': m, 's': s, 'e': e, 'e_assoc': eAB, 'vol_a': volAB}
calc = pcsaft_den(t, 101325, pyargs, phase='liq')
assert abs((calc - ref) / ref * 100) < 2
if print_result:
print('\n########## Test with water ##########')
print('----- Density at 274 K and 101325 Pa -----')
print(' Reference:', ref, 'mol m^-3')
print(' PC-SAFT:', calc, 'mol m^-3')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
# Acetic acid
m = np.asarray([1.3403])
s = np.asarray([3.8582])
e = np.asarray([211.59])
volAB = np.asarray([0.075550])
eAB = np.asarray([3044.4])
pyargs = {'x': x, 'm': m, 's': s, 'e': e, 'e_assoc': eAB, 'vol_a': volAB}
ref = 17240. # source: DIPPR correlation
calc = pcsaft_den(305, 101325, pyargs, phase='liq')
assert abs((calc - ref) / ref * 100) < 2
if print_result:
print('\n########## Test with acetic acid ##########')
print('----- Density at 305 K and 101325 Pa -----')
print(' Reference:', ref, 'mol m^-3')
print(' PC-SAFT:', calc, 'mol m^-3')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
# Dimethyl ether
m = np.asarray([2.2634])
s = np.asarray([3.2723])
e = np.asarray([210.29])
dpm = np.asarray([1.3])
dip_num = np.asarray([1.0])
pyargs = {'x': x, 'm': m, 's': s, 'e': e, 'dipm': dpm, 'dip_num': dip_num}
ref = 16110. # source: DIPPR correlation
calc = pcsaft_den(240, 101325, pyargs, phase='liq')
assert abs((calc - ref) / ref * 100) < 2
if print_result:
print('\n########## Test with dimethyl ether ##########')
print('----- Density at 240 K and 101325 Pa -----')
print(' Reference:', ref, 'mol m^-3')
print(' PC-SAFT:', calc, 'mol m^-3')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
# Binary mixture: methanol-cyclohexane
#0 = methanol, 1 = cyclohexane
x = np.asarray([0.0550, 0.945])
m = np.asarray([1.5255, 2.5303])
s = np.asarray([3.2300, 3.8499])
e = np.asarray([188.90, 278.11])
volAB = np.asarray([0.035176, 0.])
eAB = np.asarray([2899.5, 0.])
k_ij = np.asarray([[0, 0.051], [0.051, 0]])
pyargs = {
'x': x,
'm': m,
's': s,
'e': e,
'e_assoc': eAB,
'vol_a': volAB,
'k_ij': k_ij
}
ref = 9506.1 # source: J. Canosa, A. Rodríguez, and J. Tojo, “Liquid−Liquid Equilibrium and Physical Properties of the Ternary Mixture (Dimethyl Carbonate + Methanol + Cyclohexane) at 298.15 K,” J. Chem. Eng. Data, vol. 46, no. 4, pp. 846–850, Jul. 2001.
calc = pcsaft_den(298.15, 101325, pyargs, phase='liq')
assert abs((calc - ref) / ref * 100) < 2
if print_result:
print(
'\n########## Test with methanol-cyclohexane mixture ##########')
print('----- Density at 298.15 K and 101325 Pa -----')
print(' Reference:', ref, 'mol m^-3')
print(' PC-SAFT:', calc, 'mol m^-3')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
# NaCl in water
# 0 = Na+, 1 = Cl-, 2 = H2O
x = np.asarray([0.010579869455908, 0.010579869455908, 0.978840261088184])
m = np.asarray([1, 1, 1.2047])
s = np.asarray([2.8232, 2.7599589, 0.])
e = np.asarray([230.00, 170.00, 353.9449])
volAB = np.asarray([0, 0, 0.0451])
eAB = np.asarray([0, 0, 2425.67])
k_ij = np.asarray([[0, 0.317, 0], [0.317, 0, -0.25], [0, -0.25, 0]])
z = np.asarray([1., -1., 0.])
ref = 55507.23 # source: Rodriguez H.; Soto A.; Arce A.; Khoshkbarchi M.K.: Apparent Molar Volume, Isentropic Compressibility, Refractive Index, and Viscosity of DL-Alanine in Aqueous NaCl Solutions. J.Solution Chem. 32 (2003) 53-63
t = 298.15 # K
s[2] = 2.7927 + 10.11 * np.exp(-0.01775 * t) - 1.417 * np.exp(
-0.01146 * t) # temperature dependent segment diameter for water
k_ij[0, 2] = -0.007981 * t + 2.37999
k_ij[2, 0] = -0.007981 * t + 2.37999
dielc = dielc_water(t)
pyargs = {
'x': x,
'm': m,
's': s,
'e': e,
'e_assoc': eAB,
'vol_a': volAB,
'k_ij': k_ij,
'z': z,
'dielc': dielc
}
calc = pcsaft_den(t, 101325, pyargs, phase='liq')
assert abs((calc - ref) / ref * 100) < 2
if print_result:
print('\n########## Test with aqueous NaCl ##########')
print('----- Density at 298.15 K and 101325 Pa -----')
print(' Reference:', ref, 'mol m^-3')
print(' PC-SAFT:', calc, 'mol m^-3')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
# Propane
x = np.asarray([1.])
m = np.asarray([2.0020])
s = np.asarray([3.6184])
e = np.asarray([208.11])
pyargs = {'x': x, 'm': m, 's': s, 'e': e}
t = 85.525 # K
p = 1.7551e-4 # Pa
ref = 16621.0 # source: reference EOS in CoolProp
calc = pcsaft_den(t, p, pyargs, phase='liq')
assert abs((calc - ref) / ref * 100) < 2
if print_result:
print('########## Test with propane ##########')
print('----- Density at {} K and {} Pa -----'.format(t, p))
print(' Reference:', ref, 'mol m^-3')
print(' PC-SAFT:', calc, 'mol m^-3')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
t = 85.525 # K
p = 1.39e-4 # Pa
ref = 1.9547e-7 # source: reference EOS in CoolProp
calc = pcsaft_den(t, p, pyargs, phase='vap')
assert abs((calc - ref) / ref * 100) < 2
if print_result:
print('----- Density at {} K and {} Pa -----'.format(t, p))
print(' Reference:', ref, 'mol m^-3')
print(' PC-SAFT:', calc, 'mol m^-3')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
t = 293 # K
p = 833240 # Pa
ref = 11346.0 # source: reference EOS in CoolProp
calc = pcsaft_den(t, p, pyargs, phase='liq')
assert abs((calc - ref) / ref * 100) < 2
if print_result:
print('----- Density at {} K and {} Pa -----'.format(t, p))
print(' Reference:', ref, 'mol m^-3')
print(' PC-SAFT:', calc, 'mol m^-3')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
t = 430 # K
p = 2000000 # Pa
ref = 623.59 # source: reference EOS in CoolProp
calc = pcsaft_den(t, p, pyargs, phase='liq')
assert abs((calc - ref) / ref * 100) < 2
if print_result:
print('----- Density at {} K and {} Pa -----'.format(t, p))
print(' Reference:', ref, 'mol m^-3')
print(' PC-SAFT:', calc, 'mol m^-3')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
def test_cp(print_result=False):
"""Test the heat capacity function to see if it is working correctly."""
# Benzene
x = np.asarray([1.])
m = np.asarray([2.4653])
s = np.asarray([3.6478])
e = np.asarray([287.35])
cnsts = np.asarray(
[55238., 173380, 764.25, 72545,
2445.7]) # constants for Aly-Lee equation (obtained from DIPPR)
pyargs = {'x': x, 'm': m, 's': s, 'e': e}
ref = 140.78 # source: Equation of state from Polt et al. (1992) (available at https://webbook.nist.gov/chemistry/fluid/)
p = 100000.
t = 330.
rho = pcsaft_den(t, p, pyargs, phase='liq')
calc = pcsaft_cp(t, rho, cnsts, pyargs)
assert abs((calc - ref) / ref * 100) < 3
if print_result:
print('\n########## Test with benzene ##########')
print('----- Heat capacity at 330 K -----')
print(' Reference:', ref, 'J mol^-1 K^-1')
print(' PC-SAFT:', calc, 'J mol^-1 K^-1')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
# Toluene
x = np.asarray([1.])
m = np.asarray([2.8149])
s = np.asarray([3.7169])
e = np.asarray([285.69])
cnsts = np.asarray(
[58140., 286300, 1440.6, 189800,
650.43]) # constants for Aly-Lee equation (obtained from DIPPR)
pyargs = {'x': x, 'm': m, 's': s, 'e': e}
ref = 179.79 # source: Equation of state from Polt et al. (1992) (available at https://webbook.nist.gov/chemistry/fluid/)
p = 100000.
t = 370.
rho = pcsaft_den(t, p, pyargs, phase='liq')
calc = pcsaft_cp(t, rho, cnsts, pyargs)
assert abs((calc - ref) / ref * 100) < 3
if print_result:
print('\n########## Test with toluene ##########')
print('----- Heat capacity at 370 K -----')
print(' Reference:', ref, 'J mol^-1 K^-1')
print(' PC-SAFT:', calc, 'J mol^-1 K^-1')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')
# # Acetic acid
# # print('\n########## Test with acetic acid ##########')
# m = np.asarray([1.3403])
# s = np.asarray([3.8582])
# e = np.asarray([211.59])
# volAB = np.asarray([0.075550])
# eAB = np.asarray([3044.4])
# cnsts = np.asarray([40200., 136750, 1262, 70030, 569.7]) # constants for Aly-Lee equation (obtained from DIPPR)
# pyargs = {'x':x, 'm':m, 's':s, 'e':e, 'e_assoc':eAB, 'vol_a':volAB}
#
# ref = 130.3 # source: DIPPR
# p = 100000.
# t = 325.
# rho = pcsaft_den(t, p, pyargs, phase='liq')
# calc = pcsaft_cp(t, rho, cnsts, pyargs)
# """ Note: Large deviations occur with acetic acid and water. This behavior
# has been observed before and was described in R. T. C. S. Ribeiro, A. L.
# Alberton, M. L. L. Paredes, G. M. Kontogeorgis, and X. Liang, “Extensive
# Study of the Capabilities and Limitations of the CPA and sPC-SAFT Equations
# of State in Modeling a Wide Range of Acetic Acid Properties,” Ind. Eng.
# Chem. Res., vol. 57, no. 16, pp. 5690–5704, Apr. 2018. """
# # print('----- Heat capacity at 325 K -----')
# # print(' Reference:', ref, 'J mol^-1 K^-1')
# # print(' PC-SAFT:', calc, 'J mol^-1 K^-1')
# # print(' Relative deviation:', (calc-ref)/ref*100, '%')
# Dimethyl ether
m = np.asarray([2.2634])
s = np.asarray([3.2723])
e = np.asarray([210.29])
dpm = np.asarray([1.3])
dip_num = np.asarray([1.0])
cnsts = np.asarray(
[57431., 94494, 895.51, 65065,
2467.4]) # constants for Aly-Lee equation (obtained from DIPPR)
pyargs = {'x': x, 'm': m, 's': s, 'e': e, 'dipm': dpm, 'dip_num': dip_num}
ref = 102.2 # source: DIPPR correlation
p = 100000.
t = 240.
rho = pcsaft_den(t, p, pyargs, phase='liq')
calc = pcsaft_cp(t, rho, cnsts, pyargs)
assert abs((calc - ref) / ref * 100) < 3
if print_result:
print('\n########## Test with dimethyl ether ##########')
print('----- Heat capacity at 240 K -----')
print(' Reference:', ref, 'J mol^-1 K^-1')
print(' PC-SAFT:', calc, 'J mol^-1 K^-1')
print(' Relative deviation:', (calc - ref) / ref * 100, '%')