Carbon Dioxide - Fenghour, JPCRD, 1998
$\eta^0 = \dfrac{1.00697\sqrt{T}}{\sigma^2\mathfrak{S}(T^*)}$
$\mathfrak{S}(T^*)=\exp\left(\sum_{i=0}^{4}a_i[\ln T^*]^i\right)$
$\Delta\eta = d_{11}\rho + d_{21}\rho^2+\frac{d_{64}\rho^6}{(T^*)^3}+d_{81}\rho^8+\frac{d_{82}\rho^8}{T^*}$
$\Delta\eta = d_{11}\rho_c\delta + d_{21}\rho_c^2\delta^2+\frac{d_{64}\rho_c^6(\varepsilon/k)^3\delta^6\tau^3}{(T_c)^3}+\rho_c^8d_{81}\delta^8+\frac{\rho_c^8d_{82}\delta^8(\varepsilon/k)\tau}{T_c}$
In [10]:
# Conversion back to C*sqrt(M*T) - back calculate C from constant in paper - using sigma = 1 nm so it cancels out
1.00697e-6/(44.0098)**0.5
Out[10]:
In [4]:
# Convert the coefficients back to being a function of tau and delta rather than Tstar and rho
e_k = 251.196 # K
rhoc = 0.0440098*10624.9 # 467.69 # kg/m^3
Tc = 304.107 # K
d11 = 0.4071119e-2
d21 = 0.7198037e-4
d64 = 0.2411697e-16
d81 = 0.2971072e-22
d82 = -0.1627888e-22
d11s = d11*rhoc/1e6
d21s = d21*rhoc**2/1e6
d64s = d64*rhoc**6*e_k**3/Tc**3/1e6
d81s = d81*rhoc**8/1e6
d82s = d82*rhoc**8*e_k/Tc/1e6
print rhoc
print [d11s, d21s, d64s, d81s, d82s]
In [18]:
from math import log, exp, sqrt
def f(T):
Tstar = T/251.196
a = [0.235156, -0.491266, 0.05211155, 0.05347906, -0.01537102]
s = 0
for i in range(5):
s += a[i]*log(Tstar)**i
denom = exp(s)
return 1.00697*sqrt(T)/denom
f(800)
Out[18]:
In [32]:
from math import log, exp, sqrt
def f(T):
Tstar = T/386.0 #-
M = 17.03026 #kg/kmol
sigma = 0.2957 #nm
a = [4.99318220,-0.61122364,0.0,0.18535124,-0.11160946]
s = 0
for i in range(5):
s += a[i]*log(Tstar)**i
denom = exp(s)*sigma**2
return 0.021357*sqrt(M*T)/denom
f(300)
Out[32]:
In [23]:
import numpy as np
-0.5*np.array(range(13))
Out[23]:
$\Delta\eta = \displaystyle\sum_i \left(\displaystyle\sum_{j}\frac{d_{ij}}{(T^*)^j}\right)\rho^i= \displaystyle\sum_i \displaystyle\sum_{j}d_{ij}\frac{(\rho_c\delta)^i(\varepsilon/k)^j\tau^j}{(T_c)^j}$
In [6]:
# rho is units of mol/L, so convert the density to mol/L (poorly documented in paper)
D = [(3,0,0.17366936e-8),
(3,1,-0.64250359e-8),
(2,2,2.19664285e-7),
(4,2,1.67668649e-10),
(4,3,-1.49710093e-10),
(2,4,-0.83651107e-7),
(4,4,0.77012274e-10)]
e_k = 386.0
rhoc = 225.0/17.03026
Tc = 405.4
v = []
for i,j,dij in D:
v.append(dij*rhoc**i*(e_k/Tc)**j)
print v
$\Delta\eta = c_1\rho+c_2T\rho +a_1\rho+a_2\rho^2+a_3\rho^3 + \frac{a_0}{\rho-\rho_0}+\frac{a_0}{\rho_0}$
$\Delta\eta = c_1\rho_c\delta+c_2T_c\tau^{-1}\rho_c\delta +a_1\rho_c\delta+a_2\rho_c^2\delta^2+a_3\rho_c^3\delta^3 + \frac{a_0/\rho_c}{\delta-\delta_0}+\frac{a_0/\rho_c}{\delta_0}$
In [11]:
rhoc = 550
Tc = 456.831
rho0 = 1828.263
delta0 = rho0/rhoc
c1 = rhoc*-2.226486e-2/1e6
c2 = rhoc*Tc*5.550623e-5/1e6
a0 = 3.222951e5/rhoc/1e6
a1 = -1.009812e-1*rhoc/1e6
a2 = 6.161902e-5*rhoc**2/1e6
a3 = -8.84048e-8*rhoc**3/1e6
print 'a',(c1,c2,a1,a2,a3)
print 'a0',(a0)
print 'delta0',(delta0)
In [5]:
import numpy as np
_E = [-0.0737927, 0.517924, -0.308875, 0.108049, -0.408387]
E = (np.array(_E)*51.12e-6).tolist()
E
Out[5]:
In [7]:
# Conversion back to C*sqrt(M*T) - back calculate C from constant in paper - using sigma = 1 nm so it cancels out
0.2169614e-6/(66.05)**0.5
Out[7]:
In [8]:
368/66.05
Out[8]:
In [6]:
import numpy as np
Tc = 318.7232
c = np.array([0.118561, -0.378103, 0.416428, -0.165295, 0.0245381])
t = np.array([0,0.25,0.5,0.75,1])
print (c*(1/Tc)**t/1e6).tolist()
In [2]:
import numpy as np
rhom = 459.0300696
b = np.array([0,9.047e-3,5.784e-5,1.309e-7,-2.422e-10,9.424e-14,3.933e-17])/1e6
t = np.array(range(7))
print (b*rhom**t).tolist()
In [14]:
# Conductivity
import numpy as np
rhom = 459.0300696
b = np.array([3.576e-2,-9.045e-6,4.343e-8,-3.705e-12])/1e3
t = np.array(range(1,5))
print (b*rhom**t).tolist()
In [5]:
# Viscosity
import numpy as np
rhom = 453.43094
b = np.array([0,-3.038e-3,2.927e-4,-9.559e-7,1.739e-9,-1.455e-12,4.756e-16])/1e6
t = np.array(range(7))
print (b*rhom**t).tolist()
In [15]:
# Conductivity
import numpy as np
rhom = 453.43094
b = np.array([2.715e-2,4.963e-5,-4.912e-8,2.884e-11])/1e3
t = np.array(range(1,5))
print (b*rhom**t).tolist()
In [7]:
import numpy as np
rhom = 482.162772
b = np.array([0,2.260e-3,1.786e-4,-4.202e-7,8.489e-10,-8.670e-13,3.566e-16])/1e6
t = np.array(range(7))
print (b*rhom**t).tolist()
In [16]:
# Conductivity
import numpy as np
rhom = 482.162772
b = np.array([3.222e-2, 2.569e-5, -2.693e-8, 2.007e-11])/1e3
t = np.array(range(1,5))
print (b*rhom**t).tolist()
In [1]:
import numpy as np
rhom = 490.74
b = np.array([0,5.308e-4,2.234e-4,-6.742e-7,1.411e-9,-1.388e-12,5.274e-16])/1e6
t = np.array(range(7))
print (b*rhom**t).tolist()
In [1]:
# Conductivity
import numpy as np
rhom = 490.74
b = np.array([2.799e-2, 3.065e-5, -3.644e-8, 2.609e-11])/1e3
t = np.array(range(1,5))
print (b*rhom**t).tolist()
In [5]:
# Un reduced values
b1 = 1.836526
b2 = 5.126143
b3 = -1.436883
b4 = 0.626144
# Reducing value
lambda_red = 2.055e-3
print [b1*lambda_red, b2*lambda_red, b3*lambda_red, b4*lambda_red]
rhomass_red = 5.049886*102.032
print rhomass_red
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