Thermal conductivity¶

Overview¶

The polykin.properties.thermal_conductivity module provides methods to estimate the thermal conductivity of pure gases, gas mixtures, and liquid mixtures.

	Gas	Liquid
DIPPR equations	DIPPR100, DIPPR102	DIPPR100
Estimation methods	—	—
Mixing rules	KVMX2_Wassilijewa	KLMX2_Li
Pressure correction	KVPC_Stiel_Thodos, KVMXPC_Stiel_Thodos	—

Pure gases¶

The thermal conductivity of gases increases with pressure, but the effect is fairly small at low and moderate pressures. In the the low pressure domain, which extends from 1 mbar to 10 bar, the effect of pressure is often neglected. For pressures above 10 bar, the effect of pressure on thermal conductivity becomes progressiveley more important and should be considered.

Temperature correlations at low pressure¶

The thermal conductivity of low-pressure gases increases with temperature. Experimental data is usually correlated with DIPPR-100 or DIPPR-102.

In [1]:

# %pip install polykin
from polykin.properties.equations import DIPPR100

# %pip install polykin from polykin.properties.equations import DIPPR100

Let us define correlations for the thermal conductivity of ethylene and propylene.

In [2]:

# Parameters from Reid, Prausnitz, Pooling (1988)
kv_ethylene  = DIPPR100(-1.760e-2, 1.200e-4, 3.335e-8, -1.366e-11, 
                        Tmin=200, Tmax=1270, symbol='k', unit='W/(m·K)',
                        name='ethylene')
kv_propylene = DIPPR100(-7.584e-3, 6.101e-5, 9.966e-8, -3.840e-11,
                        Tmin=175, Tmax=1270, symbol='k', unit='W/(m·K)',
                        name='propylene')

# Parameters from Reid, Prausnitz, Pooling (1988) kv_ethylene = DIPPR100(-1.760e-2, 1.200e-4, 3.335e-8, -1.366e-11, Tmin=200, Tmax=1270, symbol='k', unit='W/(m·K)', name='ethylene') kv_propylene = DIPPR100(-7.584e-3, 6.101e-5, 9.966e-8, -3.840e-11, Tmin=175, Tmax=1270, symbol='k', unit='W/(m·K)', name='propylene')

In [3]:

from polykin import plotequations

_ = plotequations([kv_ethylene, kv_propylene],
                  title="Gas thermal conductivity")

from polykin import plotequations _ = plotequations([kv_ethylene, kv_propylene], title="Gas thermal conductivity")

Pressure correction¶

The increase of thermal conductivity with respect to the low-pressure value can be estimated with the method of Stiel and Thodos. The primary argument of this method is the molar volume, which can be estimated with an equation of state, like PengRobinson.

In [4]:

from polykin.properties.thermal_conductivity import KVPC_Stiel_Thodos
from polykin.thermo.eos import PengRobinson
import numpy as np

from polykin.properties.thermal_conductivity import KVPC_Stiel_Thodos from polykin.thermo.eos import PengRobinson import numpy as np

Let us estimate the residual thermal conductivity of ethylene at 350 K and 100 bar.

In [5]:

# Ethylene constants
M = 28.05e-3 # kg/mol
Tc = 282.4   # K
Pc = 50.4e5  # Pa
Zc = 0.280
w = 0.089

# Molar volume estimate
eos = PengRobinson(Tc, Pc, w)
v = eos.v(T=350., P=100e5, y=np.array([1.])) # m³/mol
v

# Ethylene constants M = 28.05e-3 # kg/mol Tc = 282.4 # K Pc = 50.4e5 # Pa Zc = 0.280 w = 0.089 # Molar volume estimate eos = PengRobinson(Tc, Pc, w) v = eos.v(T=350., P=100e5, y=np.array([1.])) # m³/mol v

Out[5]:

array([0.00018441])

In [6]:

KVPC_Stiel_Thodos(v[0], M, Tc, Pc, Zc)

KVPC_Stiel_Thodos(v[0], M, Tc, Pc, Zc)

Out[6]:

0.016870676177529628

In this case, the pressure correction is of the same magnitude as the low-pressure value itself.

Gas mixtures¶

Mixing rules at low pressre¶

The thermal conductivity of low-pressure gas mixtures can be estimated from the corresponding pure-component values using Wassilijewa's mixing rule.

In [7]:

from polykin.properties.thermal_conductivity import KVMX2_Wassilijewa

from polykin.properties.thermal_conductivity import KVMX2_Wassilijewa

Let us estimate the thermal conductivity of a 50:50 mol% mixture of ethylene and propylene at 350 K at 1 bar.

In [8]:

T = 350. # K
k = [kv_ethylene(T), kv_propylene(T)] # pure kv(T), W/(m.K)

T = 350. # K k = [kv_ethylene(T), kv_propylene(T)] # pure kv(T), W/(m.K)

In [9]:

KVMX2_Wassilijewa(y=[0.5, 0.5], k=k, M=[28.05e-3, 42.08e-3])

KVMX2_Wassilijewa(y=[0.5, 0.5], k=k, M=[28.05e-3, 42.08e-3])

Out[9]:

0.025935290438256965

Pressure correction¶

The increase of thermal conductivity with respect to the low-pressure value can be estimated with the method of Stiel and Thodos. The primary argument of this method is the molar volume, which can be estimated with an equation of state.

In [10]:

from polykin.properties.thermal_conductivity import KVMXPC_Stiel_Thodos

from polykin.properties.thermal_conductivity import KVMXPC_Stiel_Thodos

Let us estimate the residual thermal conductivity of a 50 mol% mixture of ethylene and propylene at 350 K and 100 bar.

In [11]:

# [ethylene, propylene] constants
M = [28.05e-3, 42.08e-3] # kg/mol
Tc = [282.4, 364.9]      # K
Pc = [50.4e5, 46.0e5]    # Pa
Zc = [0.280, 0.274]
w = [0.089, 0.144]

# Molar volume estimate
eos = PengRobinson(Tc, Pc, w)
y = np.array([0.5, 0.5])           # molmol
v = eos.v(T=350., P=100e5, y=y)    # m³/mol
v

# [ethylene, propylene] constants M = [28.05e-3, 42.08e-3] # kg/mol Tc = [282.4, 364.9] # K Pc = [50.4e5, 46.0e5] # Pa Zc = [0.280, 0.274] w = [0.089, 0.144] # Molar volume estimate eos = PengRobinson(Tc, Pc, w) y = np.array([0.5, 0.5]) # molmol v = eos.v(T=350., P=100e5, y=y) # m³/mol v

Out[11]:

array([0.00011174])

In [12]:

KVMXPC_Stiel_Thodos(v[0], y, M, Tc, Pc, Zc, w)

KVMXPC_Stiel_Thodos(v[0], y, M, Tc, Pc, Zc, w)

Out[12]:

0.03832154985250473

Pure liquids¶

Most organic liquids have thermal conductivities in the range 0.1-0.2 W/(m·K) at temperatures below the normal boiling point.

Temperature correlations¶

The thermal conductivity decreases with temperature. Experimental data is usually correlated with DIPPR-100.

In [13]:

from polykin.properties.equations import DIPPR100

from polykin.properties.equations import DIPPR100

Let us define correlations for the thermal conductivity of styrene and butadiene.

In [14]:

# Parameters from Reid, Prausnitz, Pooling (1988)
kl_styrene   = DIPPR100(2.696e-1, -3.384e-4, 1.675e-8, Tmin=243, Tmax=623,
                        symbol='k', unit='W/(m·K)', name='styrene')
kl_butadiene = DIPPR100(3.007e-1, -7.837e-4, 4.916e-7, Tmin=164, Tmax=393,
                        symbol='k', unit='W/(m·K)', name='1,3-butadiene')

# Parameters from Reid, Prausnitz, Pooling (1988) kl_styrene = DIPPR100(2.696e-1, -3.384e-4, 1.675e-8, Tmin=243, Tmax=623, symbol='k', unit='W/(m·K)', name='styrene') kl_butadiene = DIPPR100(3.007e-1, -7.837e-4, 4.916e-7, Tmin=164, Tmax=393, symbol='k', unit='W/(m·K)', name='1,3-butadiene')

In [15]:

_ = plotequations([kl_styrene, kl_butadiene],
                  title="Liquid thermal conductivity")

_ = plotequations([kl_styrene, kl_butadiene], title="Liquid thermal conductivity")

Liquid mixtures¶

Mixing rules¶

The thermal conductivity of liquid mixtures can be estimated from the corresponding pure-component values using Li's mixing rule.

Li's method requires the densities of the components as well, which are usually correlated with DIPPR-100 or DIPPR-105.

In [16]:

from polykin.properties.equations import DIPPR105
from polykin.properties.thermal_conductivity import KLMX2_Li

from polykin.properties.equations import DIPPR105 from polykin.properties.thermal_conductivity import KLMX2_Li

Let us estimate de thermal conductivity of a 50:50 wt% mixture of styrene and butadiene at 350 K.

In [17]:

# Parameters from Perry's
dl_styrene   = DIPPR105(0.7397, 0.2603, 636, 0.3009, Tmin=243, Tmax=636,
                        symbol=r'\rho', unit='kmol/m³', name='styrene')
dl_butadiene = DIPPR105(1.2384, 0.2725, 425.17, 0.28813, Tmin=164, Tmax=425,
                        symbol=r'\rho', unit='kmol/m³', name='1,3-butadiene')

# Parameters from Perry's dl_styrene = DIPPR105(0.7397, 0.2603, 636, 0.3009, Tmin=243, Tmax=636, symbol=r'\rho', unit='kmol/m³', name='styrene') dl_butadiene = DIPPR105(1.2384, 0.2725, 425.17, 0.28813, Tmin=164, Tmax=425, symbol=r'\rho', unit='kmol/m³', name='1,3-butadiene')

In [18]:

w = [0.5, 0.5]                                        # mass fractions
T = 350.                                              # temperature, K
k = [kl_styrene(T), kl_butadiene(T)]                  # pure kl(T), W/(m·K)
rho = [dl_styrene(T)*104.152, dl_butadiene(T)*54.092] # pure rhol(T), kg/m³

w = [0.5, 0.5] # mass fractions T = 350. # temperature, K k = [kl_styrene(T), kl_butadiene(T)] # pure kl(T), W/(m·K) rho = [dl_styrene(T)*104.152, dl_butadiene(T)*54.092] # pure rhol(T), kg/m³

In [19]:

KLMX2_Li(w, k, rho)

KLMX2_Li(w, k, rho)

Out[19]:

0.10808664126899356