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properties.py
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properties.py
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# import matplotlib.pyplot as plt
import numpy as np
k_B = 8.6e-5
# plt.rc('text', usetex=True)
# plt.rc('font', family='serif', size=12)
# ##### Tungsten ######
#
# Taken from (P.Tolias, 2017)
def Cp_W(T): # units in J/(kg*K)
return (
21.868372
+ 8.068661e-03 * T
- 1e-06 * T**2
+ 1.075862e-09 * T**3
+ 1.406637e04 / T**2
)
def rho_W(T): # units in kg/m**3
"""
(adjusted by factor 1000 as orginial equation in g/cm**3)
"""
T_W_0 = 293.15
return (
19250
- 2.66207e-01 * (T - T_W_0)
- 3.0595e-06 * (T - T_W_0) ** 2
- 9.5185e-09 * (T - T_W_0) ** 3
)
def thermal_cond_W(T): # units in W/(m*K)
return (
149.441
- 45.466e-03 * T
+ 13.193e-06 * T**2
- 1.484e-09 * T**3
+ 3.866e06 / (T + 1) ** 2
)
# (T+1) is there to avoid dividing by 0
# taken from (Frauenfelder, R. 1969)
D_0_W = 2.4e-7 # Diffusivity pre-exponential factor (m^(2).s^(-1))
E_D_W = 0.39 # Diffusivity activation energy (eV)
S_0_W = 1.87e24 # Solubility pre-exponential factor (m^(-3).Pa^(-0.5))
E_S_W = 1.04 # Solutbiility activation energy (eV)
atom_density_W = 6.3222e28
def D_W(T):
return D_0_W * np.exp(-E_D_W / k_B / T)
def S_W(T):
return S_0_W * np.exp(-E_S_W / k_B / T)
# ##### EUROfer ######
#
# Values taken from Materials properties handbook
def Cp_eurofer(T): # units in J/(kg*K)
return -139.66 + 3.4777 * T - 0.0063847 * T**2 + 4.0984e-06 * T**3
# def Cp_eurofer_2(T): # units in J/(kg*K)(Mergia)
# return 2.696*T - 0.00496*T**2 + 3.335e-06*T**3
def rho_eurofer(T): # units in kg/m**3
return 7852.102143 - 0.331026405 * T
def thermal_cond_eurofer(T): # units in W/(m*K)
return 5.4308 + 0.13565 * T - 2.3862e-04 * T**2 + 1.3393e-07 * T**3
# taken from (Chen, 2021)
D_0_eurofer = 3.15e-08 # Diffusivity pre-exponential factor (m2/s)
E_D_eurofer = 0.0622 # Diffusivity activation energy (eV)
S_0_eurofer = 2.4088e23 # Solubility pre-exponential factor (atom/m3 Pa^-0.5)
E_S_eurofer = 0.3026 # Solutbiility activation energy (eV)
atom_density_eurofer = 8.409e28 # (m-3)
trap_density_eurofer = 4.5e23 # (m-3)
trap_energy_eurofer = 0.7804 # (eV)
# recombination coefficient from Liu Journal of Nuclear Materials (2021)
Kr_0_eurofer = 1.4143446334700682e-26
E_Kr_eurofer = -0.25727457261201786
# recombination coefficient from Braun (1980)
# Kr_0_eurofer = 5.9680e-17
# E_Kr_eurofer = 0.888
# recombination coefficient from Esteban (2000)
# Kr_0_eurofer = 4.7127e-31
# E_Kr_eurofer = 2.471
# recombination coefficient instantaneus
# Kr_0_eurofer = 0
# E_Kr_eurofer = 1
def D_eurofer(T):
return D_0_eurofer * np.exp(-E_D_eurofer / k_B / T)
def S_eurofer(T):
return S_0_eurofer * np.exp(-E_S_eurofer / k_B / T)
# ##### LiPi ######
#
# Values taken from (D.Martelli et al, 2019)
def Cp_lipb(T): # units in J/(kg*K)
"""
adjusted by factor 1000 as orginial equation in J/(g*K)
for values of temperature 508K < T < 800K,
"""
return 195 - 9.116e-03 * T
def rho_lipb(T): # units in kg/(m**3)
return 10520.35 - 1.19051 * T
def thermal_cond_lipb(T): # units in W/(m*K)
"""
adjusted by factor 100 as original equation in W/(cm*K))
"""
return 9.14779235 + 0.019631 * T
def visc_lipb(T): # units (Pa s)
return (
0.01555147189
- 4.827051855e-05 * T
+ 5.641475215e-08 * T**2
- 2.2887e-11 * T**3
)
def beta_lipb(T): # units in K-1
return 1.1221e-04 + 1.531e-08 * T
rho_0_lipb = 9808.2464435 # value T = 300K, units in kg/(m**3)
# taken from (Reiter, 1990)
D_0_lipb = 4.03e-08 # Diffusivity coefficient pre-exponential factor
E_D_lipb = 0.2021 # Diffusivity coefficient activation energy (eV)
# taken from (Aiello, 2008)
S_0_lipb = 1.427214e23 # Solubility coefficient pre-exponential factor
E_S_lipb = 0.133 # Solutbiility coefficient activation energy (eV)
def D_lipb(T):
return D_0_lipb * np.exp(-E_D_lipb / k_B / T)
def S_lipb(T):
return S_0_lipb * np.exp(-E_S_lipb / k_B / T)
# ##### Al2O3 - Permeation barrier ##### #
# taken from ()
D_0_al2o3 = 7.35e-07 # Diffusivity coefficient pre-exponential factor
E_D_al2o3 = 1.899 # Diffusivity coefficient activation energy (eV)
S_0_al2o3 = 2.519e25 # Solubility coefficient pre-exponential factor
E_S_al2o3 = 0.902 # Solutbiility coefficient activation energy (eV)
def D_al2o3(T):
return D_0_al2o3 * np.exp(-E_D_al2o3 / k_B / T)
def S_al2o3(T):
return S_0_al2o3 * np.exp(-E_S_al2o3 / k_B / T)
# Al taken from (Hydrogen Permeation Measurements on Alumina - E.Serra 2005)
D_0_al = 9.7e-08 # Diffusivity coefficient pre-exponential factor
E_D_al = 0.829 # Diffusivity coefficient activation energy (eV)
S_0_al = 9.133e19 # Solubility coefficient pre-exponential factor
E_S_al = 0.234 # Solutbiility coefficient activation energy (eV)
# ##### Plotting Data ##### #
if __name__ == "__main__":
def tick_function(X):
V = 1000 / (X)
return ["%.0f" % z for z in V]
T = np.arange(500, 900, step=1)
red_W = (171 / 255, 15 / 255, 26 / 255)
grey_eurofer = (130 / 255, 130 / 255, 130 / 255)
darkening_factor_green = 0.75
green_lipb = (
darkening_factor_green * 146 / 255,
darkening_factor_green * 196 / 255,
darkening_factor_green * 125 / 255,
)
# ##### Thermophyical properties ######################################## #
# ##### Density ##### #
plt.figure()
plt.plot(T, rho_W(T), label="W", color=red_W)
plt.plot(T, rho_eurofer(T), label="Eurofer", color=grey_eurofer)
plt.plot(T, rho_lipb(T), label="LiPb", color=green_lipb)
plt.xlabel(r"T (K)")
plt.ylabel(r"$\rho$ (Kg m$^{-3}$)")
# ##### Specific heat capacity ##### #
plt.figure()
plt.plot(T, Cp_W(T), label="W", color=red_W)
plt.plot(T, Cp_eurofer(T), label="Eurofer", color=grey_eurofer)
plt.plot(T, Cp_lipb(T), label="LiPb", color=green_lipb)
plt.xlabel(r"T (K)")
plt.ylabel(r"$c_p$ (J kg$^{-1}$ K$^{-1}$)")
# #### Thermal conductivity ##### #
plt.figure()
plt.plot(T, thermal_cond_W(T), label="W", color=red_W)
plt.plot(T, thermal_cond_eurofer(T), label="Eurofer", color=grey_eurofer)
plt.plot(T, thermal_cond_lipb(T), label="LiPb", color=green_lipb)
plt.xlabel(r"T (K)")
plt.ylabel(r"$\lambda$ (W m$^{-1}$ K$^{-1}$)")
# #### Viscosity ##### #
plt.figure()
plt.plot(T, visc_lipb(T), label="LiPb", color=green_lipb)
plt.xlabel(r"T (K)")
plt.ylabel(r"Viscosity (Pa s)")
# #### Thermal expansion coefficient ##### #
plt.figure()
plt.plot(T, beta_lipb(T), label="LiPb", color=green_lipb)
plt.xlabel(r"T (K)")
plt.ylabel(r"Thermal expansion coefficient")
plt.ylim(bottom=0, top=0.0002)
# #### Hydrogen transport properties ################################### #
# #### Diffusivity ##### #
# plt.figure(figsize=(6.4, 4.8/2))
fig, axs = plt.subplots(2, 1, sharex=True)
plt.sca(axs[0])
plt.plot(1000 / T, D_eurofer(T), label="Eurofer", color=grey_eurofer)
plt.plot(1000 / T, D_lipb(T), label="PbLi", color=green_lipb)
plt.plot(1000 / T, D_W(T), label="W", color=red_W)
plt.annotate("EUROFER", (0.02 + 1000 / T[0], D_eurofer(T[0])), color=grey_eurofer)
plt.annotate("LiPb", (0.02 + 1000 / T[0], D_lipb(T[0])), color=green_lipb)
plt.annotate("W", (0.02 + 1000 / T[0], D_W(T[0])), color=red_W)
# plt.xlabel(r"1000/T (K)")
plt.ylabel(r"Diffusivity (m$^{2}$ s$^{-1}$)")
plt.yscale("log")
plt.minorticks_on()
plt.grid(which="minor", alpha=0.3)
plt.grid(which="major", alpha=0.7)
# #### Solubility ##### #
plt.sca(axs[1])
# plt.figure(figsize=(6.4, 4.8/2))
# plt.plot(1000/T, S_W(T), label="W", color=red_W)
plt.plot(1000 / T, S_al2o3(T), label="Barrier", color="black")
plt.plot(1000 / T, S_eurofer(T), label="Eurofer", color=grey_eurofer)
plt.plot(1000 / T, S_lipb(T), label="LiPb", color=green_lipb)
plt.xlabel(r"1000/T (K)")
plt.ylabel(r"Solubility (m$^{-3}$Pa$^{-0.5}$)")
plt.yscale("log")
plt.annotate("EUROFER", (0.02 + 1000 / T[0], S_eurofer(T[0])), color=grey_eurofer)
plt.annotate("LiPb", (0.02 + 1000 / T[0], S_lipb(T[0])), color=green_lipb)
plt.annotate("W", (0.02 + 1000 / T[0], S_W(T[0])), color=red_W)
plt.xlim(right=2.2)
for fig_num in plt.get_fignums():
plt.figure(fig_num)
if plt.gca().get_yscale() == "linear":
plt.ylim(bottom=0)
# plt.yscale("log")
# plt.legend()
plt.minorticks_on()
plt.grid(which="minor", alpha=0.3)
plt.grid(which="major", alpha=0.5)
plt.tight_layout()
plt.show()