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composition of the cover gas: where: - .. math::`K_{H,A} = P_{A}^{\circ} \cdot \gamma_{A}` the Henry constant. + - :math:`P_{A}^{\circ}` is the vapour pressure of compound A if it was in equilibrium with its own liquid phase. + + - :math:`P_{A}` is the vapour pressure of compound A in the gas phase of the solution. + + - :math:`X_{A}` is the molar fraction of compound A in the solution. + + - :math:`\gamma_{A} = \frac{P_{A}}{P^{\circ}}` is the thermodynamic activity coefficient of compound A. + + - :math:`P^{\circ}` is the ambient pressure, equal to 1atm. + + - :math:`K_{H,A} = P_{A}^{\circ} \cdot \gamma_{A}` is the Henry constant of compound A. .. _exampleofuse: @@ -315,234 +325,254 @@ The following laws are then used to estimate the composition of the cover gas: Example of use -------------- +In this section, three examples are provided to show how the properties of irradiation-induced impurities can be exploited: -This example illustrates how the Henry constant correlation of polonium in LBE can be used to estimate -the partial pressure of such a contaminant at different temperature values. +- **Vapour pressure of polonium as function of temperature** -In this example, we consider a generic volume of LBE subjected to varying temperature conditions and utilise -the Henry constant correlation to estimate the concentration of polonium in the vapor phase. + A generic volume of LBE subjected to varying temperature conditions is considered and + the Henry constant correlation is used to estimate the vapour pressure of polonium in the + cover gas. -The Henry constant of polonium over lead-bismuth eutectic :math:`K_{H(Po(LBE))}` can be expressed using empirical -correlations, derived from experimental measurements. The correlation is implemented in -:py:mod:`lbh15.properties.lbe_thermochemical_properties.lbe_contamination.py` in the class -*LBEPoloniumHenryConstantOhno2006*. + According to the Henry's Law, the partial pressure of polonium in LBE is given by the following expression: -According to the Henry's Law, the partial pressure of polonium in LBE is given by the following expression: + .. math:: P_{Po(LBE)} = K_{H,Po(LBE)} \cdot X_{Po(LBE)}, + :label: henry_po -.. math:: P_{Po(LBE)} = K_{H,Po(LBE)} \cdot X_{Po(LBE)} + where :math:`X_{Po(LBE)}` is the molar fraction of polonium in liquid LBE, + generally between :math:`10^{-12}` and :math:`10^{-10}`. -With :math:`X_{Po(LBE)}` the molar fraction of Polonium in liquid LBE, generally between :math:`10^{-12}` and :math:`10^{-10}`. + The correlation by Ohno2006 is exploited for the computing the Henry constant of polonium, + and the molar fraction is supposed to be :math:`10^{-12}`. -On the range [723.0, 1023.0 K] given by the Handbook, with molar fraction of :math:`10^{-12}`, the vapour pressure of Polonium in liquid LBE shows this behaviour 9 (see :numref:`pressPo`): + To compute and plot the behaviour of polonium vapour pressure, the following code is + to be used (only relevant lines are retained, neglecting the ones for defining plot features): -.. raw:: latex + .. code-block:: python - \clearpage + import numpy as np + from lbh15.properties.lbe_thermochemical_properties.lbe_contamination import LBEPoloniumHenryConstantOhno2006 + from lbh15.lbe import LBE -.. _pressPo: -.. figure:: figures/test_K_Po.png - :width: 500 - :align: Center - - Evolution of Polonium vapour pressure in liquid LBE as a function of temperature. + # Define Molar fraction of Po in LBE + X = 10**(-12) + # Generate temperature values + T_values = np.linspace(LBEPoloniumHenryConstantOhno2006().range[0], + LBEPoloniumHenryConstantOhno2006().range[1], 100) -Using the following code: + # Generate list of liq + liqs = [LBE(T=temp) for temp in T_values] -.. code-block:: python + # Calculate pressures using Henry's law + P_values = [liq.K_Po * X for liq in liqs] + - import numpy as np - import matplotlib.pyplot as plt - from lbh15.properties.lbe_thermochemical_properties.lbe_contamination import LBEPoloniumHenryConstantOhno2006 - from lbh15.lbe import LBE + .. raw:: latex - # Define initial parameters - T_start = 398.0 - X = 10**(-12) + \clearpage - # Create the LBE liquid object - liq = LBE(T=T_start) + .. _pressPo: - # Instantiate the object for Polonium Henry constant - K_Po = LBEPoloniumHenryConstantOhno2006() + .. figure:: figures/K_Po_LBE.png + :width: 500 + :align: Center - # Generate temperature values within the valid range of the correlation - T_values = np.linspace(K_Po.range[0], K_Po.range[1], 100) + Behaviour of polonium vapour pressure in liquid LBE as a function of temperature. - # Calculate pressures using Henry's law - P_values = np.array([K_Po.correlation(T) * X for T in T_values]) - # Plot the results - plt.plot(T_values, P_values, label=f'X = {X}') - plt.xlabel('Temperature (K)') - plt.ylabel('Pressure (Pa)') - plt.title('Pressure Profile using Henry\'s Law for Polonium in LBE') - plt.legend() - plt.grid(True) - plt.show() + +- **Vapour pressure of polonium as function of its molar fraction** -It is now possible to determine the concentration of polonium in the cover gas by applying the ideal gas law: + A generic volume of LBE at :math:`T = 750K` is considered and + the Henry constant correlation is used to estimate the vapour pressure of polonium + in the cover gas. According to the Henry's Law, the partial pressure of polonium in LBE is given in :eq:`henry_po`. + In the following code, the behaviour of the vapour pressure of polonium in LBE is calculated; in figure :numref:`pressPoX`, the result + is shown: -.. math:: C_{Po(g)} = \frac{P_{Po(LBE)}}{R \cdot T} + .. code-block:: python -The results (:numref:`concPo`) show a significant increase in polonium partial pressure and associated concentration with increasing temperature. + import numpy as np + from lbh15.lbe import LBE -.. code-block:: python + # Generate Molar fraction values + X_values = np.linspace(1e-10, 1e-12, 500) - import numpy as np - import matplotlib.pyplot as plt - from lbh15.properties.lbe_thermochemical_properties.lbe_contamination import LBEPoloniumHenryConstantOhno2006 - from lbh15.lbe import LBE + # Generate liquid at T = 750K + liq=LBE(T = 750) - # Define initial parameters - T_start = 398.1 - X = 10**(-12) + # Calculate pressures using Henry's law + P_values = liq.K_Po * X_values - # Create the LBE liquid object - liq = LBE(T=T_start) - # Instantiate the object for Polonium Henry constant - K_Po = LBEPoloniumHenryConstantOhno2006() + .. _pressPoX: - # Generate temperature values within the valid range of the correlation - T_values = np.linspace(K_Po.range[0], K_Po.range[1], 100) + .. figure:: figures/K_Po_LBE_X.png + :width: 500 + :align: Center - # Calculate concentration of Polonium using Ideal Gas' law - C_values = np.array([(K_Po.correlation(T) * X) / (8.314 * T) for T in T_values]) + Behaviour of polonium vapour pressure in liquid LBE as function of polonium molar fraction. - # Plot the results - plt.plot(T_values, C_values, label=f'X = {X}') - plt.xlabel('Temperature (K)') - plt.ylabel('Concentration (mol/L)') - plt.title('Concentration profile using Ideal Gas\' Law of Polonium in the cover gas') - plt.legend() - plt.grid(True) - plt.show() -.. _concPo: -.. figure:: figures/test_C_Po.png - :width: 500 - :align: Center - - Evolution of Polonium concentration in the cover gas as a function of temperature. +- **Concentration of polonium in the cover gas as function of temperature** + + To determine the concentration of polonium in the cover gas, the ideal gas law should be applied: + + .. math:: C_{Po(g)} = \frac{P_{Po(LBE)}}{R \cdot T} + + The following code implements such law, and in figure :numref:`_concPo` the result is shown. In this case, instead of a liquid metal instance, + an instance of the required property is used. + + .. code-block:: python + + import numpy as np + from scipy.constants import R + from lbh15.properties.lbe_thermochemical_properties.lbe_contamination import LBEPoloniumHenryConstantOhno2006 + + # Define initial parameters + X = 10**(-12) + + # Instantiate for K_Po + K_Po = LBEPoloniumHenryConstantOhno2006() + + # Generate temperature values + T_values = np.linspace(K_Po.range[0], K_Po.range[1], 100) + + # Calculate concentration of Polonium + C_values = np.array([(K_Po.correlation(T) * X) / (R * T) for T in T_values]) + + .. _concPo: + + .. figure:: figures/Po_concentration.png + :width: 500 + :align: Center + + Behaviour of polonium concentration in the cover gas as function of temperature. Ranges of temperature --------------------- -The temperature ranges over which the various correlations are valid are not consistently stated in the Handbook :cite:`Agency2015`. -This section provides a justification for the temperature ranges implemented in lbh15, in cases where precise data are not -suggested by Handbook :cite:`Agency2015`. +The temperature ranges over which the various correlations are valid are not always consistently stated in the Handbook :cite:`Agency2015`. +This section provides a justification of the temperature ranges implemented in *lbh15* for the correlations whose validity range is not +precisely indicated in the Handbook :cite:`Agency2015`. -*Impurities in Lead:* +- *Impurities in Lead:* -- *LeadPoloniumActivityCoefficientLi1998*: + - *LeadPoloniumActivityCoefficientLi1998*: - For the chemical activity coefficient of polonium given by Li in 1998, the same range as that given for the vapour pressure correlation - by Abakumov in 1994 is used, since Li refers to this correlation in his work. + For the chemical activity coefficient of polonium given by Li in 1998, the same range as that given for the vapour pressure correlation + by Abakumov in 1994 is used, since Li refers to this correlation in his work. -.. + .. -- *LeadPoloniumHenryConstant*: + - *LeadPoloniumHenryConstant*: - For the Henry constant of polonium given by the Handbook, the same range as that given for the vapour pressure correlation by Abakumov in - 1994 and for the activity coefficient correlation by Li 1998, since it is determined from these two correlations using Henry\'s law. + For the Henry constant of polonium given by the Handbook, the same range as that given for the vapour pressure correlation by Abakumov in + 1994 and for the activity coefficient correlation by Li 1998 is selected, since it is determined by exploiting this two correlations + through the Henry\'s law. -.. + .. -- *LeadIodineActivityCoefficient*: + - *LeadIodineActivityCoefficient*: - In the absence of any information, the temperature range of lead is selected for the purposes of coherence. + For the iodine activity coefficient given by the Handbook, as lower limit 673K is chosen since the iodine exhibits a + positive deviation from Raoult's law below it. As upper limit, the boiling temperature of lead is chosen, due to absence + of related information. -.. + .. +- *Impurities in Bismuth:* + - *BismuthPoloniumActivityCoefficient*: -*Impurities in Bismuth:* + For the chemical activity coefficient of polonium determined by the Handbook, the same range as the + one used in the experiment of Bradley in 1968 is selected, since it is the experiment upon which the + Handbook calculates the correlation. -- *BismuthPoloniumActivityCoefficient*: + .. - For the chemical activity coefficient of polonium determined by the Handbook, the same range as the - one used in the experiment of Bradley in 1968 is selected, since it is the experiment upon which the - Handbook calculates the correlation. + - *BismuthIodineVapourPressureCubicciotti1959*: -.. + For the vapour pressure or Iodine, this range has been chosen since the correlation overestimates the + real vapor pressure at any T of bismuth. -- *BismuthIodineVapourPressureCubicciotti1959*: + .. - For the vapour pressure or Iodine, this range has been chosen since the correlation overestimates the - real vapor pressure at any T of bismuth. + - *BismuthCaesiumActivityCoefficientGverdtsiteli1984*: -.. + The Handbook states that the correlation has been verified up to 1 000K. + Given that the melting temperature of caesium is considerably lower than that of bismuth, this is why + the melting temperature of bismuth is chosen as lower limit of the validity range. -- *BismuthCaesiumActivityCoefficientGverdtsiteli1984*: + .. - The Handbook states that the correlation has been verified up to 1 000K. - Given that the melting temperature of caesium is considerably lower than that of bismuth, this is why - the melting temperature of bismuth is chosen as lower limit of the validity range. -.. +- *Impurities in LBE:* + - *LBEPoloniumActivityCoefficientOhno2006*: -*Impurities in LBE:* + The correlation is derived from (5.14) and (5.23), the range is obtained by taking the highest lower limit of + the two ranges as a lower limit, and the lowest higher limit of the two ranges as a higher limit. -- *LBEPoloniumActivityCoefficient*: + .. - The correlation is derived from two other correlations (5.14) and (5.16), which provide two distinct ranges. - The correlation 5.14 has a range of [641-877K], whereas the 5.16 is valid on the range [913, 1023K]. - Fig. 5.2.4 at page 258 plots and compares the behavior of several correlations for the activity coefficient of - Po in Pb and Bi alloys over the temperature range [500 K, 1000 K]. Also, the correlation represented by the property - LBEPoloniumActivityCoefficient is plotted. But the accompanying explanatory text explicitly states that this correlation - is based on measurements made above 913 K, so its extrapolation to lower temperatures should not be trusted. This is why - the validity range of (5.16) is applied also to this correlation in lbh15. + - *LBEPoloniumActivityCoefficient*: -.. + The correlation is derived from two other correlations (5.14) and (5.16), which provide two distinct ranges. + The correlation 5.14 has a range of [641-877K], whereas the 5.16 is valid on the range [913, 1023K]. + Fig. 5.2.4 at page 258 plots and compares the behavior of several correlations for the activity coefficient of + Po in Pb and Bi alloys over the temperature range [500 K, 1000 K]. Also, the correlation represented by the property + LBEPoloniumActivityCoefficient is plotted. But the accompanying explanatory text explicitly states that this correlation + is based on measurements made above 913 K, so its extrapolation to lower temperatures should not be trusted. This is why + the validity range of (5.16) is applied also to this correlation in lbh15. + + .. -- *LBEIodineHenryConstantNeuhausen2005*: + - *LBEIodineHenryConstantNeuhausen2005*: - As stated at pag. 287 of the HANDBOOK, iodine is retained in LBE for temperature values below 700 K. - This is the reason why 700 K is implemented as the lower limit of the validity range of this property. - About the higher limit of this range, the same value indicated for LBEIodineHenryConstantKnacke1991 is - used since the HANDBOOK states that the correlation by Neuhausen shows the same behavior as the one by - Knacke for temperature values that are higher than 700 K. + As stated at pag. 287 of the HANDBOOK, iodine is retained in LBE for temperature values below 700 K. + This is the reason why 700 K is implemented as the lower limit of the validity range of this property. + About the higher limit of this range, the same value indicated for LBEIodineHenryConstantKnacke1991 is + used since the HANDBOOK states that the correlation by Neuhausen shows the same behavior as the one by + Knacke for temperature values that are higher than 700 K. -.. - -- *Experimental and under review*: + .. - For the following impurities in LBE, the nominal ranges in Table 5.2.4 om page 301 are stated to be “around” - certain temperatures. To ensure the greatest consistency, in line with the ranges presented by these correlations, - it has been decided that the term "around" will be characterised as :math:`± 0.05 * \delta_{T}`, where :math:`\delta_{T} = T_b0 - T_m0`, - :math:`T_b0` and :math:`T_m0` being the boiling and melting temperatures of LBE, respectively. - Here the classes concerned by this approximation: + - *Experimental and under review*: - - *LBEMercuryHenryConstant*: around 603K + For the following impurities in LBE, the nominal ranges in Table 5.2.4 om page 301 are stated to be “around” + certain temperatures. To ensure the greatest consistency, in line with the ranges presented by these correlations, + it has been decided that the term "around" will be characterised as :math:`± 0.05 * \delta_{T}`, where :math:`\delta_{T} = T_b0 - T_m0`, + :math:`T_b0` and :math:`T_m0` being the boiling and melting temperatures of LBE, respectively. + Here the classes concerned by this approximation: - - *LBEMercuryActivityCoefficient*: around 603K + - *LBEMercuryHenryConstant*: around 603K - - *LBECadmiumHenryConstant*: around 773K + - *LBEMercuryActivityCoefficient*: around 603K - - *LBECadmiumActivityCoefficient*: around 773K + - *LBECadmiumHenryConstant*: around 773K - - *LBEThalliumHenryConstant*: around 773K + - *LBECadmiumActivityCoefficient*: around 773K - - *LBEThalliumActivityCoefficient*: around 773K + - *LBEThalliumHenryConstant*: around 773K - - *LBERubidiumVapourPressureLandolt1960*: around 873K + - *LBEThalliumActivityCoefficient*: around 773K - - *LBERubidiumActivityCoefficient*: around 873K + - *LBERubidiumVapourPressureLandolt1960*: around 873K -.. + - *LBERubidiumActivityCoefficient*: around 873K + + .. .. _tutorials: