Fuel Property Prediction Model ============================== **FuelLib** utilizes the group contribution method (GCM), as developed by Constantinou and Gani\ :footcite:p:`constantinou_new_1994` \ :footcite:p:`constantinou_estimation_1995` in the mid-1990s, to provide a systematic approach for estimating the thermodynamic properties of pure organic compounds. The GCM decomposes molecules into structural groups, each contributing to a target property based on predefined group values. By summing these contributions, the GCM accurately predicts essential properties, including the acentric factor, normal boiling point, liquid molar volume at standard conditions (298 K) and more. This predictive capability is particularly useful for complex mixtures such as synthetic aviation turbine fuels (SATFs), where experimental thermodynamic data is limited. `FuelLib` provides SATF developers with a means to estimate these critical properties without extensive physical testing, thereby aiding in the identification of promising fuel compositions before committing to large-scale production. `FuelLib` builds on `Pavan B. Govindaraju's Matlab implementation `_, and includes gas chromatography data (GC x GC) for various jet fuels from the National Jet Fuel Combustion Program\ :footcite:p:`colket_overview_2017` (NJFCP) Air Force Research Laboratory\ :footcite:p:`edwards_reference_2017` \ :footcite:p:`edwards_jet_2020` (AFRL) and Vozka et al.\ :footcite:p:`vozka_impact_2018`. Additionally, `FuelLib` includes correlations for the thermodynamic properties of mixture such as density, viscosity, vapor pressure, surface tension, and thermal conductivity. :ref:`tab-GCM-properties` outlines the properties for the *i-th* compound in a mixture, which depends on the *k-th* first-order and *j-th* second-order group contributions. .. _tab-GCM-properties: Table of GCM properties ----------------------- .. table:: GCM properties of the *i-th* component in a mixture. The subscript *stp* denotes a standard pressure assumption. :widths: auto :align: center ==================================== ===================== =========================================== ==================== =========================================================== Property Units Group Contributions Units Description ==================================== ===================== =========================================== ==================== =========================================================== :math:`M_{w,i}` kg/mol :math:`m_{w1k}` g/mol Molecular weight. :math:`T_{c,i}` K :math:`t_{c1k}`, :math:`t_{c2j}` 1 Critical temperature\ :footcite:p:`constantinou_new_1994`. :math:`p_{c,i}` Pa :math:`p_{c1k}`, :math:`p_{c2j}` bar\ :sup:`-0.5` Critical pressure\ :footcite:p:`constantinou_new_1994`. :math:`V_{c,i}` m\ :sup:`3`\ /mol :math:`v_{c1k}`, :math:`v_{c2j}` m\ :sup:`3`\ /kmol Critical volume\ :footcite:p:`constantinou_new_1994`. :math:`T_{b,i}` K :math:`t_{b1k}`, :math:`t_{b2j}` 1 Normal boiling point\ :footcite:p:`constantinou_new_1994`. :math:`T_{m,i}` K :math:`t_{m1k}`, :math:`t_{m2j}` 1 Normal melting point\ :footcite:p:`constantinou_new_1994`. :math:`\Delta H_{f,i}` J/mol :math:`h_{f1k}`, :math:`h_{f2j}` kJ/mol Enthalpy of formation at 298 K\ :footcite:p:`constantinou_new_1994`. :math:`\Delta G_{f,i}` J/mol :math:`g_{f1k}`, :math:`g_{f2j}` kJ/mol Standard Gibbs free energy at 298 K\ :footcite:p:`constantinou_new_1994`. :math:`\Delta H_{v,\textit{stp},i}` J/mol :math:`h_{v1k}`, :math:`h_{v2j}` kJ/mol Enthalpy of vaporization at 298 K\ :footcite:p:`constantinou_new_1994`. :math:`\omega_i` 1 :math:`\omega_{1k}`, :math:`\omega_{2j}` 1 Acentric factor\ :footcite:p:`constantinou_estimation_1995`. :math:`V_{m,\textit{stp},i}` m\ :sup:`3`\ /mol :math:`v_{m1k}`, :math:`v_{m2j}` m\ :sup:`3`\ /kmol Liquid molar volume at 298 K\ :footcite:p:`constantinou_estimation_1995`. :math:`C_{p,i}` J/mol/K :math:`C_{pA1_k}`, :math:`C_{pA2_k}`,... J/mol/K Specific heat capacity\ :footcite:p:`nielsen_molecular_1998` \ :footcite:p:`poling_properties_2001`. ==================================== ===================== =========================================== ==================== =========================================================== .. _eq-GCM-properties: Equations for GCM properties ---------------------------- The properties of each compound in a mixture can be calculated as the sum of contributions from the first- and second-order groups that make up the compound. For a given mixture, let :math:`\mathbf{N}` be an :math:`N_c \times N_{g_1}` matrix that represents the number of first-order groups in each compound, where :math:`N_c` is the number of compounds in the mixture and :math:`N_{g_1}` is the total number of first-order groups as defined by Constantinou and Gani\ :footcite:p:`constantinou_new_1994,constantinou_estimation_1995`. Similarly, let :math:`\mathbf{M}` be an :math:`N_c \times N_{g_2}` matrix that specifies the number of second-order groups in each compound, where :math:`N_{g_2}` is the total number of second-order groups. The total number of groups :math:`N_g = N_{g_1} + N_{g_2} = 121`. The GCM properties for the *i-th* compound in the mixture are calculated as follows\ :footcite:p:`constantinou_new_1994` \ :footcite:p:`constantinou_estimation_1995` \ :footcite:p:`poling_properties_2001`: .. math:: \begin{align*} M_{w,i} &= \bigg[\sum_{k = 1}^{N_{g_1}}\mathbf{N}_{ik}m_{w1k} \bigg] \times 10^{-3}, \\ T_{c,i} &= 181.28 \ln \bigg[ \sum_{k=1}^{N_{g_1}} \mathbf{N}_{ik} t_{c1k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} t_{c2j} \bigg],\\ p_{c,i} &= \Bigg( \bigg[ \sum_{k=1}^{N_{g_1}} \mathbf{N}_{ik} p_{c1k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} p_{c2j} + 0.10022\bigg]^{-2} + 1.3705\Bigg)\times 10^{5}, \label{eq:gcm-pc}\\ V_{c,i} &= \Bigg( \bigg[ \sum_{k=1}^{N_{g_1}} \mathbf{N}_{ik} v_{c1k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} v_{c2j} \bigg] -0.00435 \Bigg)\times 10^{-3}, \\ T_{b,i} &= 204.359 \ln \bigg[ \sum_{k = 1}^{N_{g_1}} \mathbf{N}_{ik} t_{b1k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} t_{b2j}\bigg],\\ T_{m,i} &= 102.425 \ln \bigg[ \sum_{k = 1}^{N_{g_1}} \mathbf{N}_{ik} t_{m1k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} t_{m2j}\bigg],\\ \Delta H_{f,i} &= \Bigg( \bigg[ \sum_{k = 1}^{N_{g_1}} \mathbf{N}_{ik} h_{f1k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} h_{f2j} \bigg] + 10.835\Bigg) \times 10^3,\\ \Delta G_{f,i} &= \Bigg( \bigg[ \sum_{k = 1}^{N_{g_1}} \mathbf{N}_{ik} g_{f1k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} g_{f2j} \bigg] -14.828 \Bigg) \times 10^3,\\ \Delta H_{v,\textit{stp},i} &= \Bigg( \bigg[ \sum_{k = 1}^{N_{g_1}} \mathbf{N}_{ik} h_{v1k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} h_{v2j} \bigg] + 6.829\Bigg) \times 10^3, \\ \omega_i &= 0.4085 \ln \bigg( \Big[ \sum_{k=1}^{N_{g_1}} \mathbf{N}_{ik} \omega_{1k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} \omega_{2j} + 1.1507\Big]^{1/0.5050} \bigg), \label{eq:gcm-omega}\\ V_{m,\textit{stp},i} &= \Bigg( \bigg[ \sum_{k=1}^{N_{g_1}} \mathbf{N}_{ik} v_{m1k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} v_{m2j} \bigg] + 0.01211 \Bigg)\times 10^{-3}, \\ C_{p,i} & =\bigg[\sum_{k=1}^{N_{g_1}} \mathbf{N}_{ik} C_{pA1_k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} C_{pA2_j} -19.7779\bigg] \nonumber \\ & +\bigg[\sum_{k=1}^{N_{g_1}} \mathbf{N}_{ik} C_{pB1_k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} C_{pB2_j} + 22.5981\bigg] \theta \nonumber\\ & +\bigg[\sum_{k=1}^{N_{g_1}} \mathbf{N}_{ik} C_{pC1_k} + \sum_{j=1}^{N_{g_2}} \mathbf{M}_{ij} C_{pC2_j} - 10.7983\bigg] \theta^2 \\ \theta &= \frac{T - 298.15}{700} \end{align*} .. _eq-GCM-correlations: Equations for individual compound correlations ---------------------------------------------- This section presents correlations for physical properties that leverage the individual compound properties defined in :ref:`eq-GCM-properties`. These correlations make it possible to evaluate physical properties at non-standard temperatures and pressures, given that group contribution properties are only defined at standard conditions. Unless noted otherwise in the individual correlation, all units are assumed to be SI: length (m), mass (kg), time (s), temperature (K), mole (mol). The :ref:`tab-reduced-temps` are used throughout this section for each compound *i*, provided :math:`T` in K unless noted otherwise. .. _tab-correlation-qtys: .. table:: Derived quantities and temperature corrections :widths: auto :align: center ============================= ===================== =============================================================== Property Units Description ============================= ===================== =============================================================== :math:`\nu_i` m\ :sup:`2`\ /s Kinematic viscosity\ :footcite:p:`viswanath_viscosity_2007`. :math:`L_{v,\textit{stp},i}` J/kg Latent heat of vaporization at 298 K\ :footcite:p:`govindaraju_group_2016`. :math:`L_{v,i}` J/kg Temperature-adjusted latent heat of vaporization at 298 K\ :footcite:p:`govindaraju_group_2016`. :math:`V_{m,i}` m\ :sup:`3`\ /mol Temperature-adjusted liquid molar volume\ :footcite:p:`rackett_equation_1970` \ :footcite:p:`yamada_saturated_1973` \ :footcite:p:`govindaraju_group_2016`. :math:`\rho_i` kg/m\ :sup:`3` Density :math:`C_{\ell,i}` J/kg/K Liquid specific heat capacity\ :footcite:p:`govindaraju_group_2016`. :math:`p_{sat,i}` Pa Saturated vapor pressure\ :footcite:p:`lee_generalized_1975` \ :footcite:p:`ambrose_vapour_1989`. :math:`\sigma_i` N/m Surface tension\ :footcite:p:`brock_surface_1955`. :math:`\lambda_i` W/m/K Thermal conductivity\ :footcite:p:`poling_properties_2001`. ============================= ===================== =============================================================== .. _tab-reduced-temps: .. table:: Reduced temperature quantities :widths: auto :align: center ============================= ========================================= ====================================================== Symbol Definition Description ============================= ========================================= ====================================================== :math:`T_{r,i}` :math:`\frac{T}{T_{c,i}}` Reduced temperature. :math:`T_{r,b,i}` :math:`\frac{T_{b,i}}{T_{c,i}}` Reduced boiling point temperature. :math:`T_{r,\textit{stp},i}` :math:`\frac{298 \text{ (K)}}{T_{c,i}}` Reduced standard temperature. ============================= ========================================= ====================================================== Kinematic viscosity ^^^^^^^^^^^^^^^^^^^ .. automethod:: FuelLib.fuel.viscosity_kinematic :noindex: The kinematic viscosity of the *i-th* compound of the fuel, .. math:: \nu_i = \frac{\mu_i}{\rho_i}, is calculated from Dutt's equation (Eq. 4.23 in Viscosity of Liquids\ :footcite:p:`viswanath_viscosity_2007`) provided :math:`T` in :math:`^{\circ}` C: .. math:: \begin{align*} \nu_i = 10^{-6} \times \exp \bigg\{-3.0171 + \frac{442.78 + 1.6452 \,T_{b,i}}{T + 239 - 0.19 \,T_{b,i}} \bigg\}. \end{align*} Latent heat of vaporization ^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. automethod:: FuelLib.fuel.latent_heat_vaporization :noindex: The latent heat of vaporization for each compound at standard pressure and temperature is calculated from the enthalpy of vaporization as: .. math:: L_{v,\textit{stp},i} = \frac{\Delta H_{v,\textit{stp},i}}{M_{w,i}}. The heat of vaporization for each compound is then adjusted for variations in temperature\ :footcite:p:`govindaraju_group_2016`: .. math:: L_{v,i} = L_{v,\textit{stp},i} \bigg(\frac{1 - T_{r,i}}{1-T_{r,b,i}} \bigg)^{0.38}. Liquid molar volume ^^^^^^^^^^^^^^^^^^^ .. automethod:: FuelLib.fuel.molar_liquid_vol :noindex: The liquid molar volume is calculated at a specific temperature :math:`T` using the generalized Rackett equation\ :footcite:p:`rackett_equation_1970` \ :footcite:p:`yamada_saturated_1973` with an updated :math:`\phi_i` parameter\ :footcite:p:`govindaraju_group_2016`: .. math:: V_{m,i} = V_{m,\textit{stp},i} Z^{\phi_i}_{c,i}, where .. math:: \begin{align*} Z_{c,i} &= 0.29056 - 0.08775 \omega_i, \\ \phi_i &= \begin{cases} (1 - T_{r,i})^{2/7} - (1 - T_{r,\textit{stp},i})^{2/7}, & \text{ if } T \leq T_{c,i} \\ - (1 - T_{r,\textit{stp},i})^{2/7}, & \text{ if } T > T_{c,i} \end{cases}. \label{eq:phi} \end{align*} Density ^^^^^^^ .. automethod:: FuelLib.fuel.density :noindex: The density of the *i-th* compound is given by .. math:: \rho_i = \frac{M_{w,i}}{V_{m,i}}. Liquid specific heat capacity ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. automethod:: FuelLib.fuel.Cl :noindex: The liquid specific heat capacity for each compound at standard pressure temperature is calculated from the specific heat capacity as: .. math:: C_{\ell,i} = \dfrac{C_{p,i}}{M_{w,i}} Saturated vapor pressure ^^^^^^^^^^^^^^^^^^^^^^^^ .. automethod:: FuelLib.fuel.psat :noindex: The saturated vapor pressure for each compound is calculated as a function of temperature using either the Lee–Kesler method\ :footcite:p:`lee_generalized_1975` or the Ambrose-Walton method\ :footcite:p:`ambrose_vapour_1989`. Both methods solve .. math:: \ln p_{r,\text{sat},i} = f_i^{(0)} + \omega_i f_i^{(1)} + \omega_i^2 f_i^{(2)} for the reduced saturated vapor pressure for each compound, :math:`p_{r,\text{sat},i} = p_{\text{sat},i}/p_{c,i}`. The default method in `FuelLib` is the Lee-Kesler method, as it is more stable at higher temperatures. The Lee-Kesler\ :footcite:p:`lee_generalized_1975` method defines .. math:: \begin{align*} f_i^{(0)} &= 5.92714 - \frac{6.09648}{T_{r,i}} - 1.28862 \ln T_{r,i} + 0.169347 \, T_{r,i}^6, \\ f_i^{(1)} &= 15.2518 - \frac{15.6875}{T_{r,i}} - 13.4721 \ln T_{r,i} + 0.43577 \, T_{r,i}^6, \\ f_i^{(2)} &= 0, \end{align*} The Ambrose-Walton\ :footcite:p:`ambrose_vapour_1989` correlation sets: .. math:: \begin{align*} f_i^{(0)} &= \frac{- 5.97616\tau_i + 1.29874\tau_i^{1.5} - 0.60394\tau_i^{2.5} - 1.06841\tau_i^{5}}{T_{r,i}}, \\ f_i^{(1)} &= \frac{- 5.03365\tau_i + 1.11505\tau_i^{1.5} - 5.41217\tau_i^{2.5} - 7.46628\tau_i^{5},}{T_{r,i}}, \\ f_i^{(2)} &= \frac{- 0.64771\tau_i + 2.41539\tau_i^{1.5} - 4.26979\tau_i^{2.5} - 3.25259\tau_i^{5}}{T_{r,i}}, \end{align*} with :math:`\tau_i = 1 - T_{r,i}`. Surface tension ^^^^^^^^^^^^^^^ .. automethod:: FuelLib.fuel.surface_tension :noindex: Surface tension for each compound is approximated using the relation: .. math:: \sigma_i = p_{c,i}^{2/3} T_{c,i}^{1/3} Q_i (1 - T_{r,i})^{11/9}, provided :math:`p_{c,i}` in bar. The :math:`Q_i` term is defined by Brock and Bird\ :footcite:p:`brock_surface_1955` (default in FuelLib) as .. math:: Q_i = 0.1196 \bigg[1 + \frac{T_{r,b,i} \log(p_{c,i}/1.01325)}{1 - T_{r,b,i}}\bigg] - 0.279, or by Curl and Pitzer\ :footcite:p:`poling_properties_2001` \ :footcite:p:`curl_volumetric_1958` \ :footcite:p:`pitzer_thermodynamics_1995` as .. math:: Q_i = \frac{1.86 + 1.18 \omega_i}{19.05} \bigg[ \frac{3.75 + 0.91 \omega_i}{0.291 - 0.08\omega_i} \bigg]^{2/3}. Thermal conductivity ^^^^^^^^^^^^^^^^^^^^ .. automethod:: FuelLib.fuel.thermal_conductivity :noindex: Thermal conductivity for each compound is computed according to the method of Latini et al. as summarized in Poling's\ :footcite:p:`poling_properties_2001` book: .. math:: \lambda_i = \frac{A_i(1 - T_{r,i})^{0.38}}{T_{r,i}^{1/6}}. The constant :math:`A_i` is defined by: .. math:: A_i = \frac{A^\ast T_{b,i}^\alpha}{M_{w,i}^\beta T_{c,i}^{\gamma}}, provided :math:`M_{w,i}` in g/mol. The exponents vary depending on the family of the compound as defined in :ref:`tab-thermal-conductivity-parameters`. It is assumed that: * aromatics have contain aromatic group contributions (e.g. ACCH) * cycloparaffins contain a ring (e.g. 5-membered ring) and do not contain aromatic groups * olefins contain one or more pairs of carbon atoms linked by a double bond and do not contain aromatic groups or rings * all other compounds are assumed to be saturated hydrocarbons. .. _tab-thermal-conductivity-parameters: .. table:: Thermal conductivity relation parameters :widths: auto :align: center =========== ========================== =============== =============== =============== =============== Identifier Family :math:`A^\ast` :math:`\alpha` :math:`\beta` :math:`\gamma` =========== ========================== =============== =============== =============== =============== 0 Saturated hydrocarbons 0.00350 1.2 0.5 0.167 1 Aromatics 0.0346 1.2 1.0 0.167 2 Cycloparaffins 0.0310 1.2 1.0 0.167 3 Olefins 0.0361 1.2 1.0 0.167 =========== ========================== =============== =============== =============== =============== .. _eq-mixture-properties: Equations for mixture properties from GCM ----------------------------------------- This section contains correlations for estimating physical properties of the mixture from the individual compound and physical properties defined in :ref:`eq-GCM-properties` and :ref:`eq-GCM-correlations`. These correlations make it possible to evaluate physical properties at non-standard temperatures and pressures, as the group contribution properties are only defined at standard conditions. The :ref:`tab-mixture-properties` available in `FuelLib` are listed in table below. Mass and mole fractions defined in Table :ref:`tab-mass-mole-fracs` are used throughout this section. .. _tab-mixture-properties: .. table:: Mixture properties :widths: auto :align: center =============== =============== ===================== Symbol Units Description =============== =============== ===================== :math:`\rho` kg/m\ :sup:`3` Density :math:`\nu` m\ :sup:`2`/s Kinematic viscosity :math:`p_v` Pa Vapor pressure :math:`\sigma` N/m Surface tension :math:`\lambda` W/m/K Thermal conductivity =============== =============== ===================== .. _tab-mass-mole-fracs: .. table:: Mass and mole fractions :widths: auto :align: center ============= ======================================== ================================================================================== Symbol Definition Description ============= ======================================== ================================================================================== :math:`Y_i` :math:`\frac{m_i}{\sum_{k=1}^{N_c} m_k}` Mass fraction of compound *i*. :math:`m_i` is the mass of compound *i*. :math:`X_i` :math:`\frac{n_i}{\sum_{k=1}^{N_c} n_k}` Mole fraction of compound *i*. :math:`n_i` is the number of moles compound *i*. ============= ======================================== ================================================================================== .. _conventional-mixing-rules: Conventional mixing rules ^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: FuelLib.mixing_rule :noindex: While many of the mixture properties in FuelLib have a unique mixing rule, FuelLib's *mixing_rule* function provides a general mixing rule based on the suggestions of Harstad et al\ :footcite:p:`harstad_efficient_1997`. For a given property :math:`Q` .. math:: Q = \sum_{i=1}^{N_c} \sum_{j=1}^{N_c} X_i X_j Q_{ij}, where the pseudo-property for the couple of components, :math:`Q_{ij}` is computed using an arithmetic, .. math:: Q_{ij} = \frac{Q_i + Q_j}{2}, or a geometric mean, .. math:: Q_{ij} = \sqrt{Q_i \cdot Q_j}, where :math:`Q_i` is the property of the *i-th* compound of the multicomponent mixture. Mixture density ^^^^^^^^^^^^^^^ .. automethod:: FuelLib.fuel.mixture_density :noindex: The mixture's density is calculated as: .. math:: \rho = \sum_{i=1}^{N_c}Y_i\frac{M_{w,i}}{V_{m,i}}. Mixture kinematic viscosity ^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. automethod:: FuelLib.fuel.mixture_kinematic_viscosity :noindex: The kinematic viscosity of the mixture is computed using the Kendall-Monroe\ :footcite:p:`kendall_viscosity_1917` mixing rule, with an option to use the Arrhenius\ :footcite:p:`arrhenius_uber_1887` mixing rule. The viscosity of each component. After evaluating thirty mixing rules, Hernandez et al.\ :footcite:p:`hernandez_evaluation_2021` found that both Kendall-Monroe and Arrhenius were among the most effective without relying on additional data or parameter fitting. The Kendall-Monroe rule is: .. math:: \nu_{KM}^{1/3} = \sum_{i=1}^{N_c} X_i \, \nu_i^{1/3}. The Arrhenius rule is: .. math:: \ln \nu_{Arr} = \sum_{i=1}^{N_c} X_i\ln\nu_i . Mixture vapor pressure ^^^^^^^^^^^^^^^^^^^^^^ .. automethod:: FuelLib.fuel.mixture_vapor_pressure :noindex: The vapor pressure of the mixture is calculated according to Raoult's law: .. math:: \begin{align*} p_{v} = \sum_{i = 1}^{N_c} X_i \, p_{\textit{sat},i}. \end{align*} .. automethod:: FuelLib.fuel.mixture_vapor_pressure_antoine_coeffs :noindex: Users also have the option to return the coefficients from an Antoine fit based on the mixture vapor pressure calculated from Raoult's law above. Antoine's equation is: .. math:: \begin{align*} \log_{10}(D \cdot p_{v}) = A - \frac{B}{C + T}, \end{align*} where :math:`D` is a conversion factor for converting :math:`p_v` to units of bar or dyne/cm :sup:`2` from Pa. This feature was added to provide `Pele `_ users an option for estimating these coefficients for use in CFD simulations with spray. See the `PelePhysics documentation `_ for additional information. Mixture surface tension ^^^^^^^^^^^^^^^^^^^^^^^ .. automethod:: FuelLib.fuel.mixture_surface_tension :noindex: The surface tension of the mixture is calculated using the :ref:`conventional-mixing-rules` with an arithmetic mean for the pseudo-property :math:`\sigma_{i,j}` as recommended by Hugill and van Welsenes\ :footcite:p:`hugill_surface_1986`: .. math:: \sigma = \sum_{i=1}^{N_c} \sum_{j=1}^{N_c} X_i X_j \frac{\sigma_i + \sigma_j}{2}. Mixture thermal conductivity ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. automethod:: FuelLib.fuel.mixture_thermal_conductivity :noindex: The thermal conductivity of the mixture is calculated using the power law method of Vredeveld as described in Poling\ :footcite:p:`poling_properties_2001`: .. math:: \lambda = \bigg(\sum_{i=1}^{N_c} Y_i \lambda_i^{-2} \bigg)^{-1/2}. Validation ---------- Single Component Fuels ^^^^^^^^^^^^^^^^^^^^^^ .. figure:: /figures/singleCompFuels.png :width: 600pt :align: center Properties of heptane, decane, and dodecane against predictive data from NIST Chemistry WebBook. Multi-Component Fuels ^^^^^^^^^^^^^^^^^^^^^ .. figure:: /figures/multiCompFuels.png :width: 600pt :align: center Properties of conventional jet fuels JP-8 (POSF10264), Jet A (POSF10325), and JP-5 (POSF10289) against data from the Air Force Research Laboratory\ :footcite:p:`edwards_jet_2020`. Note that the data sets for thermal conductivity are very inconsistent, but they typically show linear decreases in thermal conductivity with temperature. Fuel Blends ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. figure:: /figures/hefaBlends.png :width: 250pt :align: center Properties of three HEFA fuels produced from different feedstocks (camelina, tallow, and mixed fat) blended with Jet-A. Measurement and GCxGC data from Vozka et al.\ :footcite:p:`vozka_impact_2018`. References ---------- .. footbibliography::