.. models Models ======= The models chosen for the propellant properties database are described below. In general, if a standard `REFPROP `_ fluid is available, then `REFPROP `_ will be used as the source of the model data. Standard `REFPROP `_ models are present for ethane, ethanol, fluorine, oxygen, methane, methanol, nitrous oxide, ammonia, parahydrogen and propane. For other propellants like A50, ClF5, IRFNA, MHF3, MMH, MON10 thru 30, N2H4, N2O4, RP1 and UDMH other sources will be used. Data sources usually do not include the full temperature range from freezing point to critical point, so some extrapolation or interpolation of the data is required. `Properties of Gases and Liquids, Fifth Edition `_, will be used to select appropriate models to represent and extend the data. Vapor Pressure -------------- Vapor pressure data will be extended using the **Wagner** equation. For a number of published `REFPROP `_ fluids, the constants for the **Wagner** equation are published. It can be seen from the vapor pressure charts for those propellants that `REFPROP `_ and the **Wagner** equation agree very well (see: methane, methanol, ethane, ethanol, propane, ammonia and parahydrogen) :ref:`Gas&Liq 5th Ed Source` recommends that the **Wagner** equation may be used down to a reduced temperature of 0.5, however, all of the REFPROP examples show its continued accuracy below that temperature. .. _Wagner: .. _Wagner Fit Source: Wagner `````` The equation requires four constants to be fitted to fluid vapor pressure data. .. image:: ./_static/wagner_eqn.jpg The values of a, b, c and d have been fitted to a number of propellants for use in RocketProps.:: a b c d Methane -6.02242, 1.26652, -0.5707, -1.366 Methanol -8.63571, 1.17982, -2.4790, -1.0240 Ethane -6.47500, 1.41071, -1.1440, -1.8590 Ethanol -8.68587, 1.17831, -4.8762, 1.5880 Propane -6.76368, 1.55481, -1.5872, -2.024 NH3 -7.28322, 1.5716, -1.85672, -2.39312 PH2 -4.902616, 1.06500, 0.737305, 0.053125 N2H4 -6.7779260205203515, 0.031241419877672347, -1.75576214157275, -0.6791935603862034 N2O4 -8.437950623751371, -3.7198359775967993, 3.2683044393949223, 12.45086649764585 A50 -7.641659365830601, 3.2061563372352757, -2.9346526057929907, 2.9995620039010578 MHF3 -4.430400554283357, -4.289202857522381, 0.4548211727181293, 1.9771250142385894 MMH -6.700466713713631, -0.09747624183487236, -0.7874544884949485, -6.37220831093827 UDMH -7.335910621921567, -0.07120599848815566, 1.0118500681251754, -9.504291228049002 MON10 -8.859867572622312, -0.23569425311772815, -1.21511310516509, -2.7924844994111235 MON25 -8.626779322479342, -0.2877745218667564, -1.2609528745852392, -3.1994074964463137 MON30 -8.394164877030429, -0.3811184987718641, -1.1304799873211373, -3.5220091386630217 IRFNA -6.198317212258754, -3.187645737139488, 4.993054204407724, -14.807512940520226 .. _Edalat Source: Edalat `````` The `Edalat Vapor Pressure equation `_ can estimate vapor pressure over a range of temperatures. It is taken from the `thermo `_ project which is an open-source project on `pypi.org `_ developed by Caleb Bell and Contributors (2016-2020). It can be used to scale a sparse set of vapor pressure data (e.g. normal boiling point) or when no vapor pressure data is available, it can estimate the full range of vapor pressure. .. image:: ./_static/psat_edalat_eqn.jpg Density ------- Density data will be extended using either the **Rackett** Equation, the **Yamada and Gunn** modification of the **Rackett** Equation or the **Daubert** Equation. :ref:`Gas&Liq 5th Ed Source` defines those equations as follows. .. _Rackett: .. _Rackett Scaling Source: Rackett ``````` .. image:: ./_static/rackett_eqn.jpg .. image:: ./_static/yamada_and_gunn_eqn.jpg .. _Daubert: .. _Daubert Source: Daubert ``````` The **Daubert** Equation changed the physical quantities and constants of Eq. (4-11.1) into four adjustable parameters to give the equation below. The value of C is generally equal to Tc while A, B, and D are generally close to the values used in Eq. (4-11.3) above. For propellants like MON10, MON25 and MON30, the Daubert equation does a good job of fitting data without needing the temperature-dependent pseudocritical properties of the mixtures. .. image:: ./_static/daubert_eqn.jpg The constants A, B, C and D were curve fit for MON10, MON25 and MON30.:: A B C D MON10 184.6090958200138, 97.08205370646357, 852.8788442633142, -0.30396612785737676 MON25 148.1481569994554, 79.40853919233791, 871.4077962994412, -0.3164174025669882 MON30 179.63314607505583, 98.3583884965557, 869.460001600849, -0.29906872709771604 Heat Capacity ------------- Liquid heat capacity approximations often start by calculating a residual heat capacity. The residual heat capacity is the difference between the liquid and ideal gas heat capacities. The following excerpt from :ref:`Gas&Liq 5th Ed Source` describes one such method. .. image:: ./_static/cp_corr_states.jpg The above residual heat capacity can be added to the ideal gas heat capacity as calculated by the Joback method below. .. _Joback Source: Joback `````` The `Joback method `_ (often named Joback/Reid method) predicts eleven important and commonly used pure component thermodynamic properties from molecular structure only. The approach for ideal gas is shown below. .. image:: ./_static/joback_cp_calc.jpg Viscosity --------- .. _Squires Source: Squires ``````` Figure 9-13 from :ref:`Gas&Liq 5th Ed Source` displays the Lewis and Squires liquid viscosity vs temperature correlation. It leads to the scaling equation 9-10.3. Given a reference viscosity at a given temperature, the value at a different temperature may be calculated. .. image:: ./_static/visc_squires_chart_v2.jpg Heat of Vaporization -------------------- .. _Pitzer Hvap Source: Pitzer `````` The `Pitzer Hvap equation `_ calculates enthalpy of vaporization at arbitrary temperatures. It is taken from the `thermo `_ project which is an open-source project on `pypi.org `_ developed by Caleb Bell and Contributors (2016-2020). .. image:: ./_static/hvap_pitzer_eqn.jpg Thermal Conductivity -------------------- .. _Nicola Source: Nicola `````` The `Nicola equation `_ is taken from the `thermo `_ project which is an open-source project on `pypi.org `_ developed by Caleb Bell and Contributors (2016-2020). It estimates the thermal conductivity of a liquid as a function of temperature using a CSP method. It is a statistically derived equation using the following correlated terms. .. image:: ./_static/cond_nicola_eqn.jpg Surface Tension --------------- .. _Sastri-Rao Source: Sastri-Rao `````````` Surface tension can be scaled using equation 12-4.1 from :ref:`Gas&Liq 5th Ed Source`. The value of n varies from 0.8 for alcohols in the Sastri-Rao method to 1.22, or 11/ 9 for other compounds. .. image:: ./_static/surften_scaling_rao.jpg For propellants having some surface tension data, n was derived from curve fit. Those values are:: n N2H4 = 1.322137 N2O4 = 0.960017 A50 = 1.000717 MHF3 = 0.859508 MMH = 0.846186 UDMH = 1.018771 .. _Pitzer Surf Source: Pitzer `````` The `Pitzer surface tension equation `_ is taken from the `thermo `_ project which is an open-source project on `pypi.org `_ developed by Caleb Bell and Contributors (2016-2020). For propellants with no surface tension data available, the `Pitzer surface tension equation `_ can be used to predict surface tension. .. image:: ./_static/surften_pitzer_eqn.jpg Pseudocritical Properties ------------------------- For propellants that are mixtures of other propellants (like A50, MHF3, MON10, MON25 or MON30) the critical properties are not pure fluid properties, but a combination of the constituent properties. .. _Tc Mixture: Tc Mixture `````````` For pseudocritical temperature, :ref:`Gas&Liq 5th Ed Source` recommends the simplest mixing rule; a mole fraction average method. This rule is often called one of Kay's rules. Comparison of Tcm from Eq. (5-3.1) with values determined from other, more complicated rules shows that the differences in Tcm are usually less than 2% if, for all components the pure component critical properties are not extremely different. .. image:: ./_static/kays_rule_eqn.jpg .. _Pc Mixture: Pc Mixture `````````` From :ref:`Gas&Liq 5th Ed Source` for the pseudocritical pressure, Pcm, a mole-fraction average of pure-component critical pressures is normally unsatisfactory. This is because the critical pressure for most systems goes through a maximum or minimum with composition. The only exceptions are if all components of the mixture have quite similar critical pressures and/ or critical volumes. The simplest rule which can give acceptable Pcm values for two-parameter or three-parameter CSP is the modified rule of Prausnitz and Gunn (1958) .. image:: ./_static/prausnitz_and_gunn_eqn.jpg where **all of the mixture pseudocriticals Zcm , Tcm , and Vcm are given by mole-fraction averages** (Kay's rule) and R is the universal gas constant of **Table 4-1**. .. image:: ./_static/table_4-1.jpg .. _Omega Mixture: Omega Mixture ````````````` According to :ref:`Gas&Liq 5th Ed Source`, the mixture pseudo acentric factor is commonly given by a mole fraction average (Joffe, 1971) .. image:: ./_static/omega_mixture_eqn.jpg