# Mixture Rules¶

For each propellant mixture, RocketProps starts with two well-documented propellants and uses the properties of those two propellants to calculate expected properties of the mixture.

Mixtures of N2H4 and MMH, for example, use N2H4 and MHF3 having MMH weight fractions below 86%, and would use MHF3 and MMH for MMH weight fractions above 86%. (Note that MHF3 is a mixture of 86% MMH + 14% N2H4)

The table below shows the reference propellants used to calculate mixtures of base and additive propellants within each weight percent additive range.

Base Propellant Additive Additive Wt% Range 1st Ref Propellant 2nd Ref Propellant
N2H4 MMH 0 to 86 N2H4 MHF3
N2H4 MMH 86 to 100 MHF3 MMH
UDMH N2H4 0 to 50 UDMH A50
UDMH N2H4 50 to 100 A50 N2H4
N2O4 NO 0 to 10 N2O4 MON10
N2O4 NO 10 to 25 MON10 MON25
N2O4 NO 25 to 30 MON25 MON30
LF2 LOX 0 to 100 LF2 LOX

## Freezing Point¶

The freezing point of each mixture is determined as outlined in Mixture Freezing Points

## Acentric Factor¶

The value of the mixture Acentric Factor , omega, is calculated using a mole fraction simple mixing rule as recommended by Eqn 5-3.3 of Gas&Liq 5th Ed

$\text{omega}_{mix} = \sum_i \text{mole_frac}_i \cdot \text{omega}_i$

## Molecular Weight¶

By definition, the value of the mixture Molecular Weight, MW, is calculated using a mole fraction simple mixing rule.

$\text{MW}_{mix} = \sum_i \text{mole_frac}_i \cdot \text{MW}_i$

## Reference T & P¶

In order to create a properties summary such as the M20 Example Mixture Summary the mixture needs to have a reference temperature ($$\text{T}_{mix}$$) and pressure ($$\text{P}_{mix}$$) at which to calculate reference properties.

In case the reference $$\text{T}_i$$ and $$\text{P}_i$$ for the base propellants are different, the mixture $$\text{T}_{mix}$$ and $$\text{P}_{mix}$$ are calculated as a mole fraction simple mixing rule of the $$\text{T}_i$$ and $$\text{P}_i$$ of each base propellant.

$\text{T}_{mix} = \sum_i \text{mole_frac}_i \cdot \text{T}_i$
$\text{P}_{mix} = \sum_i \text{mole_frac}_i \cdot \text{P}_i$

Reference properties for each base propellant are calculated at $$\text{T}_{mix}$$ and $$\text{P}_{mix}$$ and are combined with an appropriate mixing rule to compute the mixture reference properties.

## Critical Temperature¶

Calculation of mixture critical temperature ($$\text{T}_{cm}$$) uses the Li correlation from Caleb Bell, Yoel Rene Cortes-Pena, and Contributors (2016-2021). Chemicals: Chemical properties component of Chemical Engineering Design Library (ChEDL) https://chemicals.readthedocs.io/chemicals.critical.html#critical-temperature-of-mixtures.

Although Kay’s rule Eqn 5-3.1 from Gas&Liq 5th Ed (i.e. simple mole fraction mixing rule) is often sufficient. Better accuracy can usually be expected from the Li correlation.

$\begin{split}T_{cm} = \sum_{i=1}^n \Phi_i T_{ci}\\ \Phi = \frac{x_i V_{ci}}{\sum_{j=1}^n x_j V_{cj}}\end{split}$
:
Li, C. C. “Critical Temperature Estimation for Simple Mixtures.” The Canadian Journal of Chemical Engineering 49, no. 5 (October 1, 1971): 709-10. doi:10.1002/cjce.5450490529.

## Critical Compressibility Factor¶

Eqn 5-3.2 from Gas&Liq 5th Ed recommends simple mole fraction mixing rule to calculate the mixture critical compressibility factor ($$\text{Z}_{cm}$$)

$\text{Z}_{cm} = \sum_i \text{mole_frac}_i \cdot \text{Z}_i$

## Critical Pressure¶

The simplest rule which can give acceptable $$\text{P}_{cm}$$ values for two-parameter or three-parameter CSP (corresponding states principle) is the modified rule of Prausnitz and Gunn (1958); Eqn 5-3.2 from Gas&Liq 5th Ed.

$\text{P}_{cm} = \frac{\text{Z}_{cm} \cdot R \cdot \text{T}_{cm}}{\text{V}_{cm}}$

## Vapor Pressure¶

The mixture vapor pressure $$\text{P}_{vapm}$$ is assumed to follow Raoult’s Law for vapor pressure.

$\text{P}_{vapm} = \sum_i \text{mole_frac}_i \cdot \text{P}_{vapi}$

## Heat Capacity¶

Mixture heat capacity $$\text{C}_{pm}$$ can be estimated with a simple mixing rule as noted in Eqn 2-55 of Perry’s Chemical Engineers’ Handbook 8th Edition by Don Green and Robert Perry

Note that in Perry’s, the Cp has units of energy/mole and therefore uses mole fraction in the simple mixing equation. Since RocketProps uses energy/mass, mass fraction is used in the simple mixing equation.

$\text{C}_{pm} = \sum_i \text{mass_frac}_i \cdot \text{C}_{pi}$

## Heat of Vaporization¶

Mixture heat of vaporization $$\text{H}_{vapm}$$ is “tricky”. Also note that, like heat capacity $$\text{C}_{pm}$$, heat of vaporization, $$\text{H}_{vapm}$$ in RocketProps uses energy/mass for $$\text{H}_{vapm}$$ and is therefore based on mass fraction, not mole fraction.

The most common use of $$\text{H}_{vapm}$$ is in evaluating propellant droplet evaporation in a thrust chamber in order to calculate thrust chamber performance and combustion stability.

When a droplet first begins to evaporate, the instantaneous $$\text{H}_{vapm}$$ is based on the mass fraction of vapor given off for each of the mixtures constituents, not the liquid mass fractions. After a droplet has completely evaporated, the net $$\text{H}_{vapm}$$ is based on the starting liquid droplet mass fraction.

Calculating the energy release characteristics in a thrust chamber is well beyond the scope of RocketProps. For that reason, RocketProps assumes complete droplet evaporation and calculates mixture heat of vaporization $$\text{H}_{vapm}$$ based on initial droplet mass fraction.

$\text{H}_{vapm} = \sum_i \text{mass_frac}_i \cdot \text{H}_{vapi}$

## Thermal Conductivity¶

Calculation of mixture thermal conductivity ($$\lambda_m$$) uses the Filippov correlation from Caleb Bell, Yoel Rene Cortes-Pena, and Contributors (2016-2021). Chemicals: Chemical properties component of Chemical Engineering Design Library (ChEDL) https://chemicals.readthedocs.io/chemicals.thermal_conductivity.html#liquid-mixing-rules.

Note that the Filippov correlation applies to binary mixtures only and that RocketProps currently only supports binary mixtures.

$\lambda_m = w_1 \lambda_1 + w_2\lambda_2 - 0.72 w_1 w_2(\lambda_2-\lambda_1)$

For future RocketProps mixtures that might move beyond binary mixtures, the DIPPR9H correlation, also from the ChEDL, is included.

$\lambda_m^{-2} = \frac{\sum_i z_i {MW}_i \lambda_i^{-2}} {\sum_i z_i {MW}_i}$

## Surface Tension¶

Calculation of mixture surface tension ($$\sigma_m$$) uses the Winterfeld_Scriven_Davis correlation from Caleb Bell, Yoel Rene Cortes-Pena, and Contributors (2016-2021). Chemicals: Chemical properties component of Chemical Engineering Design Library (ChEDL) https://chemicals.readthedocs.io/chemicals.interface.html#mixing-rules.

$\sigma_m = \sum_i \sum_j \frac{1}{V_L^{L2}}\left(x_i V_i \right) \left( x_jV_j\right)\sqrt{\sigma_i\cdot \sigma_j}$

## Viscosity¶

The Chemical Engineering Design Library (ChEDL) for liquid mixture viscosity ($$\mu_m$$) recommends logarithmic mixing with weight fractions https://chemicals.readthedocs.io/chemicals.viscosity.html#liquid-mixing-rules is:

With that in mind, mixture viscosity in RocketProps is calculated as:

$\mu_m = exp(\sum_i \text{massfrac}_i \cdot \ln(\mu_i))$