Equation Map - Fanno

stagnation_enthalpy

$h_{t} = h + \frac{G^2}{\rho^2 2}$

fanno_temperature_ratio

$\frac{T_{2}}{T_{1}} = \frac{1 + \left[ (\gamma-1)/2 \right] M_{1}^2 }{ 1 + \left[ (\gamma-1)/2 \right] M_{2}^2 }$

fanno_pressure_ratio

$\frac{p_{2}}{p_{1}} = \frac{M_{1}}{M_{2}} \left( \frac{1 + \left[ (\gamma-1)/2 \right] M_{1}^2 }{ 1 + \left[ (\gamma-1)/2 \right] M_{2}^2 } \right) ^{1/2}$

fanno_temperature_ratio

$\frac{\rho_{2}}{\rho_{1}} = \frac{M_{1}}{M_{2}} \left( \frac{1 + \left[ (\gamma-1)/2 \right] M_{2}^2 }{ 1 + \left[ (\gamma-1)/2 \right] M_{1}^2 } \right) ^{1/2}$

fanno_stagnation_pressure_ratio

$\frac{p_{t2}}{p_{t1}} = \frac{M_{1}}{M_{2}} \left( \frac{1 + \left[ (\gamma-1)/2 \right] M_{2}^2 }{ 1 + \left[ (\gamma-1)/2 \right] M_{1}^2 } \right) ^{\frac{\gamma+1}{2(\gamma-1)}}$

fanno_temperature_star_ratio

$\frac{T}{T^*} = \frac{(\gamma+1)/2 }{ 1 + \left[ (\gamma-1)/2 \right] M^2}$

fanno_pressure_star_ratio

$\frac{p}{p^*} = \frac{1}{M} \left( \frac{(\gamma+1)/2 }{ 1 + \left[ (\gamma-1)/2 \right] M^2} \right) ^{1/2}$

fanno_density_star_ratio

$\frac{\rho}{\rho^*} = \frac{1}{M} \left( \frac{ 1 + \left[ (\gamma-1)/2 \right] M^2} {(\gamma+1)/2 } \right) ^ {1/2}$

fanno_velocity_star_ratio

$\frac{V}{V^*} = \frac{M}{1} \left( \frac {(\gamma+1)/2 }{ 1 + \left[ (\gamma-1)/2 \right] M^2} \right) ^ {1/2}$

fanno_parameter

$\frac{ f(x_{2} - x_{1}) }{D_{e}} = \frac{\gamma + 1}{2\gamma} \ln \frac{ 1 + \left[ (\gamma-1)/2 \right] M_{2}^2 }{1 + \left[ (\gamma-1)/2 \right] M_{1}^2 } - \frac{1}{\gamma} \left( \frac{1}{M_{2}^2} - \frac{1}{M_{1}^2} \right) - \frac{\gamma + 1}{2\gamma} \ln \frac{M_{2}^2}{M_{1}^2}$

fanno_parameter_max

$\frac{f(x - x^*)} {D_{e}} = \frac{\gamma + 1}{2\gamma} \ln \frac{ 1 + \left[ (\gamma-1)/2 \right] M^2} {(\gamma+1)/2 } - \frac{1}{\gamma} \left( \frac{1}{M^2} - 1 \right) - \frac{\gamma + 1}{2\gamma} \ln M^2$

mach_from_fanno uses an equation solver to determine the mach number.