Jet Propulsion/1D Analysis

Ramjet
The thrust produced by a ramjet is

$$ T = \dot{m} u_{inlet} ( \sqrt{\tau_b} - 1 ) $$

where &tau;b is the total temperature ratio produced by the combustor. Metallurgy and the availability of cooling will limit the maximum temperature that can be sustained in the combustor. We can define &tau;max as the maximum total temperature ratio compared to the inlet conditions and &tau;c as the temperature ratio relating the static and total temperatures. Then

$$ \tau_{max} = \tau_b \tau_c = \tau_b \left( 1 + \frac {\gamma-1}{2} M_0^2 \right) $$

Thus as &tau;c increases with speed for a fixed maximum temperature &tau;max stays constant and &tau;b must reduce. If the theoretical frontal area of the ramjet is constant then the mass flow through the ramjet will increase linearly with the Mach number. At the same time the heat added diminishes. The thrust then is:

$$ T = A \rho (M_0 a_0)^2  \left[ \sqrt{\frac{\tau_{max}}{1 + \frac {\gamma-1}{2} M_0^2}} - 1 \right] $$

When &tau;max=&tau;c the term in brackets goes to zero and the thrust vanishes. The thrust for a given &tau;max is shown in the following figure (the dashed line is the peak thrust):