Ramjet Efficiency⚠ unverified

Understanding the Parameters

• Flight Mach number $M$ — the single performance lever. Because the ramjet's only compression comes from decelerating the free stream, the pressure ratio is the stagnation ratio $(1 + \tfrac{\gamma-1}{2}M^2)^{\gamma/ (\gamma-1)}$, which grows fast with $M$. This is why ramjets are useless below $M \approx 0.5$, workable from $M \approx 2$–$3$, and dominant up to $M \approx 5$–$6$ where the turbojet's machinery gives out.

• Specific heat ratio $\gamma$ — the compressibility of the air ($\approx 1.4$ for cold air, falling toward $\sim 1.3$ as the flow heats). It sets how much temperature (and pressure) a given Mach-number deceleration produces. The formula's $\tfrac{\gamma-1}{2}M^2$ group is exactly the stagnation-temperature rise ratio.

• The output $\eta$ — the ideal (loss-free) thermal efficiency: the fraction of heat input that the cycle can convert to work. It rises monotonically from $0$ at $M = 0$ toward $1$ at high Mach. It is a thermodynamic ceiling, not the delivered efficiency — real ramjets suffer inlet shock losses, combustor pressure loss, and incomplete expansion, and above $M \approx 5$–$6$ the classic ramjet must give way to the scramjet (supersonic combustion) to avoid dissociation and excessive stagnation temperature.

• Why it equals the Brayton efficiency. The compression is achieved by the inlet rather than a compressor, but thermodynamically it is the same isentropic pressure rise — so the ideal ramjet is just a Brayton cycle whose pressure ratio is fixed by flight speed.

Derivation (Approaching a Proof)

The ideal ramjet is a Brayton cycle: isentropic compression (in the inlet), constant-pressure heat addition (in the combustor, Combustion Chamber Temperature), isentropic expansion (in the nozzle), and — closing the cycle — heat rejection to the atmosphere. The ideal thermal efficiency of a Brayton cycle is

$$\eta = 1 - \frac{1}{r_p^{\,(\gamma-1)/\gamma}} = 1 - \frac{T_1}{T_2},$$

where $r_p$ is the compression pressure ratio and $T_2/T_1$ the corresponding temperature ratio. For a ramjet, the compression is the ram (stagnation) process — the free stream at static temperature $T_1$ and Mach $M$ is brought to rest, and the isentropic stagnation-temperature relation (Mach Number, Speed Of Sound) gives

$$\frac{T_2}{T_1} = \frac{T_0}{T_\infty} = 1 + \frac{\gamma-1}{2}M^2.$$

Substituting this temperature ratio directly into the Brayton efficiency:

$$\eta = 1 - \frac{T_1}{T_2} = 1 - \frac{1}{1 + \frac{\gamma-1}{2}M^2}. \qquad\blacksquare$$

The result is transparent: efficiency is one minus the reciprocal of the stagnation-temperature ratio. At $M \to 0$ the ratio $\to 1$ and $\eta \to 0$ (no compression, no cycle); as $M \to \infty$ the ratio diverges and $\eta \to 1$. For example at $M = 3$, $\gamma = 1.4$: $\tfrac{\gamma-1}{2}M^2 = 1.8$, so $\eta = 1 - 1/2.8 \approx 0.64$.

Dimensional check. $\tfrac{\gamma-1}{2}M^2$ is dimensionless, so $1/(1 + \cdots)$ and $\eta$ are dimensionless — an efficiency, as required. $\checkmark$

History and Development

• The engine with no moving parts. The ramjet was patented by René Lorin (France) in 1913, before any aircraft could fly fast enough to make it work. Its beauty is mechanical simplicity — no compressor, no turbine, just an inlet, a combustor, and a nozzle — but that same feature is its curse: it produces zero static thrust and must be boosted to speed before it can run, exactly as the $\eta \to 0$ at $M = 0$ limit predicts.

• The high-speed niche. Ramjets powered missiles (Bomarc, and later air-to-air and cruise missiles) and the boundary of piloted flight. Because the turbojet loses thrust and its turbomachinery cannot survive the stagnation temperatures at high Mach, the ramjet inherits the regime from about $M \approx 3$ to $M \approx 6$.

• Toward the scramjet. Above $M \approx 6$ decelerating the flow to subsonic speed for combustion produces ruinous stagnation temperatures and dissociation losses. The scramjet keeps the flow supersonic through the combustor, extending air-breathing propulsion toward hypersonic flight — the frontier that programmes like the X-43 and X-51 have probed. The efficiency ceiling here still rises with $M$, but real losses grow faster.

Related Concepts: Brayton Efficiency, Turbojet Thrust, Combustion Chamber Temperature, Mach Number, Speed Of Sound, Thermal Efficiency, Carnot Efficiency

Notes: Registry calculator ramjet-efficiency (unverified). Ideal (loss-free) Brayton thermal efficiency with compression set by ram (stagnation) heating — $\eta = 1 - 1/(1+\tfrac{\gamma-1}{2}M^2)$. Rises from $0$ at $M=0$ (no static thrust — must be boosted) toward $1$ at high $M$. A thermodynamic ceiling; real inlet/combustor losses and (above $M\approx6$) dissociation reduce it, motivating the scramjet.