Specific Range⚠ unverified

Understanding the Parameters

• What specific range is — distance per unit fuel weight, $SR = dR/(-dW)$, in metres per newton. It is the local efficiency at one instant of cruise; total range is its integral as fuel burns and weight falls. This is the quantity a flight-management computer maximises to find the most economical cruise.

• True airspeed $V$ — appears in the numerator: covering ground faster (at fixed fuel flow) means more distance per kilogram. But raising $V$ raises drag and thus fuel flow, so $SR$ has a maximum at an intermediate speed — the max-range speed.

• $L/D$ — where the registry goes wrong — physically, a higher lift-to-drag ratio means less drag, less required thrust, less fuel flow, and therefore more distance per unit fuel. Specific range must rise with $L/D$. The registry's division by $L/D$ inverts this, and is the flagged defect.

• Aircraft weight $W$ (missing from the registry) — a heavier aircraft needs more lift, hence more induced drag and thrust, so $SR$ falls as $W$ rises. This is why $SR$ improves steadily through a long cruise as fuel burns off — and why airliners "step-climb" to stay efficient. The registry form omits $W$, so it cannot show this behaviour.

• The $g$ factor — the 9.81 converts a mass-based specific fuel consumption into the weight-based fuel flow needed to express $SR$ per unit fuel weight (newtons).

Derivation (Approaching a Proof)

Specific range is defined as distance per unit weight of fuel burned:

$$SR = \frac{dR}{-dW} = \frac{V\,dt}{-dW} = \frac{V}{-\,dW/dt} = \frac{V}{\dot W_{\text{fuel}}},$$

where $\dot W_{\text{fuel}}$ is the fuel weight burned per second. For a jet with thrust-specific fuel consumption $c_t$ (weight basis), the fuel weight flow is $\dot W_{\text{fuel}} = c_t\,T$. In steady level cruise $T = D = W/(L/D)$ (because $L = W$), so

$$\dot W_{\text{fuel}} = c_t\,\frac{W}{L/D} \quad\Longrightarrow\quad SR = \frac{V}{c_t\,W/(L/D)} = \frac{V\,(L/D)}{c_t\,W}.$$

This is the correct specific range: $\propto V$, $\propto L/D$, $\propto 1/W$. Integrating $-SR\,dW$ from $W_i$ to $W_f$ recovers $R = \tfrac{V}{c_t}\tfrac{L}{D}\ln\tfrac{W_i}{W_f}$ — exactly Range Breguet, confirming specific range is its integrand.

If $c_t$ is instead given on a mass basis (kg fuel per N thrust per s), the fuel weight flow gains a factor $g$: $\dot W_{\text{fuel}} = c_t\,g\,T$, giving $SR = \dfrac{V\,(L/D)}{c_t\,g\,W}$ — which is the registry formula except that the registry has $L/D$ in the denominator and drops $W$. Setting $W = 1$ and $L/D = 1$ (the defaults) makes the registry expression numerically equal to this correct one, which is why the bug is invisible at default values.

Dimensional check (correct form, mass-based $c_t$). Take the fuel-flow denominator first. With $c_t$ in $\text{kg}/(\text{N}\cdot\text{s})$, $$c_t\,g\,W = \frac{\text{kg}}{\text{N}\cdot\text{s}}\cdot\frac{\text{m}}{\text{s}^2}\cdot\text{N} = \frac{\text{kg}\cdot\text{m}}{\text{s}^3} = \frac{\text{N}}{\text{s}}$$ (using $\text{kg}\cdot\text{m}/\text{s}^2 = \text{N}$) — a fuel weight flow, as it should be. Then $$SR = \frac{V}{c_t\,g\,W}\times\frac{L}{D} = \frac{\text{m}/\text{s}}{\text{N}/\text{s}}\times(\text{–}) = \frac{\text{m}}{\text{N}}. \checkmark$$ Distance per unit fuel weight.

History and Development

• From Breguet to the fuel gauge. Specific range is the differential heart of the 1920s Breguet analysis: recognising that range is the integral of an instantaneous efficiency turned range optimisation into a pointwise problem — "at each instant, fly the speed and altitude that maximise miles per pound of fuel."

• The cost of carrying fuel. Because $SR \propto 1/W$, fuel already aboard costs range: an aircraft burns extra fuel just to carry its later fuel. This "fuel to carry fuel" penalty, visible directly in specific range, is why long-haul design obsesses over structural weight and why tankering fuel is usually inefficient.

• Modern flight management. Every airliner's flight-management system computes specific range in real time and recommends the cost-index-optimal speed and step-climb schedule. The concept also generalises: "range factor" and "specific air range (SAR)" are the same idea under different names in operational manuals.

Related Concepts: Range Breguet, Endurance, Lift-to-Drag Ratio, Drag Force, Lift Force, Thrust, Glide Ratio

Notes: Registry form appears inverted in $L/D$ — the correct specific range is $SR = V(L/D)/(c_t g W)$, which increases with $L/D$ and decreases with weight $W$; the registry divides by $L/D$ and omits $W$ (coincides only at the all-$1.0$ defaults). It is the integrand of Range Breguet ($R = \int SR\,(-dW)$). The 9.81 converts a mass-based $c_t$ to weight-based fuel flow.