4. Nozzle Losses¶

The ideal thrust coefficient \(C_f\) derived in §1.8 assumes isentropic, perfectly expanded, single-phase, axially directed flow. Real nozzles deviate from this ideal due to four independent loss mechanisms. Each is expressed as a percentage loss and combined into an overall nozzle efficiency \(\eta_{nozzle}\).

All empirical correlations are from: Coats et al. (1975). A Computer Program for the Prediction of Solid Propellant Rocket Motor Performance, Vol. I. AFRPL-TR-75-36 (DTIC AD-A015 140).

4.1 Divergent Loss¶

A conical nozzle exhausts gas over a range of directions from \(0\) to the half-angle \(\alpha\) relative to the thrust axis. The axial momentum fraction is computed by integrating over the cone surface, assuming uniform flow across the exit plane:

\[ \lambda = \frac{\int_0^\alpha \cos\theta\,2\pi\sin\theta\,d\theta} {\int_0^\alpha 2\pi\sin\theta\,d\theta} = \frac{\left[\frac{\sin^2\theta}{2}\right]_0^\alpha}{[-\cos\theta]_0^\alpha} = \frac{1 + \cos\alpha}{2} \]

The divergence loss is \(1 - \lambda\), expressed as a fraction in [0, 1]:

\[ \boxed{\eta_{div} = \tfrac{1}{2}\,(1-\cos\alpha)} \]

Applicable to conical nozzles only. Contoured (bell) nozzles can achieve near-zero divergence loss by design. Implemented in get_nozzle_divergent_loss_fraction.

Applies to: solid and liquid motors.

4.2 Kinetics Loss¶

In the ideal \(C_f\) derivation, the combustion products are assumed to remain in shifting chemical equilibrium as they expand through the nozzle. In reality, finite-rate chemistry "freezes" the composition at some point, leaving unreacted species that carry energy out of the nozzle without contributing to thrust.

The magnitude is proportional to the gap between CEA-predicted frozen and equilibrium (shifting) specific impulses:

\[ \eta_{kin} = 0.333\left(1 - \frac{I_{sp,frozen}}{I_{sp,shifting}}\right) \times f_P \]

where the pressure correction factor accounts for the fact that higher chamber pressures accelerate reactions toward equilibrium:

\[ f_P = \begin{cases} 1 & P_0 < 200\ \text{psi} \\ \dfrac{200}{P_0[\text{psi}]} & P_0 \geq 200\ \text{psi} \end{cases} \]

Implemented in get_kinetics_loss_fraction.

Applies to: solid and liquid motors.

4.3 Boundary Layer Loss¶

Viscous effects near the nozzle wall create a low-velocity boundary layer that reduces the effective flow area and transfers heat from the gas to the wall. The governing parameter group emerges from Blasius flat-plate boundary-layer theory: the displacement thickness scales as \(\delta^* \propto P_0^{0.8} / D_t^{0.2}\) (from a Reynolds-number argument with viscosity \(\propto T^{0.7}\)).

The loss is also time-dependent: the wall heats up during the burn, reducing the heat-flux driving force and therefore the loss:

\[ \eta_{BL} = 0.01 \cdot C_1\frac{P_0^{0.8}}{D_t^{0.2}} \left[1 + 2\exp\!\left(-\frac{C_2\, P_0^{0.8}\, t}{D_t^{0.2}}\right)\right] \left[1 + 0.016\,(\varepsilon - 9)\right] \]

with pressures in psi and diameter in inches. The 0.01 factor converts the classical percent-form expression to the fraction convention used here. \(C_1\) and \(C_2\) are nozzle-material constants (e.g. \(C_1=0.003650\), \(C_2=0.000937\) for a standard graphite/phenolic nozzle). Implemented in get_boundary_layer_loss_fraction.

Applies to: solid motors only (the empirical constants \(C_1, C_2\) are calibrated against a BATES motor; the LRE state passes \(\eta_{BL} = 0\)).

4.4 Two-Phase Flow Loss¶

Solid and hybrid propellants produce condensed-phase particles (e.g. alumina \(\text{Al}_2\text{O}_3\)) in the combustion products. These particles cannot expand isentropically through the nozzle and lag behind the gas, transferring momentum and heat imperfectly to the surrounding gas.

Step 1 — Average particle size (condensation in chamber + coagulation in nozzle):

\[ d_p = 0.454\;P_0^{1/3}\;\xi^{1/3} \left(1 - e^{-0.004 L^*}\right) \left(1 + 0.045\,D_t\right) \quad [\mu\text{m}] \]

where \(\xi\) is the condensed-phase mass fraction, \(L^*\) is the characteristic chamber length [in], and \(D_t\) is the throat diameter [in].

Step 2 — Two-phase loss (tabulated power-law fit):

\[ \eta_{2p} = 0.01 \cdot C_3\,\frac{\xi^{C_4}\,d_p^{C_5}}{P_0^{0.15}\,\varepsilon^{0.08}\,D_t^{C_6}} \]

Coefficients \(C_3\)–\(C_6\) depend on \(\xi\), \(D_t\), and \(d_p\) ranges (tabulated in AFRPL-TR-75-36; the 0.01 factor converts the percent-form expression to the fraction convention used here). Implemented in get_two_phase_flow_loss_fraction.

Applies to: solid motors only. Liquid bipropellants typically produce gas-phase products with no condensed phase, so the LRE state passes \(\eta_{2p} = 0\).

4.5 Overall Nozzle Efficiency¶

The four loss fractions are additive, giving the overall efficiency applied to the ideal thrust coefficient:

\[ \boxed{\eta_{nozzle} = 1 - \left(\eta_{div} + \eta_{kin} + \eta_{BL} + \eta_{2p} + \eta_{other}\right)} \]

\[ C_{f,real} = \eta_{nozzle}\,C_{f,ideal} \]

Implemented in get_overall_nozzle_efficiency and applied via apply_thrust_coefficient_correction.

References¶

Coats, D. E., Levine, J. N., Nickerson, G. R., Tyson, T. J., Cohen, N. S., Harry, D. P. III, & Price, C. F. (1975). A Computer Program for the Prediction of Solid Propellant Rocket Motor Performance, Volume I. Technical Report AFRPL-TR-75-36 (DTIC Accession AD-A015 140). Air Force Rocket Propulsion Laboratory.
Sutton, G. P., & Biblarz, O. (2017). Rocket Propulsion Elements (9th ed.). Wiley. Ch. 3.