Abstract
Hybrid rocket motors occupy a design space between solid and liquid propulsion systems, and the sizing of their combustion chamber is complicated by the fact that the fuel-grain port diameter, and therefore the free flow area available to the oxidizer, changes continuously over the burn. This paper develops a self-contained set of theorems that together define a parametric sizing methodology for a hybrid combustion chamber, carrying a single design point through the nozzle, the propellant mass-flow split, the solid fuel-grain geometry, and a regression-rate consistency check, without collapsing the derivation into a single numerical example. Each theorem states a closed-form geometric or mass-balance relation (throat and exit sizing from the thrust coefficient and characteristic velocity of a thermochemical equilibrium analysis, divergent-length sizing from the nozzle half-angle, fuel-grain volume and cross-sectional sizing from the target burn time and grain envelope, and, most importantly, two distinct formulations of the regression rate), and is accompanied by a short proof from first principles (mass conservation, the geometry of concentric circles, or the definition of the quantity concerned). A dedicated theorem and corollary distinguish an instantaneous regression rate, obtained from the initial burning surface area, from a space- and time-averaged regression rate, obtained purely from the port-growth geometry, and establish that only the latter certifies whether an assumed initial port diameter, outer diameter and burn time are mutually consistent. The methodology is presented for a hybrid motor of this general configuration (a circular-port paraffin-wax grain oxidized by a liquid/gaseous flow through a converging–diverging nozzle), with the required input parameters (thrust, chamber pressure, mixture ratio, expansion ratio, burn time, fuel density, and grain envelope) identified together with the basis on which each is normally chosen, and with the derived relations left in closed form so that the same theorem set applies once a designer substitutes the values appropriate to their own mission, propellant, and hardware envelope.
1. Introduction
Hybrid rocket motors store their oxidizer and fuel in different physical states, typically a liquid or gaseous oxidizer flowing over a solid fuel grain, and this arrangement gives them a combination of safety, throttleability and mechanical simplicity that neither solid nor liquid systems offer on their own. The price of this arrangement is a combustion process whose geometry is not fixed: as the grain burns, the port through which the oxidizer flows widens, the local mass flux falls, and the fuel regression rate changes with it. A chamber that is sized only from a thrust and a pressure, without reference to how the grain geometry evolves over the burn, risks being either oversized (wasted mass and volume) or unable to sustain the assumed burn time.
This paper works through a sizing sequence for a paraffin-wax / nitrous-oxide (N₂O) hybrid motor that treats the nozzle, the propellant mass-flow split and the fuel-grain geometry as one coupled problem rather than three separate calculations. Paraffin wax is used here because of its well-documented tendency to form a thin, low-viscosity melt layer at the burning surface; the resulting droplet entrainment gives paraffin-based fuels regression rates several times higher than the polymeric fuels (e.g. HTPB) traditionally used in hybrid grains, at a given oxidizer mass flux (refs. 1, 2). This property is central to the present exercise, since it directly affects how large a grain is needed to sustain a given fuel flow rate over a given burn duration.
The specific contribution of this paper is not a new regression-rate correlation, nor a new nozzle theory; both of those are well established. Rather, it is a demonstration of how a hybrid-motor sizing sequence can be organized as a small set of theorems, each with a short proof from first principles, so that the internal consistency of the design is carried explicitly rather than buried inside a single arithmetic pass. In particular, Section 7 states an instantaneous regression-rate theorem, obtained from the initial burning-surface area, alongside a space- and time-averaged regression-rate theorem, obtained purely from the assumed port and outer diameters and the target burn time, and proves as a corollary that only the latter is appropriate for checking that a chosen grain geometry can be fully consumed in the intended burn window. The gap between the two is shown to be structural rather than incidental, and is a point that is easy to get wrong in a first sizing pass.
2. Nomenclature
| Symbol | Description | Units |
|---|---|---|
| F | design thrust | N |
| Pc | chamber (stagnation) pressure | bar, Pa |
| CF | thrust coefficient | – |
| C* | characteristic velocity | m/s |
| Dt, At | throat diameter, throat area | mm, mm² |
| De, Ae | exit diameter, exit area | mm, mm² |
| ε | nozzle expansion (area) ratio, Ae/At | – |
| α | nozzle divergent half-angle | deg |
| Ldiv | divergent-section length (conical nozzle) | mm |
| O/F | oxidizer-to-fuel mass ratio | – |
| ṁox, ṁf | oxidizer, fuel mass-flow rate | kg/s |
| tb | target burn time | s |
| ρf | fuel (paraffin wax) density | kg/m³ |
| Dp,i | initial fuel-grain port diameter | mm |
| Do | fuel-grain outer diameter | mm |
| tweb | fuel web thickness, (Do − Dp,i)/2 | mm |
| Ap, Ao, Af | port, outer, and annular fuel cross-sectional areas | mm² |
| Vf, mf | fuel-grain volume, total consumable fuel mass | mm³, kg |
| L | fuel-grain length | mm |
| Ab | initial burning (port) surface area | mm², m² |
| Ṃ0 | instantaneous regression rate at t = 0 | mm/s |
| r̄ | space- and time-averaged regression rate | mm/s |
| Gox | oxidizer mass flux through the port | kg/(m²·s) |
| a, n | regression-rate correlation coefficient, exponent | – |
| Tc | chamber (stagnation) temperature | K |
| Isp | specific / effective exhaust velocity | m/s, s |
Table 1. Nomenclature used throughout this paper.
3. Design Input Parameters
Every relation derived in Sections 4–7 is stated in closed symbolic form, so that it applies to any hybrid combustion chamber of the configuration considered here (a circular-port paraffin-wax grain oxidized by a liquid/gaseous flow through a converging–diverging nozzle), once the designer has specified the quantities that the theorems take as given. Table 2 lists these quantities together with the basis on which each is normally selected, rather than committing to a particular numeric example: the theorems are meant to be applied with whatever values a specific mission, propellant choice, or hardware envelope requires, and no single choice of these inputs is privileged over another. Once a designer has fixed the entries of Table 2 for their own case, the geometric and mass-flow quantities that follow (throat and exit dimensions, grain dimensions, regression rates, and so on) are obtained directly by substitution into the theorems of Sections 4–7, without re-deriving anything.
| Symbol | Description | Basis for Selection | Unit |
|---|---|---|---|
| F | design thrust | set by the mission thrust requirement | N |
| Pc | chamber (stagnation) pressure | chosen subject to structural, feed-system, and combustion-efficiency constraints | bar |
| O/F | oxidizer-to-fuel mass ratio | chosen from the performance characteristics of the selected propellant pair | – |
| ε | nozzle expansion (area) ratio | selected from a CEA sweep at the chosen (Pc, O/F), per Remark 3.1 | – |
| α | nozzle divergent half-angle | chosen as a manufacturing and performance trade-off | deg |
| tb | target burn time | set by the mission burn-time requirement | s |
| ρf | fuel density | a property of the selected fuel material | kg/m³ |
| Dp,i | initial fuel-grain port diameter | chosen from the motor-case envelope and ignition flow-area requirements | mm |
| Do | fuel-grain outer diameter | chosen from the motor-case envelope | mm |
| CF | thrust coefficient | obtained from a CEA equilibrium analysis at the chosen (Pc, O/F) | – |
| C* | characteristic velocity | obtained from the same CEA equilibrium analysis | m/s |
Table 2. Design input parameters required by the theorems of Sections 4–7, and the basis on which each is normally selected. No numeric values are fixed here; a designer substitutes the values appropriate to their own mission, propellant, and hardware envelope.
3.1 Remark on Expansion-Ratio Selection
The expansion ratio ε in Table 2 is not itself a free theorem parameter; it is selected, ahead of Theorem 2 in Section 4, by sweeping a thermochemical equilibrium analysis (e.g. NASA CEA, refs. 4, 7) over the pressure and mixture-ratio range of interest and identifying the exit condition that maximizes specific impulse, subject to any constraint on exit static pressure relative to the operating ambient. This selection step is a prerequisite to the theorems that follow, but is not itself proved as a theorem here, since it depends on the propellant equilibrium chemistry rather than on the chamber geometry.
4. Nozzle-Sizing Theorems
Let F be the design thrust, Pc the chamber pressure, and CF the thrust coefficient obtained from a thermochemical equilibrium analysis at the chosen chamber pressure and mixture ratio. Then the required throat area and throat diameter are
By definition, the thrust coefficient is CF ≡ F/(PcAt); rearranging for At gives Eq. (1) directly. Since the throat cross-section is circular, At = (π/4)Dt2, and solving for Dt gives Eq. (2).
For a nozzle expansion (area) ratio ε ≡ Ae/At selected as in Remark 3.1, the exit area and exit diameter are
Equation (4) is immediate from the definition of ε given above Eq. (3). Since the exit cross-section is circular, the same substitution used in the proof of Theorem 1 gives Ae = (π/4)De2; combining this with Eq. (4) and with At = (π/4)Dt2 (Theorem 1) and cancelling π/4 gives De2 = ε Dt2, i.e. Eq. (5).
For a conical divergent nozzle section of half-angle α connecting a throat of diameter Dt to an exit of diameter De, the divergent-section length is
Over an axial length Ldiv, the nozzle wall radius increases from Dt/2 to De/2 while subtending a half-angle α with the nozzle axis, so that tan α = [(De − Dt)/2] / Ldiv by the definition of the tangent of the half-angle; solving for Ldiv gives Eq. (6).
Figure 1. Nozzle longitudinal profile and fuel-grain cross-section, with the symbols of Theorems 1–3 and 6 indicated (schematic, not to scale).
5. Propellant Mass-Flow Theorems
For a nozzle sized by Theorem 1 at chamber pressure Pc and throat area At, the propellant mass-flow rate that a combustion chamber operating at that pressure will actually deliver is
equivalently, in terms of the effective exhaust velocity Isp = CFC*,
By definition, C* ≡ PcAt/ṁ; solving for ṁ gives Eq. (7). Substituting At = F/(CFPc) from Theorem 1 into Eq. (7) gives ṁ = F/(CFC*), which is Eq. (8) once Isp is identified with CFC*.
Any mass-flow target adopted independently of Eqs. (7)–(8) (for instance, one chosen ahead of a detailed grain design, as in Theorem 4 below) must ultimately be reconciled with this expression before the finished chamber can be said to deliver the design thrust F at the design Pc; a mismatch between the two is not an error in the geometric theorems of Sections 6–7, which hold for whatever mass flow is substituted into them, but a signal that a further design iteration on that mass flow is required (Remark 8.1).
For a total propellant mass-flow rate ṁ and a design mixture ratio O/F ≡ ṁox/ṁf, the individual oxidizer and fuel mass-flow rates are
Writing ṁ = ṁox + ṁf and substituting ṁox = (O/F)ṁf from the definition of O/F gives ṁ = (1+O/F)ṁf, which rearranges to the first part of Eq. (9); the second follows from ṁox = ṁ − ṁf.
6. Fuel-Grain Geometric Theorems
For a target burn time tb and fuel mass-flow rate ṁf from Theorem 4, the total consumable fuel mass and the corresponding solid volume at fuel density ρf are
Equation (10) follows from the definition of ṁf as an average mass-flow rate over the burn interval [0, tb]. Equation (11) follows from the definition of density, ρf ≡ mf/Vf, solved for Vf.
For a single circular-port grain of initial port diameter Dp,i and outer diameter Do, the port, outer, and annular (fuel) cross-sectional areas, and the fuel web thickness, are
Equations (12)–(13) follow from the area of a circle applied to the port and to the outer boundary of the grain. Since the two circles are concentric, the paraffin occupies exactly their set difference, giving Eq. (14). The web thickness in Eq. (15) is the common radial distance between the two circles, (Do − Dp,i)/2, i.e. half the difference of the diameters.
Given the fuel volume Vf from Theorem 5 and the annular cross-section Af from Theorem 6, and assuming a prismatic grain (constant cross-section along its axis), the required grain length is
For a prismatic solid, volume equals cross-sectional area times length by definition; substituting Vf for the volume and Af for the cross-section and solving for the length L gives Eq. (16).
7. Regression-Rate Theorems
For a fuel grain with instantaneous burning (port) surface area Ab(t), the regression rate Ṃ(t) is the rate of radial recession of the burning surface, related to the instantaneous fuel mass-generation rate through
This relation holds at every instant of the burn; Theorems 8 and 9 below apply it in two different, and generally inconsistent, senses (evaluated at the initial instant, and averaged over the whole burn), which is the source of the distinction this paper is built around.
At the initial instant of the burn, the burning (port) surface area of the grain sized by Theorem 7 is
and the instantaneous regression rate consistent with the design fuel-flow rate ṁf through Definition 1 is
The lateral surface area of a cylinder of diameter Dp,i and length L is πDp,iL, giving Eq. (18) as Ab(0). Evaluating Definition 1 at t = 0 with ṁf(0) = ṁf (the design value) and solving for Ṃ(0) ≡ Ṃ0 gives Eq. (19).
This quantity characterizes only the start of the burn; because Ab(t) is monotonically increasing for a single expanding circular port (Corollary 1 below), Ṃ0 is not, by itself, sufficient to establish whether the grain sized by Theorem 7 burns out within the target time tb; that question is answered by Theorem 9.
For a port that grows in radius from Dp,i/2 to Do/2 over the burn time tb (regardless of the detailed shape of Ṃ(t)), the mean regression rate is
By Theorem 6, the port radius must traverse the full web thickness tweb = (Do − Dp,i)/2 exactly once over the interval [0, tb]. Writing the total radial travel as the time integral of Ṃ(t),
the mean-value form on the left is the definition of the time average of Ṃ(t) over [0, tb], and the right-hand equality follows because the integral of Ṃ(t) over the burn must equal the total radial distance travelled, tweb, independent of the (unknown) detailed shape of Ṃ(t). Substituting tweb = (Do − Dp,i)/2 gives Eq. (20).
Ṃ0 (Theorem 8) and r̄ (Theorem 9) coincide only in the degenerate case in which the burning area is constant over the burn, Ab(t) = Ab(0) for all t ∈ [0, tb]. For the single circular port considered here, Ab(t) = πD(t)L with D(t) monotonically increasing from Dp,i to Do, so Ab strictly increases over the burn and Ṃ0 necessarily overstates r̄. Only r̄ should therefore be used to certify that a chosen (Dp,i, Do, tb) triple is geometrically achievable; Ṃ0 remains useful only as a reference value at ignition.
If Ab(t) is not constant, Definition 1 shows that a constant ṁf (as assumed at ignition in Theorem 8) forces Ṃ(t) to vary inversely with Ab(t); since Ab is smallest at t = 0, Ṃ(0) is the largest value Ṃ(t) would need to take under that assumption, and therefore Ṃ0 ≥ r̄ by the mean-value relation of Theorem 9, with equality only in the constant-area case excluded above.
The oxidizer mass flux through the initial port, using ṁox from Theorem 4 and Ap from Theorem 6, is
and is related to the regression rate through the classical empirical power law
Equation (22) is the definition of mass flux, flow rate per unit cross-sectional area. Equation (23) is an empirical closure, not a derived identity: the coefficient a and exponent n are specific to the fuel formulation, the oxidizer, the port geometry, and the scale of the motor, and must be obtained from subscale burner characterization; they are not produced by the geometric theorems of this paper and are deliberately left undetermined here (Remark 8.3).
8. Discussion
The theorems of Sections 4–7 are general, but three qualitative consequences of applying them to any single design point are worth stating explicitly, as remarks rather than as further theorems, since each concerns how the theorems interact with a choice made outside the geometric framework itself.
8.1 Mass-Flow Self-Consistency
Remark (Iteration on ṁ). Proposition 1 fixes the propellant mass-flow rate once the throat area (Theorem 1) and the thermochemical pair (CF, C*) are chosen; nothing prevents a designer from provisionally adopting a different, round mass-flow value for the earlier stages of grain sizing (Theorems 4–7), particularly before a nozzle has been finalized. Whenever this is done, Proposition 1 supplies the exact criterion for closing the design: the fuel mass, fuel volume and grain length obtained from Theorems 5–7 scale linearly with the assumed ṁf, so any later correction to bring the assumed mass flow into agreement with Eq. (7)/(8) can be applied as a single multiplicative rescaling of those quantities, rather than as a re-derivation.
8.2 Expansion Ratio and Ambient Matching
Remark (Optimum vs. Working ε). Theorem 2 accepts ε as a given input (Remark 3.1); it does not certify that the value used is optimal. If a thermochemical sweep shows specific impulse still increasing at the upper bound of the tested expansion-ratio range, the adopted ε is a practical working value rather than a demonstrated optimum, and a wider sweep is needed before treating it as final. Separately, if the exit static pressure implied by the chosen ε at the design Pc is materially below the ambient pressure under which the motor will be tested, the nozzle will be overexpanded in that test, and the ideal CF used in Theorem 1 will not be realized there; a distinction between flight and ground-test expansion ratios is then warranted, and is a nozzle-selection question that sits outside the scope of Theorems 1–3.
8.3 Limitations
Remark (Scope of the Geometric Theorems). Three simplifications underlie Theorems 5–10 and are worth stating explicitly. First, Theorem 9 treats the port as a single cylindrical bore with a uniform radius at every axial station; real paraffin grains can regress non-uniformly along their length and depart from a circular port cross-section as the burn progresses, particularly once melt-layer entrainment becomes significant, so Theorem 9 should be read as a mean-geometry consistency check rather than a detailed burn-back model. Second, Corollary 1 establishes only that Ṃ0 and r̄ differ, not the detailed shape of Ṃ(t) in between; recovering that shape requires integrating Definition 1 with an assumed or measured area and flux history, which is outside the present framework. Third, Theorem 10 states the form of the regression-rate correlation but leaves its coefficient a and exponent n undetermined, since these are properties of the specific fuel batch, port geometry and test scale and must come from a subscale burner characterization rather than from geometric reasoning; the theorem set in this paper sizes a grain that is geometrically consistent with a target burn time, but does not, by itself, predict what burn time a given grain would actually achieve without that empirical closure.
9. Conclusion
This paper has organized the sizing of a hybrid rocket combustion chamber (nozzle, propellant mass-flow split, and solid fuel-grain geometry) as a set of ten theorems and one definition, each with a short proof from mass conservation, elementary circle geometry, or the definition of the quantity concerned. The theorems apply to a paraffin-wax / N₂O motor of the general configuration considered throughout, with the required input parameters identified in Table 2 together with the basis on which each is normally chosen, and are stated and proved in closed symbolic form, so that the same derivations apply unchanged to any hybrid motor of this configuration once its own design data are substituted. The central methodological result is the pair formed by Theorem 8 and Theorem 9, together with Corollary 1: an instantaneous regression rate obtained from the initial burning area is shown, in general, to overstate the space- and time-averaged regression rate that actually certifies whether a chosen initial port diameter, outer diameter and burn time are geometrically achievable, and only the latter should be used for that certification. Proposition 1 supplies a second, independent consistency requirement, between whatever mass flow is adopted for grain sizing and the mass flow implied by the already-fixed nozzle throat area and thermochemical state, and Section 8 discusses how a design that provisionally violates either requirement can be brought back into agreement without re-deriving the underlying geometry. Together, the theorem set is offered as a reusable methodology rather than as a single worked answer: what is fixed is the set of relations and the two consistency checks they support, not any particular set of numbers.
References
- Karabeyoglu, M. A., Altman, D., and Cantwell, B. J., “Combustion of Liquefying Hybrid Propellants: Part 1, General Theory,” Journal of Propulsion and Power, Vol. 18, No. 3, 2002, pp. 610–620.
- Karabeyoglu, M. A., Zilliac, G., Cantwell, B. J., DeZilwa, S., and Castellucci, P., “Scale-Up Tests of High Regression Rate Paraffin-Based Hybrid Rocket Fuels,” Journal of Propulsion and Power, Vol. 20, No. 6, 2004, pp. 1037–1045.
- Sutton, G. P., and Biblarz, O., Rocket Propulsion Elements, 9th ed., John Wiley & Sons, Hoboken, NJ, 2016.
- Gordon, S., and McBride, B. J., “Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications,” NASA Reference Publication 1311, NASA Lewis Research Center, Cleveland, OH, 1994.
- Humble, R. W., Henry, G. N., and Larson, W. J. (eds.), Space Propulsion Analysis and Design, McGraw-Hill, New York, 1995.
- Zakirov, V., and Sweeting, M., “Nitrous Oxide as a Rocket Propellant,” Acta Astronautica, Vol. 48, No. 5–12, 2001, pp. 353–362.
- NASA Glenn Research Center, Chemical Equilibrium with Applications (CEA), CEARUN online interface, https://cearun.grc.nasa.gov/.