Choked (Sonic) Mass Flow Rate Calculator
Solve the choked-flow relation ṁ = (pc · At / √Tc) · √(γ/R) · (2/(γ+1))(γ+1)/(2(γ−1)) for any of its six unknowns, with step-by-step math, a live nozzle schematic, and companion propulsion metrics including c*, thrust, and Isp.
Linear in pc and At
ṁ ∝ pc · At. Doubling the chamber pressure or the throat area doubles the mass flow. This is how engines are throttled and how thrust is scaled: for a fixed At, more chamber pressure means more mass flow and more thrust.
Inverse √Tc
ṁ ∝ 1/√Tc. A hotter chamber has a lower mass flow for the same pc and At, because the gas expands more and has a lower density at the throat. This couples to the engine balance: hot propellants need a larger throat to pass the same flow.
Propellant enters via γ and R
The factor √(γ/R)·(2/(γ+1))^((γ+1)/(2(γ−1))) is a pure function of γ (since √(γ/R) folds in R). Together with Tc, it defines the characteristic velocity c* = √(R·Tc)/Γ, and ṁ = pc·At/c*. Every propellant has a characteristic c* that engineers can look up.
Choking is what makes this valid
The formula only applies once the nozzle is choked — i.e., when pc/pa exceeds the critical ratio (≈ 1.89 for γ = 1.20). Below that, ṁ drops and depends on pa. Above it, the throat is at M = 1 and the mass flow is fixed by the chamber conditions alone — that is exactly what the formula expresses.
How to Use This Calculator
The choked-flow relation links six quantities: the mass flow rate, chamber pressure, throat area, chamber temperature, specific heat ratio, and gas constant. Pick which one you want the tool to solve, then fill in the others.
1. Choose what to solve for
Pick ṁ, pc, At, Tc, γ, or R — the tool rearranges the relation automatically. γ requires numerical solution because it appears in two coupled factors.
2. Use realistic presets
Mass flow presets span 0.5–2,500 kg/s. Chamber pressure presets span 0.5–20 MPa. Throat area presets span 0.005–0.8 m². Tc presets span 2,000–3,800 K. γ presets span 1.15–1.40. R presets span 280–430 J/(kg·K).
3. Read the result and steps
The result card shows the solved quantity with engineering notation. The step-by-step panel walks through the arithmetic plus c*, the sonic throat density, and the sonic velocity at the throat.
4. Check the nozzle schematic
The cross-section diagram shows the chamber, the convergent–divergent nozzle, the sonic throat with At, and the mass flow label — with all current values overlaid.
5. Switch to Advanced for depth
Advanced mode adds a chart of ṁ vs. throat area, a 1% sensitivity check, and companion metrics including c*, sonic density, sonic velocity, thrust per unit area, and total propellant mass for a reference burn time.
6. Cross-check with the reference table
The reference section lists typical c* values for common propellant combinations, plus a lookup grid of ṁ across the pc–At envelope.
What This Tool Offers
Built around the classical choked-flow relation used in rocket nozzle analysis, engine sizing, and propellant feed design.
Six-way solving
Solve for mass flow, chamber pressure, throat area, chamber temperature, γ, or R — including numerical solution for γ.
Live nozzle schematic
A convergent–divergent cross-section with chamber, sonic throat, exit plane, and mass flow label, all updating live with the current values.
Dual comparison charts
ṁ vs. chamber pressure at three throat areas, plus ṁ vs. throat area at three chamber pressures.
Companion propulsion metrics
Advanced mode reports characteristic velocity c*, sonic throat density, sonic velocity, throat-to-chamber density ratio, thrust estimate for a reference CT, and total propellant mass for a reference burn.
Built-in sensitivity check
See how much ṁ moves for a 1% change in each input. The √-law exponents in Tc and R, and the linear exponents in pc and At, are shown explicitly.
Reference tables
Typical c* values for LOX/RP-1, LOX/LH2, solids, and cold gas, plus a lookup grid of ṁ across the pc–At envelope.
Reference Tables
Mass flow across typical engine envelopes, plus typical c* values for common propellant combinations.
Mass flow quick lookup (kg/s, γ = 1.25, R = 350 J/kg·K, Tc = 3200 K)
ṁ = (pc·At/√Tc)·√(γ/R)·(2/(γ+1))(γ+1)/(2(γ−1)) for various chamber pressures and throat areas.
| pc (MPa) | At = 0.005 m² | At = 0.02 m² | At = 0.05 m² | At = 0.2 m² | At = 0.8 m² |
|---|
Typical Propellant Combinations
Representative values. Real engines lose a few percent to incomplete combustion and heat loss.
| Propellant | γ | R (J/kg·K) | Tc (K) | c* (m/s) | Γ factor |
|---|---|---|---|---|---|
| Solid (APCP) | 1.18 | 280 | 2800 | 1590 | 0.657 |
| LOX / RP-1 (kerosene) | 1.22 | 320 | 3500 | 1760 | 0.651 |
| N2O4 / MMH | 1.24 | 340 | 3300 | 1710 | 0.648 |
| LOX / LH2 | 1.20 | 430 | 3400 | 2280 | 0.653 |
| Cold gas N2 | 1.40 | 297 | 300 | 490 | 0.685 |
| Cold gas He | 1.67 | 2077 | 300 | 1710 | 0.727 |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| ṁ | Choked mass flow rate | 0.5 – 2,500 | kg/s |
| pc | Chamber stagnation pressure | 0.5 – 30 | MPa |
| At | Throat area | 0.001 – 2 | m² |
| Tc | Chamber stagnation temperature | 2,500 – 3,800 | K |
| γ | Ratio of specific heats | 1.15 – 1.40 (up to 1.67 for He) | — |
| R | Specific gas constant | 250 – 430 (2,077 for He) | J/(kg·K) |
What Does the Choked Mass Flow Rate Measure?
The choked (sonic) mass flow rate is the mass of propellant flowing through the nozzle throat per unit time when the nozzle is choked — that is, when the throat has reached the speed of sound and can no longer pass more flow regardless of how low the exit pressure drops. Once choked, the flow rate depends only on the chamber conditions (pc, Tc), the throat area, and the gas properties (γ, R). This is one of the most useful relations in rocket engineering because it lets designers size the throat for a desired mass flow (and hence thrust) using only the chamber state.
Choked Mass Flow Rate Formula
The defining relationship is:
An equivalent and often more useful form uses the characteristic velocity c*:
Γ = √γ · ( 2/(γ+1) )(γ+1)/(2(γ−1))
Rearranged to solve for chamber pressure, throat area, or chamber temperature:
Choked Mass Flow Variables
| Symbol | Meaning | Notes |
|---|---|---|
| ṁ | Choked mass flow rate | Fixed by chamber conditions and throat geometry once M=1 at the throat |
| pc | Chamber stagnation pressure | Set by the feed system and combustion; the primary throttle |
| At | Throat area | At = π rt²; sized to pass the required ṁ at the design pc |
| Tc | Chamber stagnation temperature | Adiabatic flame temperature; fixed by propellant chemistry |
| γ | Ratio of specific heats | Fixed by propellant chemistry; appears in two coupled factors |
| R | Specific gas constant | R = Ru/M; lower molecular weight gives higher R |
| c* | Characteristic velocity | c* = pc·At/ṁ; a propellant figure of merit |
How Each Parameter Moves the Mass Flow
The relation is linear in pc and At, inverse-half in Tc, and depends on γ and R through a coupled factor. A 1% increase in pc or At raises ṁ by 1%. A 1% increase in Tc lowers ṁ by 0.5%. The dependence on γ is more complex because γ appears both in √γ and in the (2/(γ+1))(γ+1)/(2(γ−1)) factor. Over the range of real propellants (γ = 1.15 to 1.40), increasing γ at fixed R and Tc changes the mass flow by only a few percent. In contrast, R appears as √(1/R), so a high-R propellant (like hydrogen) produces a lower mass flow for the same pc and At — which is why LH2 engines use larger throat areas for a given thrust.
Choked Mass Flow Calculation Examples
pc · At = 10,000,000 × 0.05 = 500,000 N
√Tc = √3200 = 56.569
(pc·At)/√Tc = 500,000 / 56.569 = 8,839.5
√(γ/R) = √(1.25/350) = √0.0035714 = 0.059762
(2/(γ+1))^((γ+1)/(2(γ−1))) = (2/2.25)^(2.25/0.5) = 0.88889^4.5 = 0.6537
ṁ = 8,839.5 × 0.059762 × 0.6537 = 345.5 kg/s
c* = pc·At/ṁ = 500,000 / 345.5 = 1,447 m/s
c* = √(R·Tc)/Γ ≈ √(320 × 3400)/0.651 = √1,088,000 / 0.651 = 1043.1 / 0.651 = 1602 m/s
At = ṁ · c* / pc = 100 × 1602 / 7,000,000 = 0.0229 m²
Equivalent throat radius rt = √(At/π) = √(0.0229/3.1416) = 0.0854 m ≈ 8.5 cm
Frequently Asked Questions
Choked flow means the nozzle throat has reached Mach 1 — the local flow velocity equals the local speed of sound. Once this happens, the mass flow through the throat is fixed by the chamber conditions and cannot increase, even if the exit pressure drops further. Information cannot travel upstream past the throat (the flow is supersonic downstream), so the chamber cannot "know" what the exit pressure is. This is why the mass flow rate depends only on pc, Tc, At, γ, and R — not on pe or pa.
Because the sonic velocity at the throat is proportional to √Tc (the speed of sound is a = √(γRT)). A hotter chamber has a higher sound speed and a lower gas density (for a fixed pressure), so the same throat area passes less mass flow. Physically, the gas at the throat is moving faster but is thinner, and the density effect wins because it enters linearly while the velocity effect enters only via the sound speed scaling. The result is that ṁ ∝ 1/√Tc.
c* (pronounced "c-star") is the characteristic velocity of the propellant combination. It is defined as c* = pc·At/ṁ, which means it is a measure of how much mass flow a given chamber pressure and throat area can push through. Equivalently, ṁ = pc·At/c*. c* depends only on the propellant chemistry (through γ, R, Tc) and not on the nozzle expansion. It is tabulated for every propellant combination and is the fundamental performance figure of merit for the combustion chamber and throat. Typical values: 1,550–1,650 m/s for solids, 1,700–1,800 m/s for LOX/RP-1, 2,200–2,400 m/s for LOX/LH2.
The critical pressure ratio is the chamber-to-ambient ratio at which the throat first reaches sonic conditions. It is pc/pa = ((γ+1)/2)^(γ/(γ−1)). For γ = 1.20 this is about 1.83; for γ = 1.40 (air), about 1.89. Once pc/pa exceeds this value, the nozzle is choked and the mass flow is given by the formula on this page. Below it, the throat is subsonic and mass flow depends on the exit-to-ambient pressure ratio.
Because that is the only way to have supersonic flow downstream. In a converging duct, flow accelerates only until M = 1 at the minimum section. To accelerate further to supersonic speeds, the duct must diverge — which is exactly what the divergent section of a rocket nozzle does. If the throat were not sonic, the flow would decelerate in the divergent section and the engine would produce no useful thrust. Rocket engines are therefore always designed to operate with a choked throat.
The choked mass flow rate sets the required propellant flow to the chamber. Combined with the mixture ratio (O/F for liquid bipropellants), it tells you the required oxidizer flow ṁox = ṁ · MR/(1+MR) and fuel flow ṁf = ṁ/(1+MR). These in turn set the pump sizes, the valve and line diameters, and the tank pressurisation requirements. For the Space Shuttle Main Engine at 100% throttle, ṁ ≈ 500 kg/s with an O/F of about 6, so ṁox ≈ 430 kg/s and ṁf ≈ 70 kg/s — values that dictate the entire engine plumbing layout.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| mdot | Mass flow rate | kg/s |
| p_c | Chamber pressure | Pa |
| A_t | Throat area | m2 |
| T_c | Chamber temperature | K |
| γ | Ratio of specific heats | |
| R | Specific gas constant | J/(kg*K) |
What it means
When the flow reaches sonic speed at the throat, the mass flow rate is choked and cannot increase further for a given upstream condition. This formula gives the mass flow rate through a nozzle under choked conditions. It is derived from the continuity equation and isentropic relations. This is crucial for rocket and jet engine performance: the mass flow determines thrust. Understanding choked flow is essential for propulsion system design and for analysing nozzle operation.
Worked example
Choked Mass Flow – Two Examples
Real‑World| Parameter | Value |
|---|---|
| p_c | 7×10⁶ Pa |
| A_t | 0.05 m² |
| T_c | 3500 K |
| γ | 1.2 |
| R | 350 |
| Parameter | Value |
|---|---|
| p_c | 1×10⁷ |
| A_t | 0.06 |
| T_c | 3300 |
| γ | 1.22 |
| R | 400 |
Common mistakes
- Choked (sonic) mass flow rate: ṁ = (p_c·A_t / √T_c) · √(γ/R) · (2/(γ+1))^((γ+1)/(2(γ−1))).
- p_c: Chamber pressure (Pa).
- A_t: Throat area (m²).
- T_c: Chamber temperature (K).
- Valid when flow is sonic at throat (M=1).
- Mass flow is maximum for given p_c and T_c.
Applications
Choked (sonic) mass flow rate, ṁ = (p_c·A_t/√(T_c))·√(γ/R)·(2/(γ+1))^((γ+1)/(2(γ−1))), gives the maximum mass flow through a nozzle throat when the flow is sonic. It is used to size rocket engine throats, to compute chamber pressure, and to design turbopumps. Engineers use this to ensure that the throat area is adequate for the required mass flow, and to set operating conditions. By applying the choked flow equation, aerospace engineers can design stable, predictable rocket engines.
- Rocket engine throat sizing and design
- Chamber pressure and propellant flow rate determination
- Turbopump and feed system design
- Flow rate measurement and control in engine testing
- Nozzle design and performance analysis
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