Engineering task and calculation objective
The L2.4 module calculates critical mass flow rates through nozzles, valves and pipe fittings according to the VDI Heat Atlas (VDI-Wärmeatlas), 12th edition 2019 — the standard German reference for heat transfer and fluid flow engineering. Critical (choked) flow occurs when the fluid reaches the speed of sound in the narrowest cross-section: the mass flow rate then no longer increases when the back pressure is lowered further. This limit determines the maximum possible throughput of a restriction and is therefore a central quantity in safety engineering.
In practice, the calculation of the critical mass flow rate is needed above all for the sizing of safety valves, rupture disks and blowdown lines, for leak-rate and incident analyses (vessel depressurization through a hole or a severed line), and for throttling devices in steam and refrigeration systems. Two-phase flows are particularly demanding: if a subcooled or saturated liquid expands across the restriction, it partially vaporizes (flashing), and the critical mass flow rate lies far below that of a pure liquid flow.
The module provides calculation approaches by various authors that differ in their assumptions — single-phase or two-phase flow, thermodynamic equilibrium or delayed evaporation, homogeneous or slip-affected mixture, subcooled, saturated or superheated inlet state. Complete fluid property data are a prerequisite for the calculation.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define the inlet state and geometry: The stagnation pressure and temperature or vapor quality at the inlet as well as the narrowest flow cross-section of the nozzle, valve or pipe fitting are specified. The correct classification of the inlet state is decisive: subcooled liquid, saturated two-phase mixture or superheated vapor.
- Provide the fluid property data: The expansion calculation requires the densities of the gas and liquid phases, the enthalpy of vaporization, the vapor-pressure curve and the heat capacities. Without consistent property data along the expansion path, no reliable calculation of the critical mass flow rate is possible.
- Select the calculation model: Depending on the flow regime, an author model is chosen: isentropic nozzle flow for single-phase gases and vapors, homogeneous equilibrium models (HEM) for two-phase mixtures in equilibrium, or approaches with delayed evaporation or slip for short nozzles and subcooled inlet. The models differ above all in how quickly the fluid can follow the pressure drop thermodynamically.
- Search for the critical state in the narrowest cross-section: Along the expansion from the stagnation state, the mass flux is evaluated as a function of the pressure in the narrowest cross-section; its maximum defines the critical pressure and the critical mass flux. For ideal gases this leads to the well-known critical pressure ratio; for two-phase flow the maximum search is performed numerically.
- Determine the critical mass flow rate and discharge reduction: The maximum mass flow rate follows from the critical mass flux and the narrowest cross-section; real contraction and friction are accounted for via a discharge coefficient. Comparing the author models reveals the spread of the prediction and supports a conservative design.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Mass flow in a nozzle | Qm,D | kg/s |
| Diameter of the nozzle of the narrowest flow cross section (nozzle neck) | dth | m |
| Diameter of the safety valve at the inlet | d0 | m |
| Cross sectional area of the narrowest flow cross section | Ath | m² |
| Specific volume in the narrowest flow cross section of the nozzle | νth | m³/kg |
| Acceleration of gravity | g | m/s² |
| Angle from the horizontal | θ | ° |
| Loss coefficient in relation to the reference condition | ζv,ref | - |
| Pressure at the inlet | p0 | Pa |
| Counter Pressure | pb | Pa |
| Temperature at the inlet to the nozzle | T0 | K |
| Spezifisches Volume of the mixture at the inlet of the nozzle | v0 | m³/kg |
| Dimensionless specific volume: Specific volume in the narrowest flow cross-section relative to the specific volume at the inlet | v*th | - |
| Pressure ratio at the inlet of the nozzle | ηth | - |
| Pressure ratio of counter pressure to pressure at inlet | ηb | - |
| Pressure ratio in inlet cross section | ηin | - |
| Dimensionless mass flow | C | - |
| Isentropic exponent | κ | - |
| Flow mass content | ẋ | - |
| Specific volume of the gas | vg | m³/kg |
| Specific volume of the liquid | vl | m³/kg |
| Compressibility factor | ω | - |
| Calculation according to: | Literatur | – |
| Compressibility factor for a vapor / liquid mixture in thermodynamic equilibrium N=1 | ωN=1 | - |
Calculation options
Calculation according to:
4.1.1 Leung (HEM) · 4.1.2 Henry und Fauske · 4.1.3 Darby · 4.1.4 Diener Schmidt (HNE-DS) · 4.1.5 API 520 (2-pointt-ω-method) · 4.1.6 Schmidt (HNE-DS) · 4.1.7 Schmidt Claramunt (HNE-CSE)
Evaporation or condensation
Evaporation · Condensation
Author for constants of the two-phase multiplier
Chrisholm · Lockhart Martinelli · 2
Worked example
As a single-phase limiting case, the critical mass flow rate of air through an ideal nozzle is calculated in this worked example. Air flows from a vessel at 10 bar (abs) and 20 °C through a nozzle with a narrowest cross-section of 100 mm². The back pressure is atmospheric, so the flow is safely supercritical. Find the maximum (critical) mass flow rate for isentropic flow (discharge coefficient 1).
Given values
| Stagnation pressure p0 | 10 bar (abs) |
| Stagnation temperature T0 | 20 °C = 293.15 K |
| Narrowest cross-section A | 100 mm² = 1.0·10⁻⁴ m² |
| Isentropic exponent κ (air) | 1.4 |
| Specific gas constant R (air) | 287.1 J/(kg·K) |
Solution
Check the critical pressure ratio
p*/p0 = (2/(κ+1))κ/(κ−1) = (2/2.4)3.5 = 0.528
Critical pressure in the narrowest cross-section: p* = 0.528 · 10 bar = 5.28 bar. The back pressure of 1 bar lies far below this — the nozzle is choked, the flow is critical.
Critical mass flux
For isentropic flow of an ideal gas, in the choked state:
ṁ/A = p0 · √(κ/(R·T0)) · (2/(κ+1))(κ+1)/(2(κ−1))
Numerical values: √(κ/(R·T0)) = √(1.4/(287.1 · 293.15)) = 4.078·10⁻³ s/m
(2/2.4)3.0 = 0.5787
ṁ/A = 10·10⁵ Pa · 4.078·10⁻³ s/m · 0.5787 = 2,360 kg/(m²·s)
Critical mass flow rate
ṁ = 2,360 kg/(m²·s) · 1.0·10⁻⁴ m² = 0.236 kg/s ≈ 850 kg/h
Lowering the back pressure further below 5.28 bar no longer increases the mass flow rate — the nozzle is the throughput-limiting restriction.
Result
| Critical pressure ratio p*/p0 | 0.528 |
| Critical mass flux | 2,360 kg/(m²·s) |
| Critical mass flow rate ṁ | 0.236 kg/s ≈ 850 kg/h |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
How do I recognize whether a flow is critical?
Critical flow occurs when the pressure ratio of back pressure to stagnation pressure falls below the critical pressure ratio. For ideal gases it is (2/(κ+1))^(κ/(κ−1)) — about 0.528 for air, which means the nozzle is choked at back pressures below roughly 53 % of the upstream pressure. For two-phase flow, the critical pressure ratio is model-dependent and must be determined from the maximum condition of the mass flux.
Why do the various author models deliver different critical mass flow rates?
The models differ in their assumptions about thermodynamic and mechanical equilibrium: the homogeneous equilibrium model assumes complete, instantaneous evaporation and equal phase velocities, and for short nozzles with subcooled inlet it often predicts mass flow rates that are too low, because real evaporation sets in with a delay. Non-equilibrium and slip models correct for this. The scatter between the models is a realistic measure of the prediction uncertainty.
Why is the critical mass flow rate of a two-phase mixture so much smaller than that of a liquid?
Even small vapor fractions drastically lower the density and above all the speed of sound of the mixture — the speed of sound of a water-steam mixture can lie far below that of either single phase. As a result, the choking condition is reached at low velocities. Anyone who calculates a flashing liquid as an incompressible flow via Bernoulli substantially overestimates the throughput — a dangerous misjudgment for safety valves.
What role does the length of the restriction play?
In very short nozzles and orifices, the liquid has hardly any residence time to evaporate; the flow behaves approximately like a liquid flow with a high mass flow rate (non-equilibrium). With increasing length, thermodynamic equilibrium establishes itself and the mass flow rate approaches the HEM value. The geometry — nozzle, valve or long pipe fitting — is therefore a central criterion for selecting the calculation model.