Introduction & Context

Steam-jet ejectors are critical components in process engineering, serving as vacuum pumps that utilize the Venturi effect to convert the pressure energy of a motive fluid into kinetic energy. This kinetic energy creates a low-pressure zone, entraining a secondary suction fluid. These devices are essential in applications such as vacuum distillation, steam surface condensers, and chemical vapor recovery, where they provide a reliable, low-maintenance solution for maintaining sub-atmospheric pressures without moving parts.

Methodology & Formulas

The calculation follows a thermodynamic approach based on isentropic expansion, momentum conservation, and pressure recovery through a diffuser.

1. Motive Nozzle Dynamics

The motive steam velocity V_m is determined by the expansion regime. For subcritical flow, the velocity is derived from the isentropic expansion equation:

\[ V_m = \sqrt{2 \cdot \left( \frac{\gamma}{\gamma - 1} \right) \cdot R \cdot T_m \cdot \left[ 1 - \left( \frac{P_s}{P_m} \right)^{\frac{\gamma - 1}{\gamma}} \right]} \]

In choked flow conditions, the velocity is limited to the sonic velocity at the critical pressure P_c:

\[ V_m = \sqrt{\gamma \cdot R \cdot T_c} \]

2. Momentum and Energy Balance

The mixed stream velocity V_mix is calculated by applying the conservation of momentum, assuming the suction fluid velocity is negligible:

\[ V_{mix} = \frac{\dot{m}_m \cdot V_m}{\dot{m}_m + \dot{m}_s} \]

The mixed temperature T_mix is derived from the energy balance of the two streams:

\[ T_{mix} = \frac{\dot{m}_m \cdot C_p \cdot T_m + \dot{m}_s \cdot C_p \cdot T_s}{(\dot{m}_m + \dot{m}_s) \cdot C_p} \]

3. Pressure Recovery

The final discharge pressure P_d is calculated by converting the kinetic energy of the mixed stream back into pressure energy within the diffuser, adjusted by the diffuser efficiency η_d:

\[ P_d = P_s + \left( \eta_d \cdot 0.5 \cdot \rho_{mix} \cdot V_{mix}^2 \right) \]

Operational Regimes and Validation Criteria

Parameter	Condition	Engineering Implication
Flow Regime	P_s / P_m ≤ P_{critical_ratio}	Choked flow; requires Convergent-Divergent nozzle design.
Compression Ratio	P_d / P_s > 5.0	High compression; multi-stage ejector system required.
Entrainment Ratio	ω < 0.1 or ω > 0.5	Outside typical empirical efficiency range; check design geometry.

How do I calculate the required motive steam pressure and flow for a given evacuation duty?

Begin with the absolute pressures at the start (P1) and end (P2) of the evacuation period and the vessel volume V.
Calculate the mass of gas to be removed: m = (P1 – P2) V / (R T), correcting for any condensable vapours.
Select an ejector performance curve from the vendor that intersects the design suction pressure at the desired capacity.
Read the corresponding motive steam flow; if the curve is for 6 bar(g) and your header is 8 bar(g), scale flow by √(ΔP_actual/ΔP_curve).
Add 10–15% margin to the calculated steam flow for wear and fouling.

Why does the ejector curve show kg air-equivalent instead of ACFM, and how do I convert?

Steam-jet curves are normalized to 20 °C dry air so one family of curves fits any gas.
Convert actual gas flow to air-equivalent using: W_air-eq = W_actual √(MW_actual / 29).
If water vapour is present, add the non-condensable portion only (total minus vapour pressure at suction temperature).

Can I optimise the number of stages, or is lower always better?

Single-stage ejectors are cheapest down to ≈ 100 mbar; below that multi-stage reduces motive steam and cooling-water demand.
Add a condenser between stages to knock out motive steam and cut the load on downstream ejectors.
Below 5 mbar a steam-jet/liquid-ring hybrid may be more economical—evaluate total utilities and capital.
Always compare annual steam cost against the extra price of each additional stage.

How do back-pressure and after-condenser water temperature affect achievable vacuum?

The last stage discharges against a back-pressure equal to after-condenser water vapour pressure plus any non-condensable partial pressure.
Raise cooling-water temperature from 30 °C to 35 °C and the vapour pressure climbs from 42 to 56 mbar, moving the operating point to a higher absolute pressure.
If the ejector curve shows a 5 mbar shut-off, confirm the condenser can handle the motive steam load at summer water temperature; otherwise the device will hunt or stall.
Size the tail pipe to keep velocity <30 m s⁻¹ to minimise back-pressure from friction.

Worked Example: Steam-Jet Ejector Performance Analysis

In a chemical processing facility, a steam-jet ejector is utilized to maintain a vacuum in a process vessel. The system must entrain 50.0 kg/h of suction vapor using high-pressure motive steam. The following calculation verifies the discharge pressure capability of the ejector under the specified operating conditions.

Knowns:

Motive Steam Pressure: 2.0 bar
Motive Steam Temperature: 170.0 °C
Suction Vapor Pressure: 1.2 bar
Suction Vapor Temperature: 60.0 °C
Suction Mass Flow Rate: 50.0 kg/h
Entrainment Ratio (Target): 0.3
Specific Gas Constant (Steam): 461.5 J/(kg·K)
Isentropic Efficiency: 0.8
Heat Capacity Ratio: 1.3

Step-by-Step Calculation:

Calculate the required motive steam mass flow rate based on the entrainment ratio:
\[ M_{motive} = \frac{M_{suction}}{RW} = \frac{50.0}{0.3} = 166.667 \text{ kg/h} \]
Convert mass flow rates to SI units (kg/s):
\[ M_{suction,s} = 0.014 \text{ kg/s}, \quad M_{motive,s} = 0.046 \text{ kg/s}, \quad M_{total,s} = 0.060 \text{ kg/s} \]
Determine the motive steam velocity at the nozzle exit using the isentropic expansion formula:
\[ V_{motive} = 443.955 \text{ m/s} \]
Calculate the mixture temperature after momentum exchange:
\[ T_{mix} = 417.765 \text{ K} \]
Calculate the mixture velocity based on the conservation of momentum:
\[ V_{mix} = 341.504 \text{ m/s} \]
Determine the recovered discharge pressure by applying the efficiency factor to the kinetic energy conversion:
\[ P_{recovered} = 149035.480 \text{ Pa} \]

Final Answer:

The calculated discharge pressure for the steam-jet ejector is 1.490 bar. Given the target discharge pressure of 1.5 bar, the system operates at a compression ratio of 1.25, indicating that the ejector is performing within the expected design parameters for this process application.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle