Ejector (G)

Ejector in a gas network

Since R2023a

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Libraries:
Simscape / Fluids / Gas / Turbomachinery

Description

The Ejector (G) block models an ejector in a gas network. The ejector forces the high-pressure primary flow through a nozzle with a specific throat area, which creates a low-pressure jet that draws in the secondary flow. The expansion of the primary jet creates a throat for the secondary flow. The block assumes that the two flows mix under uniform pressure and the combined flow exits through a diffuser to recover pressure. The primary and secondary flows must be the same gas.

You can use an ejector as a pump or compressor to drive a secondary fluid. Unlike a pump or compressor, there is no external source of mechanical energy, and rather the high pressure primary fluid is the energy source that drives the secondary fluid.

Ejector Geometry

The high-pressure primary flow enters the ejector at location p and flows through a convergent-divergent nozzle with throat area S_pt, which is the value of the Primary nozzle throat area parameter. The ejector chokes the primary flow at the throat (location pt) and accelerates the flow to a low-pressure supersonic jet at the primary nozzle exit (location pn). The low pressure of the supersonic primary flow jet draws in the secondary flow at location s around the nozzle.

The supersonic primary flow jet continues to expand and decrease in pressure beyond the nozzle exit until it reaches its largest area at the expanded primary flow jet (location pe). The expansion of the supersonic primary flow jet creates a fictitious aerodynamic throat for the secondary flow at location st, which is at the same location as pe. The secondary flow also chokes at the aerodynamic throat. After passing the aerodynamic throat, the primary and secondary flow mix under uniform pressure (location m). If the mixed flow is still supersonic, the flow causes a shock in the mixing chamber at location s and the diffuser (location d) decelerates the mixed flow to raise the outlet pressure.

This schematic describes the ejector where:

• Location p is the primary inlet.

• Location s is the secondary inlet.

• Location pt is the primary nozzle throat.

• Location pn is the primary nozzle exit.

• Location pe is the expanded primary flow jet.

• Location st is the secondary aerodynamic throat.

• Location m is the mixed flow.

• Location ms is the shock in the mixing chamber.

• Location d is the diffuser outlet.

Mass Flow Rate

The ejector accelerates the primary gas flow through the nozzle. The mass flow rate of the primary flow is

$${\dot{m}}_p=S_{pt}\sqrt{\left(\frac{2\gamma }{\gamma -1}\right){\eta }_p{p}_p{\rho }_p\left[{\left(\frac{p_m}{p_p}\right)}^{\frac{2}{\gamma }}-{\left(\frac{p_m}{p_p}\right)}^{\frac{\gamma +1}{\gamma }}\right]},$$

where:

• S_pt is the value of the Primary nozzle throat area parameter.

• $$\gamma =\frac{c_p}{c_v}$$ is the ratio of heat capacities where c_p is the specific heat at constant pressure and c_v is the specific heat at constant volume.

• η_p is the value of the Efficiency for primary flow through nozzle parameter.

• p_p is the pressure at the primary flow inlet.

• p_m is the pressure after the primary and secondary flow mix.

• ρ_p is the density at the primary flow inlet.

The primary flow draws the secondary gas flow into the suction chamber and accelerates it towards the fictitious aerodynamic throat created by the expanding primary jet. The mass flow rate of the secondary flow is

$${\dot{m}}_s=S_{st}\sqrt{\left(\frac{2\gamma }{\gamma -1}\right){\eta }_s{p}_s{\rho }_s\left[{\left(\frac{p_m}{p_s}\right)}^{\frac{2}{\gamma }}-{\left(\frac{p_m}{p_s}\right)}^{\frac{\gamma +1}{\gamma }}\right]},$$

where:

• S_st is the area of the fictitious aerodynamic throat. The block calculates the value based on the expansion of the primary flow jet.

• η_s is the value of the Efficiency for secondary suction flow parameter.

• p_s is the pressure at the secondary flow inlet.

• ρ_s is the density at the secondary flow inlet.

When $$\frac{p_m}{p_p}<B_{crit},$$ where $$B_{crit}={\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma }{\gamma -1}},$$ the primary flow becomes choked and the mass flow rate reaches the maximum value

$${\dot{m}}_p^{*}=S_{pt}\sqrt{{\eta }_p{p}_p{\rho }_p\gamma {\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}.}$$

When $$\frac{p_m}{p_s}<B_{crit},$$ the secondary flow becomes choked and the mass flow rate reaches the maximum value

$${\dot{m}}_s{}^{*}=S_{st}\sqrt{{\eta }_s{p}_s{\rho }_s\gamma {\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}}.$$

Primary and Secondary Flow Relations

The flow area expansion of the supersonic primary flow jet comes from the relation

$$\frac{S_{pe}}{S_{pt}}={\Psi }_{pe}\sqrt{\frac{\left(\frac{\gamma -1}{2}\right){\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}}{{\left(\frac{p_m}{p_p}\right)}^{\frac{2}{\gamma }}-{\left(\frac{p_m}{p_p}\right)}^{\frac{\gamma +1}{\gamma }}}},$$

where ψ_pe is the value of the Efficiency for primary flow expansion parameter.

The block uses the area of the expanded supersonic primary flow jet, S_pe, to calculate the flow area of the fictitious secondary flow aerodynamic throat, S_st

$$\frac{S_{st}}{S_{pt}}=\frac{S_m}{S_{pt}}-\max \left\{\frac{S_{pe}}{S_{pt}},\frac{S_{pm}}{S_{pt}}\right\},$$

where:

• $$\frac{S_m}{S_{pt}}$$ is the value of the Area ratio of mixing chamber to throat parameter.

• $$\frac{S_{pn}}{S_{pt}}$$ is the value of the Area ratio of nozzle exit to throat parameter.

The primary flow velocity of the expanded jet is

$${\text{v}}_{pe}=\sqrt{\frac{2\gamma }{\gamma -1}\left(\frac{p_p}{{\rho }_p}\right)\left[1-{\left(\frac{p_m}{p_p}\right)}^{\frac{\gamma -1}{\gamma }}\right]}.$$

The secondary flow velocity at the aerodynamic throat is

$${\text{v}}_{st}=\sqrt{\frac{2\gamma }{\gamma -1}\left(\frac{p_s}{{\rho }_s}\right)\left[1-{\left(\frac{p_m}{p_s}\right)}^{\frac{\gamma -1}{\gamma }}\right]}.$$

The secondary flow velocity is limited to the choked value. When $$\frac{p_m}{p_s}<B_{crit},$$, the secondary flow reaches the maximum value of

$${\text{v}}_{st}^{*}=\sqrt{\frac{2\gamma }{\gamma +1}\left(\frac{p_s}{{\rho }_s}\right)}.$$

Mixing Model

After the fictitious aerodynamic throat, the primary and secondary flows mix under uniform pressure. The mixed flow temperature is

$${\text{T}}_m=\frac{{\dot{m}}_p{T}_p+{\dot{m}}_s{T}_s}{{\dot{m}}_m}-\frac{\gamma -1}{2\gamma R}{v}_m^2,$$

where:

• T_p is the primary flow inlet temperature. The block assumes that this value is equal to the primary flow stagnation temperature because kinetic energy at the inlet is negligible relative to the inside of the ejector.

• T_m is the mixed flow temperature.

• T_s is the secondary flow inlet temperature. The block assumes that this value is equal to the secondary flow stagnation temperature because kinetic energy at the inlet is negligible relative to the inside of the ejector.

• R is the specific gas constant.

• v_m is the flow velocity after the primary and secondary flow mix.

• $$\dot{m}$$_m is the mass flow rate after the primary and secondary flow mix, meaning $${\dot{m}}_m={\dot{m}}_p+{\dot{m}}_s.$$

The momentum balance in the mixing region is

$$p_m{S}_m+{\dot{m}}_m{v}_m=p_m{S}_{pe}+{\dot{m}}_p{v}_{pe}+p_m{S}_{st}+{\dot{m}}_s{v}_{st},$$

where S_m is the cross-sectional area after the primary and secondary flow mix.

The pressure terms drop out due to the uniform pressure mixing assumption, and the mixed flow velocity is

$${\text{v}}_m={\psi }_m\frac{{\dot{m}}_p{v}_{pe}+{\dot{m}}_s{v}_{st}}{{\dot{m}}_m},$$

where ψ_m is the value of the Efficiency for mixing parameter.

The mixed flow Mach number is

$${\text{M}}_m=\frac{v_m}{\sqrt{\gamma R{T}_m}}.$$

Mixed Diffuser Flow Relations

If the flow is supersonic after mixing, meaning M_M > 1, a shock wave occurs. The pressure rise after the shock is

$$\frac{p_{ms}}{p_m}=1+\frac{2\gamma }{\gamma +1}\left(M_m^2-1\right),$$

where M_m is the Mach number after the primary and secondary flow mix.

The Mach number after the shock is

$${\text{M}}_{ms}^2=\frac{\frac{\gamma -1}{\gamma +1}\left(M_m^2-1\right)+1}{\frac{2\gamma }{\gamma +1}\left(M_m^2-1\right)+1}.$$

After the shock, the diffuser decelerates the flow to raise the pressure. The block assumes that the kinetic energy at the diffuser outlet is negligible and decelerates the flow to stagnation pressure at p_d:

$$\frac{p_d}{p_{ms}}={\left(1+\frac{\gamma -1}{2}{M}_{ms}^2\right)}^{\frac{\gamma }{\gamma -1}}.$$

If M_M ≤ 1, there is no shock and

$$\begin{array}{l}\frac{p_{ms}}{p_m}=1\\ M_{ms}^2=M_m^2.\end{array}$$

The relationship between the mixing pressure, p_m, and the diffuser outlet pressure, p_d, is

$$\frac{p_d}{p_m}=\{\begin{array}{cc}\frac{{\left[\frac{\gamma +1}{2}{M}_m^2\right]}^{\frac{\gamma }{\gamma -1}}}{{\left[\frac{2\gamma }{\gamma +1}\left(M_m^2-1\right)+1\right]}^{\frac{1}{\gamma -1}}},& M_m>1\\ {\left(1+\frac{\gamma -1}{2}{M}_m^2\right)}^{\frac{\gamma }{\gamma -1}},& M_m\le 1\end{array}$$

Critical Mode Operation

In normal operation, the primary flow is high pressure and is choked during operation. The block can also operate if the secondary flow is choked. If both the primary and secondary flows are choked, the block operates in critical mode.

The critical mixing pressure, p_m^*, is a low pressure threshold that causes choking in both primary and secondary flow:

$${\text{p}}_m^{*}=\min \left\{p_p,p_s\right\}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma }{\gamma -1}}.$$

The block uses the same equations as above, with p_m replaced by p_m^*, to calculate the critical diffuser outlet pressure, p_d^*. If the actual outlet pressure is greater than the critical diffuser outlet pressure, then the ejector is subcritical and the block relates the mixing pressure to diffuser outlet pressure. If the actual outlet pressure is less than the critical diffuser outlet pressure, then the ejector is critical and the block limits the mixing pressure to the critical mixing pressure. This means that:

$${\text{p}}_m=\{\begin{array}{cc}\frac{p_d}{\left(\frac{p_d}{p_m}\right)}& p_d\ge p_d^{*}\\ p_m^{*}& p_d<p_d^{*}\end{array}$$

Assumptions and Limitations

• The block uses isentropic ideal gas relations to derive the model equations.

• The block accounts for losses due to friction, mixing, and expansion waves and shocks at the nozzle exit using empirical loss coefficients.

• After the primary flow exits the nozzle, the flows do not mix until the primary flow has fully expanded.

• After expansion, the flows mix under uniform pressure.

• Kinetic energy at the primary flow inlet, secondary flow inlet, and diffuser outlet are negligible compared to the kinetic energy of the flow inside the ejector.

• The flow is steady and one-dimensional.

• The flow is adiabatic.

• Model results with reversed flow may not be accurate.

Ports

Conserving

A — Primary inlet

gas

Gas conserving port associated with associated with the primary nozzle inlet.

B — Outlet

gas

Gas conserving port associated with the diffuser outlet.

S — Secondary inlet

gas

Gas conserving port associated with the secondary inlet.

Parameters

Primary nozzle throat area — Area of primary nozzle throat

1e-4 m^2 (default) | positive scalar

Area of the throat for the primary flow nozzle.

Area ratio of nozzle exit to throat — Ratio of areas of nozzle exit to nozzle throat

3 (default) | positive scalar

Ratio of the area of the nozzle exit to the area of the nozzle throat. This parameter limits how big the secondary aerodynamic throat can be.

Area ratio of mixing chamber to throat — Ratio of areas of mixing chamber to throat

8 (default) | positive scalar

Ratio of the area of the mixing chamber to the area of the throat. The difference between the mixing chamber area and the expanded jet area of the primary flow creates a fictitious aerodynamic throat for the secondary flow.

Minimum area ratio of secondary throat to primary throat — Limit on ratio of aerodynamic areas of secondary throat to primary throat

0.1 (default) | positive scalar

Minimum aerodynamic area ratio for the secondary throat to the primary throat. The block does not allow the area of the fictitious aerodynamic throat to go below this value. To avoid hitting this minimum, increase the value of the Area ratio of mixing chamber to throat parameter.

Report when secondary throat area falls below minimum — Notification when secondary throat area is below minimum

Warning (default) | None | Error

Whether the block does nothing, generates a warning, or generates an error when the secondary throat area falls below the minimum, specified by the Minimum area ratio of secondary throat to primary throat parameter.

Efficiency for primary flow through nozzle — Efficiency factor of primary flow through nozzle

0.95 (default) | positive scalar

Empirical factor that accounts for reduction in primary mass flow rate due to losses in the primary nozzle.

Efficiency for secondary suction flow — Efficiency factor of secondary suction flow

0.85 (default) | positive scalar

Empirical factor that accounts for reduction in secondary mass flow rate due to losses in the suction of the secondary flow.

Efficiency for primary flow expansion — Efficiency factor of primary flow expansion

0.88 (default) | positive scalar

Empirical factor that accounts for reduction in the expansion of the primary flow jet due to mixing losses.

Efficiency for mixing — Efficiency factor for mixed flow velocity due to mixing

0.84 (default) | positive scalar

Empirical factor that accounts for reduction in the mixed flow velocity due to losses in the mixing chamber.

Cross-sectional area at port A — Cross-sectional area of port A

0.01 m^2 (default) | positive scalar

Cross-sectional area of port A.

Cross-sectional area at port B — Cross-sectional area of port B

0.01 m^2 (default) | positive scalar

Cross-sectional area of port B.

Cross-sectional area at port S — Cross-sectional area of port S

0.01 m^2 (default) | positive scalar

Cross-sectional area of port S.

References

[1] Huang, B. J., et al. "A 1-D analysis of ejector performance." International journal of refrigeration 22.5 (1999): 354-364.

[2] Chen, WeiXiong, et al. "A 1D model to predict ejector performance at critical and sub-critical operational regimes." International journal of refrigeration 36.6 (2013): 1750-1761.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2023a