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Gas compressor in a thermodynamic cycle

**Library:**Simscape / Fluids / Gas / Turbomachinery

The Compressor (G) block models a dynamic compressor, such as a centrifugal or axial
compressor, in a gas network. You can parameterize the block analytically or by a
tabulated compressor map. Fluid flowing from port **A** to port
**B** generates torque. Port **R** reports shaft
torque and angular velocity relative to port **C**, which is associated
with the compressor casing.

In the tabulated data parameterization, the *surge margin*, the ratio
between the surge pressure ratio at a given mass flow rate and the operating point
pressure ratio minus `1`

, is output at port
**SM**.

The compressor *design point* is the intended operational pressure ratio
over and mass flow rate through the compressor during simulation. The compressor
operating point and the point of maximum efficiency do not need to coincide.

A compressor map depicts compressor performance as a function of pressure ratio, the
compressor outlet pressure to the inlet pressure, and corrected mass flow rate. The
map plots the isentropic efficiency of the compressor between the two extremes of
choked flow and surge flow. Compressor maps use *β* lines to
assess performance at an interval across shaft speeds. Choked flow corresponds to β
= 0 and surge flow corresponds to β = 1. β lines are perpendicular to the compressor
shaft constant speed lines, *N*, which are also corrected for
pressure and temperature changes in the compressor.

Due to the large changes in pressure and temperature inside a compressor, the compressor map plots performance in terms of a corrected mass flow rate. The corrected mass flow rate is adjusted from the inlet mass flow rate with a corrected pressure and corrected temperature:

$${\dot{m}}_{A}\sqrt{\frac{{T}_{A}}{{T}_{corr}}}={\dot{m}}_{corr}\frac{{p}_{A}}{{p}_{corr}},$$

where:

$$\dot{m}$$

is the mass flow rate at port_{A}**A**.*T*is the temperature at port_{A}**A**.*T*is the_{corr}**Reference temperature for corrected flow**. When using a tabulated compressor map, the data supplier specifies this value. When using the analytical parameterization, this is the temperature at which the pressure ratio–mass flow rate relationship over a range of temperatures converges to a single trend line.$$\dot{m}$$

is the corrected mass flow rate._{corr}When

**Parameterization**is set to`Analytical`

, this is the**Corrected mass flow rate at design point**.When

**Parameterization**is set to`Tabulated`

, this is derived from the**Corrected mass flow rate table, mdot(N,beta)**.*p*is the pressure at port_{A}**A**.*p*is the_{corr}**Reference pressure for corrected flow**. When using a tabulated compressor map, the data supplier specifies this value. When using the analytical parameterization, this is the pressure at which the pressure ratio–mass flow rate relationship over a range of pressures converges to a single trend line.

The shaft torque, *τ*, is calculated as:

$$\tau =\frac{{\dot{m}}_{A}\Delta {h}_{total}}{{\eta}_{m}\omega},$$

where:

*Δh*is the total change in the fluid specific enthalpy._{total}*η*is the compressor_{m}**Mechanical efficiency**.*ω*is the relative shaft angular velocity,*ω*._{R}- ω_{C}

Reversed flow, from **B** to **A**, is outside
of the typical compressor operation mode and accurate results should not be
expected. A threshold region when flow approaches zero ensures that no torque is
generated when the flow rate is near zero or reversed.

If you do not have tabulated compressor data available, you can model the compressor pressure ratio, corrected mass flow rate, and isentropic efficiency analytically. The analytical method does not use β lines and the block does not report a surge margin.

The pressure ratio at a given shaft speed and mass flow rate is calculated as:

$$\pi =1+\left({\pi}_{D}-1\right)\left[{\tilde{N}}^{ab}+2\tilde{N}k\mathrm{ln}\left(1-\frac{\tilde{m}-{\tilde{N}}^{b}}{k}\right)\right],$$

where:

*π*is the_{D}**Pressure ratio at design point**.$$\tilde{N}$$ is the normalized corrected shaft speed,

$$\frac{N}{{N}_{D}},$$

where

*N*is the_{D}**Corrected speed at design point**.$$\tilde{m}$$ is the normalized corrected mass flow rate,

$$\frac{{\dot{m}}_{corr}}{{\dot{m}}_{D}},$$

where $$\dot{m}$$

is the_{D}**Corrected mass flow rate at design point**.*a*is the**Spine shape, a**.*b*is the**Speed line spread, b**.*k*is the**Speed line roundness, k**.

The map *spine* refers to the analytical line
denoting the nominal compressor performance. The map *speed
lines* are the shaft constant-speed lines that intersect the spine
perpendicularly. The spine and speed line variables are tunable parameters that
can be adjusted for different performance characteristics.

**Analytical Parameterization Default Compressor Map**

When **Efficiency specification** is set to
`Analytical`

, the block models variable compressor
efficiency as:

$$\eta ={\eta}_{0}\left(1-C{\left|\frac{\tilde{p}}{{\tilde{m}}^{a+\Delta a-1}}-\tilde{m}\right|}^{c}-D{\left|\frac{\tilde{m}}{{\tilde{m}}_{0}}-1\right|}^{d}\right),$$

where:

*η*is the_{0}**Maximum isentropic efficiency**.*C*is the**Efficiency contour gradient orthogonal to spine, C**.*D*is the**Efficiency contour gradient along spine, D**.*c*is the**Efficiency peak flatness orthogonal to spine, c**.*d*is the**Efficiency peak flatness along spine, d**.$$\tilde{p}$$ is the normalized corrected pressure ratio,

$$\frac{\pi -1}{{\pi}_{D}-1},$$

where

*π*is the_{D}**Corrected pressure ratio at design point**.$$\tilde{m}$$

is the normalized corrected mass flow rate at which the compressor reaches its_{0}**Maximum isentropic efficiency**.

The efficiency variables are tunable parameters that you can
adjust for different performance characteristics. *a* measures
the relationship between the operating point and the point of maximum
efficiency. When *Δa = 0*, the compressor operates at the point
of maximum efficiency.

Alternatively, you can model constant efficiency by assigning a **Constant
efficiency value**.

When **Parameterization** is set to `Tabulated`

,
the compressor isentropic efficiency, pressure ratio, and corrected mass flow rate
are a function of the corrected speed, *N*, and the map index,
*β*. The block uses linear interpolation between data points
for the efficiency, pressure ratio, and corrected mass flow rate parameters.

If the simulation conditions exceed *β* = 1, surge flow is modeled: the
pressure ratio remains at its value at *β* = 1, while the mass flow
rate continues to change. If the simulation conditions fall below
*β* = 0, choked flow is modeled: the mass flow rate remains at
its value at *β* = 0, while the pressure ratio continues to change.
To constrain the compressor performance within the map boundaries, the block
extrapolates isentropic efficiency to the nearest point.

You can choose to be notified when the operating point pressure ratio exceeds the surge
pressure ratio. Set **Report when static margin is negative** to
`Warning`

to receive a warning or to
`Error`

to stop the simulation when this occurs.

To visualize the block map, right-click the block and select **Fluids** > **Plot Compressor Map Characteristics**.

Each time you modify the block settings, click **Apply** at the
bottom of the dialog box, then click **Reload Data** on the
figure window.

**Tabulated Parameterization Default Compressor Map**

Mass is preserved over the block:

$${\dot{m}}_{A}+{\dot{m}}_{B}=0,$$

where $$\dot{m}$$* _{B}* is the mass flow rate
at port

The energy balance in the block is calculated as:

$${\Phi}_{A}+{\Phi}_{B}+{P}_{fluid}=0,$$

where:

*Φ*is the energy flow rate at port_{A}**A**.*Φ*is the energy flow rate at port_{B}**B**.*P*is the hydraulic power delivered to the fluid, which is determined from the change in total fluid specific enthalpy: $${P}_{fluid}={\dot{m}}_{A}\Delta {h}_{total}.$$_{fluid}

The shaft does not rotate under reversed flow conditions. Results during reversed flows may not be accurate.

The block only models dynamic compressors.

Successful simulation initialization requires a moderately accurate pressure input.

[1] Greitzer, E. M. et al. “N+3
Aircraft Concept Designs and Trade Studies. Volume 2: Appendices – Design Methodologies
for Aerodynamics, Structures, Weight, and Thermodynamic Cycles.” *NASA
Technical Report*, 2010.

[2] Kurzke, Joachim. "How to Get
Component Maps for Aircraft Gas Turbine Performance Calculations." *Volume 5:
Manufacturing Materials and Metallurgy; Ceramics; Structures and Dynamics; Controls,
Diagnostics and Instrumentation; Education; General*, American Society of
Mechanical Engineers, 1996, p. V005T16A001.

[3] Plencner, Robert M. “Plotting
component maps in the Navy/NASA Engine Program (NNEP): A method and its usage.”
*NASA Technical Memorandum*, 1989.