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Modeling Pneumatic Systems

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Intended Applications Assumptions and Limitations Fundamental Equations Network Variables Connection Constraints References

Intended Applications

The Foundation library contains basic pneumatic elements, such as orifices, chambers, and pneumatic-mechanical converters, as well as pneumatic sensors and sources. Use these blocks to model pneumatic systems, for applications such as:

• Factory automation — basic pneumatic linear/rotational actuators, valves (variable orifices), and air supply

• Robotics — robotic arms and haptic interfaces

• Gaseous transportation systems and pipelines

You can also use these blocks to model dry air and low-pressure flows, for example, for HVAC applications.

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Assumptions and Limitations

Pneumatic block models are based on the following assumptions:

• Working fluid is an ideal gas satisfying the ideal gas law.

• Specific heats at constant pressure and constant volume, c_p and c_v, are constant.

• Processes are adiabatic, that is, there is no heat transfer between components and the environment (except for components with a separate thermal port).

• Gravitational effects can be neglected, that is, underlying equations contain no head pressures due to gravity.

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Fundamental Equations

The energy balance for a control volume [1] is

where

Ecv	Control volume total energy
Qcv	Heat energy per second added to the gas through the boundary
Wcv	Mechanical work per second performed by the gas
hi, ho	Inlet and outlet enthalpies
vi, vo	Gas inlet and outlet velocities
g	Acceleration due to gravity
zi, zo	Elevations at inlet and outlet ports
mi, mo	Mass flow rates in and out of the control volume

The equation is an accounting balance for the energy of the control volume. It states that the rate of energy increase or decrease within the control volume equals the difference between the rates of energy transfer in and out across the boundary. The mechanisms of energy transfer are heat and work, as for closed systems, and the energy that accompanies the mass entering and exiting.

Pneumatic block models make several simplifying assumptions, as described previously.

The ideal gas law relates pressure, density, and temperature:

where

p	Absolute pressure
ρ	Gas density
R	Specific gas constant
T	Absolute gas temperature

Also, the specific enthalpies for an ideal gas at temperature T and constant pressure and constant volume are given by:

The pneumatic components also use the mass continuity equation:

where ρ is the density of the gas within the component. For components with no internal mass of gas, the equation simplifies to:

where G is the mass flow rate through the component.

For specific equations used in each block, see the block reference pages.

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Network Variables

The Across variables are pressure and temperature, and the Through variables are mass flow rate and heat flow. Note that these choices result in a pseudo-bond graph, because the product of pressure and mass flow rate is not power.

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Connection Constraints

Every node in a pneumatic network must have a defined temperature as well as pressure. This rule places some constraints on how you connect the pneumatic elements. In effect, every node should have a volume of fluid associated with it. When the ideal gas law is applied, this volume of fluid determines the relationship between temperature and pressure. Some elements already have a volume of fluid associated with them, and therefore having just one of these components connected to a node satisfies this condition. Such blocks include Constant Volume Pneumatic Chamber, Pneumatic Piston Chamber, Rotary Pneumatic Piston Chamber, and Pneumatic Atmospheric Reference.

An exception to the above rule (that every node must have a volume of fluid associated with it) occurs when two nodes are connected by a component for which the heat equation says that the temperatures are equal. In this case, just one of the nodes needs to be connected to a component with associated volume of fluid. Such components include the pressure and flow rate sources.

For models that represent an actual pneumatic network, these constraints should have no impact. For example, connecting two orifices in series makes no physical sense because the underlying assumption of the orifice equation is that gas is discharged into a volume of fluid. Therefore, modeling actual physical systems should automatically satisfy these constraints.

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References

[1] Moran M.J. and Shapiro H.N. Fundamentals of Engineering Thermodynamics. Second edition. New York: John Wiley & Sons, 1992.

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Modeling Best Practices

Simulating Physical Models

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