Pipe (G)

Rigid conduit for gas flow

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Libraries:
Simscape / Foundation Library / Gas / Elements

Description

The Pipe (G) block models pipe flow dynamics in a gas network. The block accounts for viscous friction losses and convective heat transfer with the pipe wall. The pipe contains a constant volume of gas. The pressure and temperature evolve based on the compressibility and thermal capacity of this gas volume. Choking occurs when the outlet reaches the sonic condition.

Caution

Gas flow through this block can choke. If a Mass Flow Rate Source (G) block or a Controlled Mass Flow Rate Source (G) block connected to the Pipe (G) block specifies a greater mass flow rate than the possible choked mass flow rate, you get a simulation error. For more information, see Choked Flow.

Mass Balance

Mass conservation relates the mass flow rates to the dynamics of the pressure and temperature of the internal node representing the gas volume:

$$\frac{\partial M}{\partial p}\cdot \frac{d{p}_I}{dt}+\frac{\partial M}{\partial T}\cdot \frac{d{T}_I}{dt}={\dot{m}}_A+{\dot{m}}_B$$

where:

• $$\frac{\partial M}{\partial p}$$ is the partial derivative of the mass of the gas volume with respect to pressure at constant temperature and volume.

• $$\frac{\partial M}{\partial T}$$ is the partial derivative of the mass of the gas volume with respect to temperature at constant pressure and volume.

• p_I is the pressure of the gas volume.

• T_I is the temperature of the gas volume.

• t is time.

• $$\dot{m}$$_A and $$\dot{m}$$_B are mass flow rates at ports A and B, respectively. Flow rate associated with a port is positive when it flows into the block.

Energy Balance

Energy conservation relates the energy and heat flow rates to the dynamics of the pressure and temperature of the internal node representing the gas volume:

$$\frac{\partial U}{\partial p}\cdot \frac{d{p}_I}{dt}+\frac{\partial U}{\partial T}\cdot \frac{d{T}_I}{dt}={\Phi }_A+{\Phi }_B+Q_H$$

where:

• $$\frac{\partial U}{\partial p}$$ is the partial derivative of the internal energy of the gas volume with respect to pressure at constant temperature and volume.

• $$\frac{\partial U}{\partial T}$$ is the partial derivative of the internal energy of the gas volume with respect to temperature at constant pressure and volume.

• Φ_A and Φ_B are energy flow rates at ports A and B, respectively.

• Q_H is heat flow rate at port H.

Partial Derivatives for Perfect and Semiperfect Gas Models

The partial derivatives of the mass M and the internal energy U of the gas volume, with respect to pressure and temperature at constant volume, depend on the gas property model. For perfect and semiperfect gas models, the equations are:

$$\begin{array}{l}\frac{\partial M}{\partial p}=V\frac{{\rho }_I}{p_I}\\ \frac{\partial M}{\partial T}=-V\frac{{\rho }_I}{T_I}\\ \frac{\partial U}{\partial p}=V\left(\frac{h_I}{ZR{T}_I}-1\right)\\ \frac{\partial U}{\partial T}=V{\rho }_I\left(c_{pI}-\frac{h_I}{T_I}\right)\end{array}$$

where:

• ρ_I is the density of the gas volume.

• V is the volume of gas.

• h_I is the specific enthalpy of the gas volume.

• Z is the compressibility factor.

• R is the specific gas constant.

• c_pI is the specific heat at constant pressure of the gas volume.

Partial Derivatives for Real Gas Model

For real gas model, the partial derivatives of the mass M and the internal energy U of the gas volume, with respect to pressure and temperature at constant volume, are:

$$\begin{array}{l}\frac{\partial M}{\partial p}=V\frac{{\rho }_I}{{\beta }_I}\\ \frac{\partial M}{\partial T}=-V{\rho }_I{\alpha }_I\\ \frac{\partial U}{\partial p}=V\left(\frac{{\rho }_I{h}_I}{{\beta }_I}-T_I{\alpha }_I\right)\\ \frac{\partial U}{\partial T}=V{\rho }_I\left(c_{pI}-h_I{\alpha }_I\right)\end{array}$$

where:

• β is the isothermal bulk modulus of the gas volume.

• α is the isobaric thermal expansion coefficient of the gas volume.

Momentum Balance

The momentum balance for each half of the pipe models the pressure drop due to momentum flux and viscous friction:

$$\begin{array}{l}{p}_A-p_I={\left(\frac{{\dot{m}}_A}{S}\right)}^2\cdot \left(\frac{1}{{\rho }_I}-\frac{1}{{\rho }_A}\right)+\Delta p_{AI}\\ p_B-p_I={\left(\frac{{\dot{m}}_B}{S}\right)}^2\cdot \left(\frac{1}{{\rho }_I}-\frac{1}{{\rho }_B}\right)+\Delta p_{BI}\end{array}$$

where:

• p is the gas pressure at port A, port B, or internal node I, as indicated by the subscript.

• ρ is the density at port A, port B, or internal node I, as indicated by the subscript.

• S is the cross-sectional area of the pipe.

• Δp_AI and Δp_BI are pressure losses due to viscous friction.

The heat exchanged with the pipe wall through port H is added to the energy of gas volume represented by the internal node via the energy conservation equation (see Energy Balance). Therefore, the momentum balances for each half of the pipe, between port A and the internal node and between port B and the internal node, are assumed to be adiabatic processes. The adiabatic relations are:

$$\begin{array}{l}{h}_A+\frac{1}{2}{\left(\frac{{\dot{m}}_A}{{\rho }_{A}S}\right)}^2=h_I+\frac{1}{2}{\left(\frac{{\dot{m}}_A}{{\rho }_{I}S}\right)}^2\\ h_B+\frac{1}{2}{\left(\frac{{\dot{m}}_B}{{\rho }_{B}S}\right)}^2=h_I+\frac{1}{2}{\left(\frac{{\dot{m}}_B}{{\rho }_{I}S}\right)}^2\end{array}$$

where h is the specific enthalpy at port A, port B, or internal node I, as indicated by the subscript.

The pressure losses due to viscous friction, Δp_AI and Δp_BI, depend on the flow regime. The Reynolds numbers for each half of the pipe are defined as:

$$\begin{array}{l}{\mathrm{Re}}_A=\frac{\left|{\dot{m}}_A\right|\cdot D_h}{S\cdot {\mu }_I}\\ {\mathrm{Re}}_B=\frac{\left|{\dot{m}}_B\right|\cdot D_h}{S\cdot {\mu }_I}\end{array}$$

where:

• D_h is the hydraulic diameter of the pipe.

• μ_I is the dynamic viscosity at internal node.

If the Reynolds number is less than the Laminar flow upper Reynolds number limit parameter value, then the flow is in the laminar flow regime. If the Reynolds number is greater than the Turbulent flow lower Reynolds number limit parameter value, then the flow is in the turbulent flow regime.

In the laminar flow regime, the pressure losses due to viscous friction are:

$$\begin{array}{l}\Delta p_{A{I}_{lam}}=f_{shape}\frac{{\dot{m}}_A\cdot {\mu }_I}{2{\rho }_I\cdot D_h^2\cdot S}\cdot \frac{L+L_{eqv}}{2}\\ \Delta p_{B{I}_{lam}}=f_{shape}\frac{{\dot{m}}_B\cdot {\mu }_I}{2{\rho }_I\cdot D_h^2\cdot S}\cdot \frac{L+L_{eqv}}{2}\end{array}$$

where:

• f_shape is the Shape factor for laminar flow viscous friction parameter value.

• L_eqv is the Aggregate equivalent length of local resistances parameter value.

In the turbulent flow regime, the pressure losses due to viscous friction are:

$$\begin{array}{l}\Delta p_{A{I}_{tur}}=f_{Darc{y}_A}\frac{{\dot{m}}_A\cdot \left|{\dot{m}}_A\right|}{2{\rho }_I\cdot D_h\cdot S^2}\cdot \frac{L+L_{eqv}}{2}\\ \Delta p_{B{I}_{tur}}=f_{Darc{y}_B}\frac{{\dot{m}}_B\cdot \left|{\dot{m}}_B\right|}{2{\rho }_I\cdot D_h\cdot S^2}\cdot \frac{L+L_{eqv}}{2}\end{array}$$

where f_Darcy is the Darcy friction factor at port A or B, as indicated by the subscript.

The Darcy friction factors are computed from the Haaland correlation:

$$\begin{array}{l}{f}_{Darc{y}_A}={\left[-1.8{\log }_{10}\left(\frac{6.9}{{\mathrm{Re}}_A}+{\left(\frac{{\epsilon }_{rough}}{3.7D_h}\right)}^{1.11}\right)\right]}^{-2}\\ f_{Darc{y}_B}={\left[-1.8{\log }_{10}\left(\frac{6.9}{{\mathrm{Re}}_B}+{\left(\frac{{\epsilon }_{rough}}{3.7D_h}\right)}^{1.11}\right)\right]}^{-2}\end{array}$$

where ε_rough is the Internal surface absolute roughness parameter value.

When the Reynolds number is between the Laminar flow upper Reynolds number limit and the Turbulent flow lower Reynolds number limit parameter values, the flow is in transition between laminar flow and turbulent flow. The pressure losses due to viscous friction during the transition region follow a smooth connection between those in the laminar flow regime and those in the turbulent flow regime.

Convective Heat Transfer

The convective heat transfer equation between the pipe wall and the internal gas volume is:

$$Q_H=Q_{conv}+\frac{k_I{S}_{surf}}{D_h}\left(T_H-T_I\right)$$

S_surf is the pipe surface area, S_surf = 4SL/D_h. Assuming an exponential temperature distribution along the pipe, the convective heat transfer is

$$Q_{conv}=\left|{\dot{m}}_{avg}\right|c_{p_{avg}}\left(T_H-T_{in}\right)\left(1-\exp \left(-\frac{h_{coeff}{S}_{surf}}{\left|{\dot{m}}_{avg}\right|c_{p_{avg}}}\right)\right)$$

where:

• T_in is the inlet temperature depending on flow direction.

• $${\dot{m}}_{avg}=\left({\dot{m}}_A-{\dot{m}}_B\right)/2$$ is the average mass flow rate from port A to port B.

• $$c_{p_{avg}}$$ is the specific heat evaluated at the average temperature.

The heat transfer coefficient, h_coeff, depends on the Nusselt number:

$$h_{coeff}=Nu\frac{k_{avg}}{D_h}$$

where k_avg is the thermal conductivity evaluated at the average temperature. The Nusselt number depends on the flow regime. The Nusselt number in the laminar flow regime is constant and equal to the value of the Nusselt number for laminar flow heat transfer parameter. The Nusselt number in the turbulent flow regime is computed from the Gnielinski correlation:

$$N{u}_{tur}=\frac{\frac{f_{Darcy}}{8}\left({\mathrm{Re}}_{avg}-1000\right){\mathrm{Pr}}_{avg}}{1+12.7\sqrt{\frac{f_{Darcy}}{8}}\left({\mathrm{Pr}}_{avg}^{2/3}-1\right)}$$

where Pr_avg is the Prandtl number evaluated at the average temperature. The average Reynolds number is

$${\mathrm{Re}}_{avg}=\frac{\left|{\dot{m}}_{avg}\right|D_h}{S{\mu }_{avg}}$$

where μ_avg is the dynamic viscosity evaluated at the average temperature. When the average Reynolds number is between the Laminar flow upper Reynolds number limit and the Turbulent flow lower Reynolds number limit parameter values, the Nusselt number follows a smooth transition between the laminar and turbulent Nusselt number values.

Choked Flow

The choked mass flow rates out of the pipe at ports A and B are:

$$\begin{array}{l}{\dot{m}}_{A_{choked}}={\rho }_A\cdot a_A\cdot S\\ {\dot{m}}_{B_{choked}}={\rho }_B\cdot a_B\cdot S\end{array}$$

where a_A and a_B is the speed of sound at ports A and B, respectively.

The unchoked pressure at port A or B is the value of the corresponding Across variable at that port:

$$\begin{array}{l}{p}_{A_{unchoked}}=\text{A}\text{.p}\\ p_{B_{unchoked}}=\text{B}\text{.p}\end{array}$$

The choked pressures at ports A and B are obtained by substituting the choked mass flow rates into the momentum balance equations for the pipe:

$$\begin{array}{l}{p}_{A_{choked}}=p_I+{\left(\frac{{\dot{m}}_{A_{choked}}}{S}\right)}^2\cdot \left(\frac{1}{{\rho }_I}-\frac{1}{{\rho }_A}\right)+\Delta p_{A{I}_{choked}}\\ p_{B_{choked}}=p_I+{\left(\frac{{\dot{m}}_{B_{choked}}}{S}\right)}^2\cdot \left(\frac{1}{{\rho }_I}-\frac{1}{{\rho }_B}\right)+\Delta p_{B{I}_{choked}}\end{array}$$

Δp_AIchoked and Δp_BIchoked are the pressure losses due to viscous friction, assuming that the choking has occurred. They are computed similar to Δp_AI and Δp_BI, with the mass flow rates at ports A and B replaced by the choked mass flow rate values.

Depending on whether choking has occurred, the block assigns either the choked or unchoked pressure value as the actual pressure at the port. Choking can occur at the pipe outlet, but not at the pipe inlet. Therefore, if p_Aunchoked ≥ p_I, then port A is an inlet and p_A = p_Aunchoked. If p_Aunchoked < p_I, then port A is an outlet and

$$p_A=\{\begin{array}{ll}{p}_{A_{unchoked}},\hfill & \text{if }{p}_{A_{unchoked}}\ge p_{A_{choked}}\hfill \\ p_{A_{choked}},\hfill & \text{if }{p}_{A_{unchoked}}<p_{A_{choked}}\hfill \end{array}$$

Similarly, if p_Bunchoked ≥ p_I, then port B is an inlet and p_B = p_Bunchoked. If p_Bunchoked < p_I, then port B is an outlet and

$$p_B=\{\begin{array}{ll}{p}_{B_{unchoked}},\hfill & \text{if }{p}_{B_{unchoked}}\ge p_{B_{choked}}\hfill \\ p_{B_{choked}},\hfill & \text{if }{p}_{B_{unchoked}}<p_{B_{choked}}\hfill \end{array}$$

Variables

To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables and Initial Conditions for Blocks with Finite Gas Volume.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see Modify Nominal Values for a Block Variable.

Assumptions and Limitations

• The pipe wall is perfectly rigid.

• The flow is fully developed. Friction losses and heat transfer do not include entrance effects.

• The effect of gravity is negligible.

• Fluid inertia is negligible.

• This block does not model supersonic flow.

Examples

Pneumatic Actuation Circuit

How the Foundation Library gas components can be used to model a controlled pneumatic actuator. The Directional Valve is a masked subsystem created from Variable Local Restriction (G) blocks to model the opening and closing of the flow paths. The Double-Acting Actuator is a masked subsystem created from Translational Mechanical Converter (G) blocks to model the interface between the gas network and the mechanical translational network.

• Open Model

Pneumatic Motor Circuit

How a pneumatic vane motor can be modeled using the Simscape™ language. The Pneumatic Motor component is built using the Simscape Foundation gas domain. It inherits from the foundation.gas.two_port_steady base class, which contains common equations that implement the upwind energy flow rate and the gas properties at the ports. The Pneumatic Motor subclass implements equations that describe behaviors specific to the component, such as the motor torque and flow rate characteristics and the mass and energy balance. The Pneumatic Motor block is inserted into the model using the Simscape Component block without the need to generate a separate library.

• Open Model

Ports

Conserving

A — Inlet or outlet

gas

Gas conserving port associated with the inlet or outlet of the pipe. This block has no intrinsic directionality.

B — Inlet or outlet

gas

Gas conserving port associated with the inlet or outlet of the pipe. This block has no intrinsic directionality.

H — Temperature of pipe wall

thermal

Thermal conserving port associated with the temperature of the pipe wall. This temperature may differ from the temperature of the gas volume.

Parameters

Geometry

Pipe length — The length of the pipe

5 m (default)

The length of the pipe along the direction of flow.

Cross-sectional area — The internal area of the pipe

0.01 m^2 (default)

The internal area of the pipe normal to the direction of the flow.

Hydraulic diameter — Diameter of an equivalent cylindrical pipe with the same cross-sectional area

0.1 m (default)

Diameter of an equivalent cylindrical pipe with the same cross-sectional area.

Friction and Heat Transfer

Aggregate equivalent length of local resistances — The combined length of all local resistances present in the pipe

0.1 m (default)

The combined length of all local resistances present in the pipe. Local resistances include bends, fittings, armatures, and pipe inlets and outlets. The effect of the local resistances is to increase the effective length of the pipe segment. This length is added to the geometrical pipe length only for friction calculations. The gas volume depends only on the pipe geometrical length, defined by the Pipe length parameter.

Internal surface absolute roughness — Average depth of all surface defects on the internal surface of the pipe

15e-6 m (default)

Average depth of all surface defects on the internal surface of the pipe, which affects the pressure loss in the turbulent flow regime.

Laminar flow upper Reynolds number limit — The Reynolds number above which flow begins to transition from laminar to turbulent

2000 (default)

The Reynolds number above which flow begins to transition from laminar to turbulent. This number equals the maximum Reynolds number corresponding to fully developed laminar flow.

Turbulent flow lower Reynolds number limit — The Reynolds number below which flow begins to transition from turbulent to laminar

4000 (default)

The Reynolds number below which flow begins to transition from turbulent to laminar. This number equals to the minimum Reynolds number corresponding to fully developed turbulent flow.

Shape factor for laminar flow viscous friction — Effect of pipe geometry on the viscous friction losses

64 (default)

Dimensionless factor that encodes the effect of pipe cross-sectional geometry on the viscous friction losses in the laminar flow regime. Typical values are 64 for a circular cross section, 57 for a square cross section, 62 for a rectangular cross section with an aspect ratio of 2, and 96 for a thin annular cross section [1].

Nusselt number for laminar flow heat transfer — Ratio of convective to conductive heat transfer

3.66 (default)

Ratio of convective to conductive heat transfer in the laminar flow regime. Its value depends on the pipe cross-sectional geometry and pipe wall thermal boundary conditions, such as constant temperature or constant heat flux. Typical value is 3.66, for a circular cross section with constant wall temperature [2].

References

[1] White, F. M., Fluid Mechanics. 7th Ed, Section 6.8. McGraw-Hill, 2011.

[2] Cengel, Y. A., Heat and Mass Transfer – A Practical Approach. 3rd Ed, Section 8.5. McGraw-Hill, 2007.

Extended Capabilities

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

Version History

Introduced in R2016b

R2024a: Numerical improvement for low-flow computation in pipes

To improve accuracy for pressure drop inside the pipe at low-speed flows, the momentum balance computations in the Pipe (G) block have been changed slightly. There may be slight differences in simulation results compared to previous releases, especially when Mach number is low.