# Pipe (G)

Rigid conduit for gas flow

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## 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:

`$\frac{\partial M}{\partial p}\cdot \frac{d{p}_{I}}{dt}+\frac{\partial M}{\partial T}\cdot \frac{d{T}_{I}}{dt}={\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}$`

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.

• pI is the pressure of the gas volume.

• TI is the temperature of the gas volume.

• t is time.

• $\stackrel{˙}{m}$A and $\stackrel{˙}{m}$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.

• ΦA and ΦB are energy flow rates at ports A and B, respectively.

• QH 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:

• ρI is the density of the gas volume.

• V is the volume of gas.

• hI is the specific enthalpy of the gas volume.

• Z is the compressibility factor.

• R is the specific gas constant.

• cpI 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:

`$\begin{array}{l}{p}_{A}-{p}_{I}={\left(\frac{{\stackrel{˙}{m}}_{A}}{S}\right)}^{2}\cdot \left(\frac{1}{{\rho }_{I}}-\frac{1}{{\rho }_{A}}\right)+\Delta {p}_{AI}\\ {p}_{B}-{p}_{I}={\left(\frac{{\stackrel{˙}{m}}_{B}}{S}\right)}^{2}\cdot \left(\frac{1}{{\rho }_{I}}-\frac{1}{{\rho }_{B}}\right)+\Delta {p}_{BI}\end{array}$`

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.

• ΔpAI and ΔpBI 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:

`$\begin{array}{l}{h}_{A}+\frac{1}{2}{\left(\frac{{\stackrel{˙}{m}}_{A}}{{\rho }_{A}S}\right)}^{2}={h}_{I}+\frac{1}{2}{\left(\frac{{\stackrel{˙}{m}}_{A}}{{\rho }_{I}S}\right)}^{2}\\ {h}_{B}+\frac{1}{2}{\left(\frac{{\stackrel{˙}{m}}_{B}}{{\rho }_{B}S}\right)}^{2}={h}_{I}+\frac{1}{2}{\left(\frac{{\stackrel{˙}{m}}_{B}}{{\rho }_{I}S}\right)}^{2}\end{array}$`

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, ΔpAI and ΔpBI, depend on the flow regime. The Reynolds numbers for each half of the pipe are defined as:

`$\begin{array}{l}{\mathrm{Re}}_{A}=\frac{|{\stackrel{˙}{m}}_{A}|\cdot {D}_{h}}{S\cdot {\mu }_{I}}\\ {\mathrm{Re}}_{B}=\frac{|{\stackrel{˙}{m}}_{B}|\cdot {D}_{h}}{S\cdot {\mu }_{I}}\end{array}$`

where:

• Dh is the hydraulic diameter of the pipe.

• μI 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:

`$\begin{array}{l}\Delta {p}_{A{I}_{lam}}={f}_{shape}\frac{{\stackrel{˙}{m}}_{A}\cdot {\mu }_{I}}{2{\rho }_{I}\cdot {D}_{h}^{2}\cdot S}\cdot \frac{L+{L}_{eqv}}{2}\\ \Delta {p}_{B{I}_{lam}}={f}_{shape}\frac{{\stackrel{˙}{m}}_{B}\cdot {\mu }_{I}}{2{\rho }_{I}\cdot {D}_{h}^{2}\cdot S}\cdot \frac{L+{L}_{eqv}}{2}\end{array}$`

where:

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

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

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

`$\begin{array}{l}\Delta {p}_{A{I}_{tur}}={f}_{Darc{y}_{A}}\frac{{\stackrel{˙}{m}}_{A}\cdot |{\stackrel{˙}{m}}_{A}|}{2{\rho }_{I}\cdot {D}_{h}\cdot {S}^{2}}\cdot \frac{L+{L}_{eqv}}{2}\\ \Delta {p}_{B{I}_{tur}}={f}_{Darc{y}_{B}}\frac{{\stackrel{˙}{m}}_{B}\cdot |{\stackrel{˙}{m}}_{B}|}{2{\rho }_{I}\cdot {D}_{h}\cdot {S}^{2}}\cdot \frac{L+{L}_{eqv}}{2}\end{array}$`

where fDarcy 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:

`$\begin{array}{l}{f}_{Darc{y}_{A}}={\left[-1.8\mathrm{log}\left(\frac{6.9}{{\mathrm{Re}}_{A}}+{\left(\frac{{\epsilon }_{rough}}{3.7{D}_{h}}\right)}^{1.11}\right)\right]}^{-2}\\ {f}_{Darc{y}_{B}}={\left[-1.8\mathrm{log}\left(\frac{6.9}{{\mathrm{Re}}_{B}}+{\left(\frac{{\epsilon }_{rough}}{3.7{D}_{h}}\right)}^{1.11}\right)\right]}^{-2}\end{array}$`

where εrough 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 gas volume is:

`${Q}_{H}={h}_{coeff}\left(\frac{4SL}{{D}_{h}}\right)\left({T}_{H}-{T}_{I}\right)$`

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

`${h}_{coeff}=\left(\frac{N{u}_{A}+N{u}_{B}}{2}\right)\frac{{k}_{I}}{{D}_{h}}$`

The Nusselt numbers, NuA and NuB, depend on the flow regime. The Nusselt numbers in the laminar flow regime are constant and equal to the Nusselt number for laminar flow heat transfer parameter value. The Nusselt numbers in the turbulent flow regime are computed from the Gnielinski correlation:

`$\begin{array}{l}N{u}_{{A}_{tur}}=\frac{\frac{{f}_{Darc{y}_{A}}}{8}\left({\mathrm{Re}}_{A}-1000\right){\mathrm{Pr}}_{I}}{1+12.7\sqrt{\frac{{f}_{Darc{y}_{A}}}{8}}\left({\mathrm{Pr}}_{I}^{2/3}-1\right)}\\ N{u}_{{B}_{tur}}=\frac{\frac{{f}_{Darc{y}_{B}}}{8}\left({\mathrm{Re}}_{B}-1000\right){\mathrm{Pr}}_{I}}{1+12.7\sqrt{\frac{{f}_{Darc{y}_{B}}}{8}}\left({\mathrm{Pr}}_{I}^{2/3}-1\right)}\end{array}$`

where Pr is the Prandtl number, Pr = cpμ/k. k is thermal conductivity.

The Nusselt numbers during the transition region follow a smooth connection between those in the laminar flow regime and those in the turbulent flow regime.

### Choked Flow

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

`$\begin{array}{l}{\stackrel{˙}{m}}_{{A}_{choked}}={\rho }_{A}\cdot {a}_{A}\cdot S\\ {\stackrel{˙}{m}}_{{B}_{choked}}={\rho }_{B}\cdot {a}_{B}\cdot S\end{array}$`

where aA and aB 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:

`$\begin{array}{l}{p}_{{A}_{choked}}={p}_{I}+{\left(\frac{{\stackrel{˙}{m}}_{{A}_{choked}}}{S}\right)}^{2}\cdot \left(\frac{1}{{\rho }_{I}}-\frac{1}{{\rho }_{A}}\right)+\Delta {p}_{A{I}_{choked}}\\ {p}_{{B}_{choked}}={p}_{I}+{\left(\frac{{\stackrel{˙}{m}}_{{B}_{choked}}}{S}\right)}^{2}\cdot \left(\frac{1}{{\rho }_{I}}-\frac{1}{{\rho }_{B}}\right)+\Delta {p}_{B{I}_{choked}}\end{array}$`

ΔpAIchoked and ΔpBIchoked are the pressure losses due to viscous friction, assuming that the choking has occurred. They are computed similar to ΔpAI and ΔpBI, 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 pAunchokedpI, then port A is an inlet and pA = pAunchoked. If pAunchoked < pI, then port A is an outlet and

Similarly, if pBunchokedpI, then port B is an inlet and pB = pBunchoked. If pBunchoked < pI, then port B is an outlet and

### Variables

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

### 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.

• Heat transfer is calculated with respect to the temperature of the fluid volume in the pipe. To model temperature gradient due to heat transfer along a long pipe, connect multiple Pipe (G) blocks in series.

## Ports

### Conserving

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

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

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

## Parameters

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#### Geometry

The length of the pipe along the direction of flow.

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

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

#### Friction and Heat Transfer

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.

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

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.

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.

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].

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.