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Pipe segment in an isothermal liquid network

**Library:**Simscape / Fluids / Isothermal Liquid / Pipes & Fittings

The Pipe (IL) block models flow in a rigid or flexible-walled pipe with losses due to
wall friction. The effects of dynamic compressibility, fluid inertia, and pipe elevation
can be optionally modeled. You can define multiple pipe segments and set the liquid
pressure for each segment. By segmenting the pipe and setting **Fluid
inertia** to `On`

, you can model events such as
water hammer in your system.

The pipe block can be divided into segments with the **Number of
segments** parameter. When the pipe is composed of a number of
segments, the pressure in each segment is calculated based on the inlet pressure and
the effect on the segment mass flow rate of the fluid compressibility and wall
flexibility, if applicable. The fluid volume in each segment remains fixed. For a
two-segment pipe, the pressure evolves linearly with respect to the pressure defined
at ports **A** and **B**. For a pipe with three or
more segments, you can specify the fluid pressure in each segment in vector or
scalar form in the **Initial liquid pressure** parameter. The
scalar form will apply a constant value over all segments.

You can model flexible walls for all cross-sectional geometries. When you set
**Pipe wall specification** to
`Flexible`

, the block assumes uniform expansion
along all directions and preserves the defined cross-sectional shape. This may
not result in physical results for noncircular cross-sectional areas undergoing
high pressure relative to atmospheric pressure. Two options are available for
modeling the volumetric expansion of a cross-sectional area:

`Cross-sectional area vs. pressure`

, where the change in volume is modeled by:$$\dot{V}=L\left(\frac{{S}_{N}+{K}_{ps}\left(p-{p}_{atm}\right)-S}{\tau}\right),$$

where:

*L*is the**Pipe length**.*S*_{nom}is the nominal pipe cross-sectional area defined for each shape.*S*is the current pipe cross-sectional area.*p*is the internal pipe pressure.*p*is the atmospheric pressure._{atm}*K*_{ps}is the**Static pressure-cross sectional area gain**.Assuming uniform elastic deformation of a thin-walled, open-ended cylindrical pipe, you can calculate

*K*_{ps}as:$${K}_{ps}=\frac{\Delta D}{\Delta p}=\frac{\pi {D}_{N}^{3}}{4tE},$$

where

*t*is the pipe wall thickness and*E*is Young's modulus.*τ*is the**Volumetric expansion time constant**.

`Hydraulic diameter vs. pressure`

, where the change in volume is modeled by:$$\dot{V}=\frac{\pi}{2}DL\left(\frac{{D}_{N}+{K}_{pd}\left(p-{p}_{atm}\right)-D}{\tau}\right),$$

where:

*D*_{N}is the nominal hydraulic diameter defined for each shape.*D*is the current pipe hydraulic diameter.*K*_{pd}is the**Static pressure-hydraulic diameter gain**. Assuming uniform elastic deformation of a thin-walled, open-ended cylindrical pipe, you can calculate*K*_{pd}as:$${K}_{pd}=\frac{\Delta D}{\Delta p}=\frac{{D}_{N}^{2}}{2tE}.$$

When flexible walls are not modeled,
*S*_{N} = *S* and
*D*_{N} = *D*.

The nominal hydraulic diameter and the **Pipe diameter**,
*d*_{circle}, are the same. The pipe
cross sectional area is: $${S}_{N}=\frac{\pi}{4}{d}_{circle}^{2}.$$

The nominal hydraulic diameter,
*D*_{h,nom}, is the difference between
the **Pipe outer diameter** and **Pipe inner
diameter**, *d*_{o} –
*d*_{i}. The pipe cross sectional area
is $${S}_{N}=\frac{\pi}{4}\left({d}_{{}_{o}}^{2}-{d}_{{}_{i}}^{2}\right).$$

The nominal hydraulic diameter is:

$${D}_{N}=\frac{2hw}{h+w},$$

where:

*h*is the**Pipe height**.*w*is the**Pipe width**.

The pipe cross sectional area is $${S}_{N}=wh.$$

The nominal hydraulic diameter is:

$${D}_{N}=2{a}_{maj}{b}_{\mathrm{min}}\frac{\left(64-16{\left(\frac{{a}_{maj}-{b}_{\mathrm{min}}}{{a}_{maj}+{b}_{\mathrm{min}}}\right)}^{2}\right)}{\left({a}_{maj}+{b}_{\mathrm{min}}\right)\left(64-3{\left(\frac{{a}_{maj}-{b}_{\mathrm{min}}}{{a}_{maj}+{b}_{\mathrm{min}}}\right)}^{4}\right)},$$

where:

*a*_{maj}is the**Pipe major axis**.*b*_{min}is the**Pipe minor axis**.

The pipe cross sectional area is $${S}_{N}=\frac{\pi}{4}{a}_{maj}{b}_{\mathrm{min}}.$$

The nominal hydraulic diameter is:

$${D}_{N}={l}_{side}\frac{\mathrm{sin}\left(\theta \right)}{1+\mathrm{sin}\left(\frac{\theta}{2}\right)}$$

where:

*l*_{side}is the**Pipe side length**.*θ*is the**Pipe vertex angle**.

The pipe cross sectional area is $${S}_{N}=\frac{{l}_{side}^{2}}{2}\mathrm{sin}\left(\theta \right).$$

The analytical Haaland correlation models losses due to wall friction either
by *aggregate equivalent length*, which accounts for
resistances due to nonuniformities as an added straight-pipe length that results
in equivalent losses, or by *local loss coefficient*, which
directly applies a loss coefficient for pipe nonuniformities.

When the **Local resistances specification** parameter is set
to `Aggregate equivalent length`

and the flow in the
pipe is lower than the **Laminar flow upper Reynolds number
limit**, the pressure loss over all pipe segments is:

$$\Delta {p}_{f,A}=\frac{\upsilon \lambda}{2{D}^{2}S}\frac{L+{L}_{add}}{2}{\dot{m}}_{A},$$

$$\Delta {p}_{f,B}=\frac{\upsilon \lambda}{2{D}^{2}S}\frac{L+{L}_{add}}{2}{\dot{m}}_{B},$$

where:

*ν*is the fluid kinematic viscosity.*λ*is the**Laminar friction constant for Darcy friction factor**, which you can define when**Cross-sectional geometry**is set to`Custom`

and is otherwise equal to 64.*D*is the pipe hydraulic diameter.*L*_{add}is the**Aggregate equivalent length of local resistances**.$$\dot{m}$$

_{A}is the mass flow rate at port**A**.$$\dot{m}$$

_{B}is the mass flow rate at port**B**.

When the Reynolds number is greater than the **Turbulent
flow lower Reynolds number limit**, the pressure loss in the pipe is:

$$\Delta {p}_{f,A}=\frac{f}{2{\rho}_{I}{S}^{2}}\frac{L+{L}_{add}}{2}{\dot{m}}_{A}\left|{\dot{m}}_{A}\right|,$$

$$\Delta {p}_{f,B}=\frac{f}{2{\rho}_{I}{S}^{2}}\frac{L+{L}_{add}}{2}{\dot{m}}_{B}\left|{\dot{m}}_{B}\right|,$$

where:

*f*is the Darcy friction factor. This is approximated by the empirical Haaland equation and is based on the**Surface roughness specification**,*ε*, and pipe hydraulic diameter:$$f={\left\{-1.8{\mathrm{log}}_{10}\left[\frac{6.9}{\mathrm{Re}}+{\left(\frac{\epsilon}{3.7{D}_{h}}\right)}^{1.11}\right]\right\}}^{-2},$$

Pipe roughness for brass, lead, copper, plastic, steel, wrought iron, and galvanized steel or iron are provided as ASHRAE standard values. You can also supply your own

**Internal surface absolute roughness**with the`Custom`

setting.*ρ*_{I}is the internal fluid density.

When the **Local resistances specification** parameter is set
to `Local loss coefficient`

and the flow in the pipe is
lower than the **Laminar flow upper Reynolds number limit**,
the pressure loss over all pipe segments is:

$$\Delta {p}_{f,A}=\frac{\upsilon \lambda}{2{D}^{2}S}\frac{L}{2}{\dot{m}}_{A}.$$

$$\Delta {p}_{f,B}=\frac{\upsilon \lambda}{2{D}^{2}S}\frac{L}{2}{\dot{m}}_{B}.$$

When the Reynolds number is greater than the
**Turbulent flow lower Reynolds number limit**, the
pressure loss in the pipe is:

$$\Delta {p}_{f,A}=\left(\frac{f\frac{L}{2}}{D}+{C}_{loss,total}\right)\frac{1}{2{\rho}_{I}{S}^{2}}{\dot{m}}_{A}\left|{\dot{m}}_{A}\right|,$$

$$\Delta {p}_{f,B}=\left(\frac{f\frac{L}{2}}{D}+{C}_{loss,total}\right)\frac{1}{2{\rho}_{I}{S}^{2}}{\dot{m}}_{B}\left|{\dot{m}}_{B}\right|,$$

where *C*_{loss,total}
is the loss coefficient, which can be defined in the **Total local loss
coefficient** parameter as either a single coefficient or the sum
of all loss coefficients along the pipe.

The Nominal Pressure Drop vs. Nominal Mass Flow Rate parameterization characterizes losses with a loss coefficient for rigid or flexible walls. When the fluid is incompressible, the pressure loss over the entire pipe due to wall friction is:

$$\Delta {p}_{f,A}={K}_{p}{\dot{m}}_{A}\sqrt{{\dot{m}}_{A}^{2}+{\dot{m}}_{th}^{2}},$$

where *K*_{p} is:

$${K}_{p}=\frac{\Delta {p}_{N}}{{\dot{m}}_{N}^{2}},$$

where:

*Δp*_{N}is the**Nominal pressure drop**, which can be defined either as a scalar or a vector.$${\dot{m}}_{N}$$ is the

**Nominal mass flow rate**, which can be defined either as a scalar or a vector.

When the **Nominal pressure drop** and
**Nominal mass flow rate** parameters are supplied as
vectors, the scalar value *K*_{p} is
determined from a least-squares fit of the vector elements.

Pressure losses due to viscous friction can also be determined from
user-provided tabulated data of the **Darcy friction factor
vector** and the **Reynolds number vector for turbulent
Darcy friction factor** parameters. Linear interpolation is
employed between data points.

For an incompressible fluid, the mass flow into the pipe equals the mass flow out of the pipe:

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

When the fluid is compressible and pipe walls are rigid, the difference between the mass flow into and out of the pipe depends on the fluid density change due to compressibility:

$${\dot{m}}_{A}+{\dot{m}}_{B}={\dot{p}}_{I}\frac{d{\rho}_{I}}{d{p}_{I}}V,$$

When the fluid is compressible and the pipe walls are flexible, the difference between the mass flow into and out of the pipe is based on the change in fluid density due to compressibility, and the amount of fluid accumulated in the newly deformed regions of the pipe:

$${\dot{m}}_{A}+{\dot{m}}_{B}={\dot{p}}_{I}\frac{d{\rho}_{I}}{d{p}_{I}}V+{\rho}_{I}\dot{V}.$$

The changes in momentum between the pipe inlet and outlet
comprises the changes in pressure due to pipe wall friction, which is modeled
according to the **Viscous friction parameterization** and pipe
elevation. For a pipe that does not model fluid inertia, the momentum balance is:

$${p}_{A}-{p}_{I}=\Delta {p}_{f,A}+{\rho}_{I}\frac{\Delta z}{2}g,$$

$${p}_{B}-{p}_{I}=\Delta {p}_{f,B}-{\rho}_{I}\frac{\Delta z}{2}g,$$

where:

*p*_{A}is the pressure at port**A**.*p*_{I}is the fluid volume internal pressure.*p*_{B}is the pressure at port**B**.*Δp*_{f}is the pressure loss due to wall friction, parameterized by the**Viscous friction losses**specification according to the respective port.*Δz*is the pipe elevation. In the case of constant-elevation pipes, this is the**Elevation gain from port A to port B**parameter; otherwise, it is received as a physical signal at port**EL**.*g*is the gravitational acceleration. In the case of a fixed gravitational constant, this is the**Gravitational acceleration**parameter; otherwise, it is received as a physical signal at port**G**.

For a pipe with modeled fluid inertia, the momentum balance is:

$${p}_{A}-{p}_{I}=\Delta {p}_{f,A}+{p}_{I}\frac{\Delta z}{2}g+{\ddot{m}}_{A}\frac{L}{2S},$$

$${p}_{B}-{p}_{I}=\Delta {p}_{f,B}-{p}_{I}\frac{\Delta z}{2}g+{\ddot{m}}_{B}\frac{L}{2S},$$

where:

$$\ddot{m}$$ is the fluid acceleration at its respective port.

*S*is the pipe cross-sectional area.

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