# Pipe (TL)

Closed conduit that transports fluid between thermal liquid components

**Libraries:**

Simscape /
Fluids /
Thermal Liquid /
Pipes & Fittings

## Description

The Pipe (TL) block represents thermal liquid flow through a pipe. The
block finds the temperature across the pipe from the differential between ports, pipe
elevation, and any additional heat transfer at port **H**.

The pipe can have a constant or varying elevation between ports **A**
and **B**. For a constant elevation differential, use the
**Elevation gain from port A to port B** parameter. You can specify
a variable elevation by setting **Elevation gain specification** to
`Variable`

. This exposes physical signal port
**EL**.

You can choose to include the effects of fluid dynamic compressibility, inertia, and wall flexibility. When the block includes these phenomena, it calculates the flow properties for each number of pipe segments that you specify.

### Pipe Geometry

Use the **Cross-sectional geometry** parameter to specify the
shape of the pipe.

**Circular**

The nominal hydraulic diameter, *D _{N}*,
and the pipe diameter,

*d*, are both equal to the is the value of the

_{circle}**Pipe diameter**parameter. The pipe cross-sectional area is $${S}_{N}=\frac{\pi}{4}{d}_{circle}^{2}.$$

**Annular**

The nominal hydraulic diameter is the difference between the **Pipe
outer diameter** and **Pipe inner diameter**
parameters *D _{N}* =

*d*–

_{outer}*d*. The pipe cross-sectional area is $${S}_{N}=\frac{\pi}{4}\left({d}_{outer}^{2}-{d}_{inner}^{2}\right).$$

_{inner}**Rectangular**

The nominal hydraulic diameter is

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

where:

*h*is the is the value of the**Pipe height**parameter.*w*is the is the value of the**Pipe width**parameter.

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

**Elliptical**

The nominal hydraulic diameter is

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

where:

*a*is the is the value of the_{maj}**Pipe major axis**parameter.*b*is the is the value of the_{min}**Pipe minor axis**parameter.

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

**Isosceles Triangular**

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*is the is the value of the_{side}**Pipe side length**parameter.*θ*is the is the value of the**Pipe vertex angle**parameter.

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

**Custom**

When the **Cross-sectional geometry** parameter is
`Custom`

, you can specify the pipe cross-sectional
area with the **Cross-sectional area** parameter. The nominal
hydraulic diameter is the value of the **Hydraulic diameter**
parameter.

### Pipe Flexibility

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 setting may not
result in physical results for noncircular cross-sectional areas undergoing high
pressure relative to atmospheric pressure. When you model flexible walls, you can
use the **Volumetric expansion specification** parameter to specify
the volumetric expansion of the pipe cross-sectional area.

When the **Volumetric expansion specification** parameter is
`Cross-sectional area vs. pressure`

, the change in
volume is

$$\dot{V}=L\left(\frac{A-S}{\tau}\right),$$

where:

$$A={S}_{N}+{K}_{ps}\left(p-{p}_{atm}\right).$$

*L*is the**Pipe length**parameter.*S*is the nominal pipe cross-sectional area defined for each shape._{N}*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 gauge pressure to cross-sectional area gain**parameter.To calculate

*K*assuming uniform elastic deformation of a thin-walled, open-ended cylindrical pipe, use_{ps}$${K}_{ps}=\frac{\Delta D}{\Delta p}=\frac{\pi {D}_{N}^{3}}{4tE},$$

where

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

When the **Volumetric expansion specification** parameter is
`Cross-sectional area vs. pressure - Tabulated`

, the
block uses the same equation for $$\dot{V}$$ as the ```
Cross-sectional area vs.
pressure
```

setting. The block calculates *A* with
the table lookup function

$$A={S}_{N}+tablelookup\left({p}_{ps},{A}_{ps},(p-{p}_{atm}),interpolation=linear,extrapolation=linear\right),$$

where *p _{ps}* is the

**Static gauge pressure vector**parameter and

*A*is the

_{ps}**Cross sectional area gain vector**parameter.

When the **Volumetric expansion specification** parameter is
`Hydraulic diameter vs. pressure`

, the change in volume is

$$\dot{V}=\frac{\pi}{2}DL\left(\frac{{D}_{static}-D}{\tau}\right),$$

where:

$${D}_{static}={D}_{N}+{K}_{pd}\left(p-{p}_{atm}\right).$$

*D*is the nominal hydraulic diameter defined for each shape._{N}*D*is the current pipe hydraulic diameter.*K*is the_{pd}**Static gauge pressure to hydraulic diameter gain**parameter. To calculate*K*assuming uniform elastic deformation of a thin-walled, open-ended cylindrical pipe, use_{ps}$${K}_{pd}=\frac{\Delta D}{\Delta p}=\frac{{D}_{N}^{2}}{2tE}.$$

When the **Volumetric expansion specification** parameter is
`Based on material properties`

, the block uses the same
equation for $$\dot{V}$$ as the `Hydraulic diameter vs. pressure`

setting but calculates *D _{static}* depending
on the value of the

**Material behavior**parameter

$${\text{D}}_{static}={D}_{N}\left(1+{\u03f5}_{hoop}\right).$$

This parameterization assumes a cylindrical thin-walled pressure vessel where $${\sigma}_{radial}=0.$$

When the **Material behavior** parameter is ```
Linear
elastic
```

,

$${\u03f5}_{hoop}=\frac{1}{E}\left[{\sigma}_{hoop}-v{\sigma}_{longitudinal}\right],$$

where:

*E*is the value of the**Young's modulus**parameter.*v*is the value of the**Poisson's ratio**parameter.$${\sigma}_{hoop}=\frac{pD}{2t}$$, where

*t*is the value of the**Pipe wall thickness**parameter.$${\sigma}_{longitudinal}=\frac{pD}{4t}.$$

When the **Material behavior** parameter is
`Multilinear elastic`

, the block calculates the von
Mises stress, *σ _{v}*, which simplifies to $${\sigma}_{v}=\sqrt{\frac{3}{4}}\frac{pD}{2t}$$, to determine the equivalent strain. The hoop strain is

$${\u03f5}_{hoop}={\u03f5}_{hoop}^{elastic}+{\u03f5}_{hoop}^{plastic}$$

$$\begin{array}{l}{\u03f5}_{hoop}^{elastic}=\frac{1}{E}\left[{\sigma}_{hoop}-v{\sigma}_{longitudinal}\right]\\ {\u03f5}_{hoo{p}_{i,j}}^{plastic}=\frac{3}{2}\left(\frac{1}{{E}_{s}}-\frac{1}{E}\right){S}_{i,j}\end{array}$$

where:

The block calculates the Young's Modulus,

*E*, from the first elements of the**Stress vector**and**Strain vector**parameters.$${E}_{S}=\frac{{\sigma}_{total}}{{\u03f5}_{total}}$$, where

*σ*and_{total}*ε*are the equivalent total stress and the equivalent total strain, respectively. The block calculates the equivalent total strain from the von Mises stress and the stress-strain curve._{total}$${\text{S}}_{i,j}={\sigma}_{i,j}-\left[\frac{{\sigma}_{hoop}+{\sigma}_{longitudinal}+{\sigma}_{radial}}{3}\right]{\delta}_{i,j},$$ where

*σ*are the elements of the Cauchy stress tensor._{i,j}

If you do not model flexible walls, *S _{N}*
=

*S*and

*D*=

_{N}*D*.

**Pipe Wall Thermal Expansion**

If you select **Pipe thermal expansion**, the block models
the thermal expansion of the pipe wall using these assumptions:

The pipe material is isotropic.

The Biot number of the pipe is less than 0.1 and the pipe can be modeled with lumped thermal capacitance.

The temperature change and pipe deformations are small enough that a first order approximation for area expansion is accurate.

When the **Material behavior** parameter is
`Cross-sectional area vs. pressure`

,
`Cross-sectional area vs. pressure - Tabulated`

, or
`Hydraulic diameter vs. pressure`

and you select
**Pipe thermal expansion**, the block adds a thermal
expansion term when calculating area or diameter.

When **Material behavior** is ```
Cross-sectional
area vs. pressure
```

,

$$A={S}_{N}+{K}_{ps}\left(p-{p}_{atm}\right)+{S}_{N}2\alpha \Delta T,$$

where:

*ɑ*is the value of the**Coefficient of thermal expansion**parameter.$$\Delta T={T}_{I}-{T}_{ref}.$$

*T*is the fluid temperature at the internal node of the block._{I}*T*is the value of the_{ref}**Thermal expansion reference temperature**parameter.

When **Material behavior** is ```
Cross-sectional
area vs. pressure - Tabulated
```

,

$$A={S}_{N}+tablelookup\left({p}_{ps},{A}_{ps},(p-{p}_{atm}),interpolation=linear,extrapolation=linear\right)+{S}_{N}2\alpha \Delta T.$$

When **Material behavior** is ```
Hydraulic diameter
vs. pressure
```

,

$${D}_{static}={D}_{N}+{K}_{pd}(p-{p}_{atm})+{D}_{N}\alpha \Delta T.$$

When the **Material behavior** parameter is
`Multilinear elastic`

and you select **Pipe
thermal expansion**, the block calculates
*D _{static}* as

$${D}_{static}={D}_{N}\left(1+{\u03f5}_{hoop}+{\u03f5}_{thermal}\right),$$

where $${\u03f5}_{thermal}=\alpha \Delta T.$$

### Heat Transfer at the Pipe Wall

You can include heat transfer to and from the pipe walls in multiple ways. There
are two analytical models: the `Gnielinski correlation`

,
which models the Nusselt number as a function of the Reynolds and Prandtl numbers
with predefined coefficients, and the ```
Dittus-Boelter correlation -
Nusselt = a*Re^b*Pr^c
```

, which models the Nusselt number as a
function of the Reynolds and Prandtl numbers with user-defined coefficients.

The ```
Nominal temperature differential vs. nominal mass flow
rate
```

, ```
Tabulated data - Colburn factor vs. Reynolds
number
```

, and ```
Tabulated data - Nusselt number vs.
Reynolds number & Prandtl number
```

are lookup table
parameterizations based on user-supplied data.

Heat transfer between the fluid and pipe wall occurs through convection,
*Q _{Conv}* and conduction,

*Q*, where the net heat flow rate,

_{Cond}*Q*is

_{H}*Q*=

_{H}*Q*+

_{Conv}*Q*.

_{Cond}Heat transfer due to conduction is:

$${Q}_{\text{Cond}}=\frac{{k}_{\text{I}}{S}_{\text{H}}}{D}\left({T}_{\text{H}}-{T}_{\text{I}}\right),$$

where:

*D*is the nominal hydraulic diameter,*D*, if the pipe walls are rigid, and is the pipe steady-state diameter,_{N}*D*, if the pipe walls are flexible._{S}*k*is the thermal conductivity of the thermal liquid, defined internally for each pipe segment._{I}*S*is the surface area of the pipe wall._{H}*T*is the pipe wall temperature._{H}*T*is the fluid temperature at the internal node of the block._{I}

Heat transfer due to convection is:

$${Q}_{\text{Conv}}={c}_{\text{p,Avg}}\left|{\dot{m}}_{\text{Avg}}\right|\left({T}_{\text{H}}-{T}_{\text{In}}\right)\left[1-\text{exp}\left(-\frac{h{S}_{\text{H}}}{{c}_{\text{p,Avg}}\left|{\dot{m}}_{\text{Avg}}\right|}\right)\right],$$

where:

*c*_{p, Avg}is the average fluid specific heat which the block calculates using a lookup table.$$\dot{m}$$

_{Avg}is the average mass flow rate through the pipe.*T*is the fluid inlet port temperature._{In}*h*is the pipe heat transfer coefficient.

The heat transfer coefficient *h* is:

$$h=\frac{\text{Nu}{k}_{\text{Avg}}}{D},$$

except when parameterizing by ```
Nominal temperature
differential vs. nominal mass flow rate
```

, where
*k _{Avg}* is the average thermal
conductivity of the thermal liquid over the entire pipe and

*Nu*is the average Nusselt number in the pipe.

**Analytical Parameterizations**

When **Heat transfer parameterization** is set to
`Gnielinski correlation`

and the flow is turbulent,
the average Nusselt number is calculated as:

$$\text{Nu}=\frac{\raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\left(\text{Re}-1000\right)\text{Pr}}{1+12.7{\left(\text{}\raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\right)}^{1/2}\left({\text{Pr}}^{\text{2/3}}-1\right)},$$

where:

*f*is the average Darcy friction factor, according to the Haaland correlation:$$f={\left\{-1.8{\text{log}}_{\text{10}}\left[\frac{6.9}{\text{Re}}+{\left(\frac{{\u03f5}_{\text{R}}}{3.7D}\right)}^{1.11}\right]\right\}}^{\text{-2}},$$

where

*ε*_{R}is the pipe**Internal surface absolute roughness**.*Re*is the Reynolds number.*Pr*is the Prandtl number.

When the flow is laminar, the data from [1]
determines how the Nusselt number depends on the **Cross-sectional
geometry** parameter:

When

**Cross-sectional geometry**is`Circular`

, the Nusselt number is 3.66.When

**Cross-sectional geometry**is`Annular`

, the block calculates the Nusselt number from tabulated data using a lookup table with linear interpolation and nearest extrapolation.$$\frac{{D}_{inner}}{{D}_{outer}}$$ Nusselt number 1/20 17.46 1/10 11.56 1/4 7.37 1/2 5.74 1 4.86 The block adjusts the calculated Nusselt number with a correction factor $$\text{F=0}\text{.86}{\left(\frac{{D}_{outer}}{{D}_{inner}}\right)}^{0.16}.$$

When

**Cross-sectional geometry**is`Rectangular`

, the block calculates the Nusselt number from tabulated data using a lookup table with linear interpolation and nearest extrapolation.$$\frac{\mathrm{min}\left(h,w\right)}{\mathrm{max}\left(h,w\right)}$$ Nusselt number 0 7.54 1/8 5.60 1/6 5.14 1/4 4.44 1/3 3.96 1/2 3.39 1 2.98 When

**Cross-sectional geometry**is`Elliptical`

, the block calculates the Nusselt number from tabulated data using a lookup table with linear interpolation and nearest extrapolation.$$\frac{{b}_{min}}{{a}_{maj}}$$ Nusselt number 1/16 3.65 1/8 3.72 1/4 3.79 1/2 3.74 1 3.66 When

**Cross-sectional geometry**is`Isosceles triangular`

, the block calculates the Nusselt number from tabulated data using a lookup table with linear interpolation and nearest extrapolation.θ Nusselt number 10π/180 1.61 30π/180 2.26 60π/180 2.47 90π/180 2.34 120π/180 2.00 When

**Cross-sectional geometry**is`Custom`

, the Nusselt number is the value of the**Nusselt number for laminar flow heat transfer**parameter.

When **Heat transfer parameterization** is set to
`Dittus-Boelter correlation`

and the flow is
turbulent, the average Nusselt number is calculated as:

$$\text{Nu}=a{\text{Re}}_{}^{b}{\text{Pr}}_{}^{c},$$

where:

*a*is the value of the**Coefficient a**parameter.*b*is the value of the**Exponent b**parameter.*c*is the value of the**Exponent c**parameter.

The block default Dittus-Boelter correlation is:

$$\text{Nu}=0.023{\text{Re}}_{}^{0.8}{\text{Pr}}_{}^{0.4}.$$

When the flow is laminar, the Nusselt number depends on the
**Cross-sectional geometry** parameter.

**Parameterization By Tabulated Data**

When the **Heat transfer parameterization** parameter is set
to ```
Tabulated data - Colburn factor vs. Reynolds
number
```

, the average Nusselt number is calculated as:

$$\text{Nu}={\text{J}}_{\text{M}}(\text{Re}){\text{RePr}}_{}^{1/3}.$$

where *J*_{M} is the
Colburn-Chilton factor.

When the **Heat transfer parameterization** parameter is set
to ```
Tabulated data - Nusselt number vs. Reynolds number &
Prandtl number
```

, the Nusselt number is interpolated from the
three-dimensional array of average Nusselt number as a function of both average
Reynolds number and average Prandtl number:

$$\text{Nu}=\text{Nu}(\text{Re},\text{Pr}).$$

When the **Heat transfer parameterization** parameter is set
to ```
Nominal temperature difference vs. nominal mass flow
rate
```

and the flow is turbulent, the heat transfer coefficient
is calculated as:

$$h=\frac{{h}_{\text{N}}{D}_{\text{N}}^{1.8}}{{\dot{m}}_{\text{N}}^{0.8}}\frac{{\dot{m}}_{\text{Avg}}^{0.8}}{{D}^{1.8}},$$

where:

$$\dot{m}$$

_{N}is the value of the**Nominal mass flow rate**parameter.$$\dot{m}$$

_{Avg}is the average mass flow rate:$${\dot{m}}_{Avg}=\frac{{\dot{m}}_{\text{A}}-{\dot{m}}_{\text{B}}}{2}.$$

*h*_{N}is the nominal heat transfer coefficient, which is calculated as:$${h}_{\text{N}}=\frac{{\dot{m}}_{\text{N}}{c}_{\text{p,N}}}{{S}_{\text{H,N}}}\text{ln}\left(\frac{{T}_{\text{H,N}}-{T}_{\text{In,N}}}{{T}_{\text{H,N}}-{T}_{\text{Out,N}}}\right),$$

where:

*S*is the nominal wall surface area._{H,N}*T*is the value of the_{H,N}**Nominal wall temperature**parameter.*T*is the value of the_{In,N}**Nominal inflow temperature**parameter.*T*is the value of the_{Out,N}**Nominal outflow temperature**parameter.

This relationship is based on the assumption that the Nusselt number is proportional to the Reynolds number:

$$\frac{hD}{k}\propto {\left(\frac{\dot{m}D}{S\mu}\right)}^{0.8}.$$

If the pipe walls are rigid, the expression for the heat transfer coefficient becomes:

$$h=\frac{{h}_{\text{N}}}{{\dot{m}}_{\text{N}}^{0.8}}{\dot{m}}_{Avg}^{0.8}.$$

### Pressure Loss Due to Friction

**Haaland Correlation**

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 value of the**Laminar friction constant for Darcy friction factor**parameter, which you can define when the**Cross-sectional geometry**parameter is`Custom`

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

_{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.

**Nominal Pressure Drop vs. Nominal Mass Flow Rate**

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

**Tabulated Data – Darcy Friction Factor vs. Reynolds Number**

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.

### Momentum Balance

The pressure differential over the pipe is due to the pressure at the pipe ports, friction at the pipe walls, and hydrostatic changes due to any change in elevation:

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

where:

*p*is the pressure at a port_{A}**A**.*p*is the pressure at a port_{B}**B**.*Δp*is the pressure differential due to viscous friction,_{f}*Δp*._{f,A}+Δp_{f,B}*g*is the value of the**Gravitational acceleration**parameter or the signal at port**G**.*Δz*the elevation differential between port**A**and port**B**, or*z*._{A}- z_{B}*ρ*is the internal fluid density, which is measured at each pipe segment. If fluid dynamic compressibility is not modeled, this is:_{I}$${p}_{\text{I}}=\frac{{p}_{\text{A}}+{p}_{\text{B}}}{2}.$$

When fluid inertia is not modeled, the momentum balance between port
**A** and internal node I is:

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

When fluid inertia is not modeled, the momentum balance between port
**B** and internal node I is:

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

When fluid inertia is modeled, the momentum balance between port
**A** and internal node I is:

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

where:

$$\ddot{m}$$

_{A}is the fluid inertia at port**A**.*L*is the value of the**Pipe length**parameter.*S*is the value of the**Nominal cross-sectional area**parameter.

When fluid inertia is modeled, the momentum balance between port
**B** and internal node I is:

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

where

$$\ddot{m}$$_{B} is the fluid inertia at port
**B**.

### Pipe Discretization

You can divide the pipe into multiple segments. If a pipe has more than one segment, the mass flow, energy flow, and momentum balance equations are calculated for each segment. Having multiple pipe segments can allow you to track changes to variables such as fluid density when fluid dynamic compressibility is modeled.

If you would like to capture specific phenomena in your application, such as water
hammer, choose a number of segments that provides sufficient resolution of the
transient. The following formula, from the Nyquist sampling theorem, provides a rule
of thumb for pipe discretization into a minimum of *N* segments:

$$N=2L\frac{f}{c},$$

where:

*L*is the**Pipe length**.*f*is the transient frequency.*c*is the speed of sound.

For some applications, you may need to connect Pipe (TL) blocks in series. For example, you may require multiple pipe segments to define a thermal boundary condition along the length of a pipe. In this case, model the pipe segments by using a Pipe (TL) block for each segment and use the thermal ports to set the thermal boundary condition.

### Mass Balance

For a rigid pipe with an incompressible fluid, the pipe mass conversation equation is:

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

where:

$$\dot{m}$$

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

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

For a flexible pipe with an incompressible fluid, the pipe mass conservation equation is:

$${\dot{m}}_{\text{A}}+{\dot{m}}_{\text{B}}={\rho}_{\text{I}}\dot{V},$$

where:

*ρ*is the thermal liquid density at internal node I. Each pipe segment has an internal node._{I}$$\dot{V}$$ is the rate of deformation of the pipe volume.

For a flexible pipe with a compressible fluid, the mass within the pipe can change with pressure and temperature. The bulk modulus and thermal expansion coefficient of the thermal liquid account for this dependence and the pipe mass conservation equation is:

$${\dot{m}}_{\text{A}}+{\dot{m}}_{\text{B}}={\rho}_{\text{I}}\dot{V}+{\rho}_{\text{I}}V\left(\frac{{\dot{p}}_{\text{I}}}{{\beta}_{\text{I}}}+{\alpha}_{\text{I}}{\dot{T}}_{\text{I}}\right),$$

where:

*p*is the thermal liquid pressure at the internal node I._{I}$$\dot{T}$$

_{I}is the rate of change of the thermal liquid temperature at the internal node I.*β*is the thermal liquid bulk modulus._{I}*α*is the liquid thermal expansion coefficient._{I}

### Energy Balance

The energy accumulation rate in the pipe at internal node I is defined as:

$$\stackrel{.}{E}={\varphi}_{\text{A}}+{\varphi}_{\text{B}}+{Q}_{\text{H}}-{\dot{m}}_{Avg}g\Delta z,$$

where:

*ϕ*is the energy flow rate at port_{A}**A**.*ϕ*is the energy flow rate at port_{B}**B**.*Q*is the heat transfer through the pipe wall._{H}

If the fluid is incompressible, the expression for energy accumulation rate is

$$\dot{E}={\rho}_{\text{I}}{c}_{{p}_{I}}V\frac{d{T}_{I}}{dt},$$

where:

*c*is the fluid specific heat at the internal node of the block._{pI}*V*is the pipe volume.

If the fluid is compressible, the expression for energy accumulation rate is

$$\dot{E}={\frac{\partial ({\rho}_{\text{I}}{u}_{I})}{\partial p}|}_{T}\frac{d{p}_{I}}{dt}V+{\frac{\partial ({\rho}_{\text{I}}{u}_{I})}{\partial T}|}_{p}\frac{d{T}_{I}}{dt}V,$$

where:

$$\begin{array}{l}{\frac{\partial ({\rho}_{\text{I}}{u}_{I})}{\partial p}|}_{T}=\left(\frac{{\rho}_{I}{h}_{I}}{{\beta}_{I}}-{T}_{I}{\alpha}_{I}\right)V\\ {\frac{\partial ({\rho}_{\text{I}}{u}_{I})}{\partial T}|}_{p}=\left({c}_{{p}_{I}}-{h}_{I}{\alpha}_{I}\right)V\end{array}$$

and *h _{I}* is the specific
enthalpy at the internal node of the block.

If the fluid is compressible and the pipe walls are flexible, the expression for energy accumulation rate is

$$\dot{E}={\frac{\partial ({\rho}_{\text{I}}{u}_{I})}{\partial p}|}_{T}\frac{d{p}_{I}}{dt}V+{\frac{\partial ({\rho}_{\text{I}}{u}_{I})}{\partial T}|}_{p}\frac{d{T}_{I}}{dt}V+{\rho}_{\text{I}}{h}_{I}\frac{dV}{dt}.$$

## Ports

### Input

### Conserving

## Parameters

## References

[1] Budynas R. G. Nisbett J. K. & Shigley J. E. (2004). Shigley's mechanical engineering design (7th ed.). McGraw-Hill.

[2] Cengel, Y.A. *Heat and Mass Transfer: A Practical Approach
*(3^{rd} edition). New York, McGraw-Hill,
2007

[3] Ju Frederick D., Butler Thomas A., Review of Proposed Failure Criteria for Ductile Materials (1984) Los Alamos National Laboratory.

[4] Hencky H (1924) Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen. Z Angew Math Mech 4:323–335

[5] Jahed H, “A Variable Material Property Approach for Elastic-Plastic Analysis of Proportional and Non-proportional Loading, (1997) University of Waterloo

## Extended Capabilities

## Version History

**Introduced in R2016a**