# Pipe (TL)

Closed conduit for the transport of fluid between thermal liquid components

**Library:**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 optionally 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 Flexibility

The block treats a flexible wall as a uniform radial expansion that maintains the
original pipe cross-sectional shape. You can specify the pipe area in the
**Nominal cross-sectional area** parameter, meaning that there
is no specified cross-sectional geometry modeled by the block. However, the block
uses the pipe hydraulic diameter in heat transfer and pressure loss
calculations.

The deformation of the pipe diameter is calculated as:

$$\dot{D}=\frac{{D}_{\text{S}}-D}{\tau},$$

where:

*D*is the post-deformation, steady-state pipe diameter, and_{S}$${D}_{\text{S}}={D}_{N}+{K}_{c}\left(p-{p}_{atm}\right),$$

where

*K*_{c}is the**Static pressure-diameter compliance**,*p*is the tube pressure, and*p*is the atmospheric pressure. Assuming elastic deformation of a thin-walled, open-ended pipe, you can calculate_{atm}*K*_{c}as:$${K}_{\text{c}}=\frac{{D}^{2}}{2tE},$$

where

*t*is the pipe wall thickness and*E*is Young's modulus.*D*is the nominal pipe diameter, or the diameter previous to deformation:_{N}$${D}_{\text{N}}=\sqrt{\frac{4S}{\pi}},$$

where

*S*is the value of the**Nominal cross-sectional area**parameter.*D*is the**Hydraulic diameter**parameter.*τ*is the**Viscoelastic pressure time constant**parameter.

### 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 look-up table
parameterizations based on user-supplied data.

Heat transfer between the fluid and pipe wall occurs through convection,
*Q*_{Conv} and conduction,
*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**Hydraulic diameter**if the pipe walls are rigid, and is the pipe steady-state diameter,*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, taken at the pipe internal node._{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.$$\dot{m}$$

_{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 Nusselt number is 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 is the **Nusselt number
for laminar flow heat transfer** parameter.

**Parameterization By Tabulated Data**

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

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

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

When **Heat transfer parameterization** 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**Nominal mass flow rate**.$$\dot{m}$$

_{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_{H,N}**Nominal wall temperature**.*T*is the_{In,N}**Nominal inflow temperature**.*T*is the_{Out,N}**Nominal outflow temperature**.

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**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.

**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 **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.

**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**Gravitational acceleration**.*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**Pipe length**.*S*is the**Nominal cross-sectional area**.

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.

In some cases, such as modeling thermal transients along a pipe, it may be better suited to your application to connect multiple Pipe (TL) blocks in series.

### 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 pipe mass conservation equation is: This dependence is captured by the bulk modulus and thermal expansion coefficient of the thermal liquid:

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

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.

### Energy Balance

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

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

where:

*ϕ*is the energy flow rate at port_{A}**A**.*ϕ*is the energy flow rate at port_{B}**B**.*ϕ*is the energy flow rate at port_{H}**H**.

The total energy is defined as:

$$E={\rho}_{\text{I}}{u}_{\text{I}}V,$$

where:

*u*is the fluid specific internal energy at node I._{I}*V*is the pipe volume.

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

$$\dot{E}={\rho}_{\text{I}}V{\left(\frac{du}{dp}\frac{dp}{dt}+\frac{du}{dT}\frac{dT}{dt}\right)}_{\text{I}}.$$

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

$$\dot{E}={\rho}_{\text{I}}V{\left(\frac{du}{dp}\frac{dp}{dt}+\frac{du}{dT}\frac{dT}{dt}\right)}_{\text{I}}+\left({\rho}_{\text{I}}{u}_{\text{I}}+{p}_{\text{I}}\right){\left(\frac{dV}{dt}\right)}_{\text{I}}.$$

## Ports

### Input

### Conserving

## Parameters

## Model Examples

## Extended Capabilities

## Version History

**Introduced in R2016a**