Pipe (TL)

R2026b

Pipe that transports fluid in thermal liquid networks

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Pipe (TL) block

To add a block to a model, double-click the canvas and start typing the block name. Then, select the block from the list.

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_circle, are both equal to the is the value of the 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_inner. The pipe cross-sectional area is $$S_N=\frac{\pi }{4}\left(d_{outer}^2-d_{inner}^2\right).$$

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

$$D_N=2a_{maj}{b}_{min}\frac{\left(64-16{\left(\frac{a_{maj}-b_{min}}{a_{maj}+b_{min}}\right)}^2\right)}{\left(a_{maj}+b_{min}\right)\left(64-3{\left(\frac{a_{maj}-b_{min}}{a_{maj}+b_{min}}\right)}^4\right)},$$

where:

• a_maj is the is the value of the Pipe major axis parameter.

• b_min is the is the value of the 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

$$D_N=l_{side}\frac{\sin \left(\theta \right)}{1+\sin \left(\frac{\theta }{2}\right)},$$

where:

• l_side is the is the value of the Pipe side length parameter.

• θ is the is the value of the Pipe vertex angle parameter.

The pipe cross-sectional area is $$S_N=\frac{l_{side}^2}{2}\sin \left(\theta \right).$$

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_N 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_atm is either the atmospheric pressure or the value of the Environment pressure parameter, depending on the value of the Environment pressure specification parameter.

• K_ps is the Static gauge pressure to cross-sectional area gain parameter.

  To calculate K_ps assuming uniform elastic deformation of a thin-walled, open-ended cylindrical pipe, use

  $$K_{ps}=\frac{\Delta D}{\Delta p}=\frac{\pi D_N^3}{4tE},$$

  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_ps is the 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_N is the nominal hydraulic diameter defined for each shape.

• D is the current pipe hydraulic diameter.

• K_pd is the Static gauge pressure to hydraulic diameter gain parameter. To calculate K_ps assuming uniform elastic deformation of a thin-walled, open-ended cylindrical pipe, use

  $$K_{pd}=\frac{\Delta D}{\Delta p}=\frac{D_N^2}{2tE}.$$

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

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

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

When the Material behavior parameter is Linear elastic,

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

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

$${ϵ}_{hoop}={ϵ}_{hoop}^{elastic}+{ϵ}_{hoop}^{plastic}$$

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

where:

• The block calculates the Young's Modulus, E, from the first elements of the Stress vector and Strain vector parameters.

• $$E_S=\frac{{\sigma }_{total}}{{ϵ}_{total}}$$, where σ_total and ε_total 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.

• $${\text{S}}_{i,j}={\sigma }_{i,j}-\left[\frac{{\sigma }_{hoop}+{\sigma }_{longitudinal}+{\sigma }_{radial}}{3}\right]{\delta }_{i,j},$$ where σ_i,j are the elements of the Cauchy stress tensor.

If you do not model flexible walls, S_N = S and D_N = D.

Pipe Wall Thermal Expansion

If you select Enable 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 Enable 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_I is the fluid temperature at the internal node of the block.

• T_ref is the value of the 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 Enable pipe thermal expansion, the block calculates D_static as

$$D_{static}=D_N\left(1+{ϵ}_{hoop}+{ϵ}_{thermal}\right),$$

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

Heat Transfer at the Pipe Wall

Heat transfer between the fluid and pipe wall occurs through convection, Q_Conv and conduction, Q_Cond, where the net heat flow rate, Q_H is 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_N, if the pipe walls are rigid, and is the pipe steady-state diameter, D_S, if the pipe walls are flexible.

• k_I is the thermal conductivity of the thermal liquid, defined internally for each pipe segment.

• S_H is the surface area of the pipe wall.

• T_H is the pipe wall temperature.

• T_I is the fluid temperature at the internal node of the block.

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_In is the fluid inlet port temperature.

• 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.

Nusselt Number Parameterizations by Correlations

The block calculates the Nusselt number depending on the flow and the setting of Heat transfer parameterization. When Heat transfer parameterization is

• Gnielinski correlation

• Nominal temperature differential vs. nominal mass flow rate

• Dittus-Boelter correlation - Nusselt = a * Re^b * Pr^c

and the flow is laminar, the block uses Nusselt number data and lookup tables to determine the Nusselt number. For the Nusselt number data that the block uses, see Algorithms. When the flow is turbulent, the block calculates the Nusselt number by using equations that depends on the correlation.

When Heat transfer parameterization is Gnielinski correlation and the flow is turbulent, the average Nusselt number is

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

where:

• f is the average Darcy friction factor, according to the Haaland correlation,

  $$f={\left\{-1.8{\text{log}}_{\text{10}}\left[\frac{6.9}{\text{Re}}+{\left(\frac{{ϵ}_{\text{R}}}{3.7D}\right)}^{1.11}\right]\right\}}^{\text{-2}},$$

  where ε_R is the pipe value of the Internal surface absolute roughness parameter.

• Re is the Reynolds number.

• Pr is the Prandtl number.

When Heat transfer parameterization is Dittus-Boelter correlation - Nusselt = a * Re^b * Pr^c and the flow is turbulent, the average Nusselt number is

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

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.

When Heat transfer parameterization is Nominal temperature difference vs. nominal mass flow rate and the flow is turbulent, the heat transfer coefficient is

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

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,

  $$h_{\text{N}}=\frac{{\dot{m}}_{\text{N}}{c}_{\text{p,N}}}{S_{\text{H,N}}}\text{ln}\left(\frac{T_{\text{H,N}}-T_{\text{In, N}}}{T_{\text{H,N}}-T_{\text{Out,N}}}\right),$$

  where:

  • S_H,N is the nominal wall surface area.

  • T_H,N is the value of the Nominal wall temperature parameter.

  • T_In,N is the value of the Nominal inflow temperature parameter.

  • T_Out,N is the value of the Nominal outflow temperature parameter.

This relationship assumes 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 is

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

When the flow is turbulent, Cross-sectional geometry is Annular, and Heat transfer walls is Inner wall or Outer wall, you can use the Nusselt number correction factor parameter to specify a Nusselt number correction term. When you set Nusselt number correction factor to

• None, the correction factor is 1.

• Petukhov and Roizen factor, the correction factor is $$F_{inner}=0.86{\left(\frac{D_{inner}}{D_{outer}}\right)}^{-0.16}.$$ [8]

• Gnielinski, the correction factor is $$F_{inner}=0.75{\left(\frac{Pr}{P{r}_w}\right)}^{0.11}{\left(\frac{D_{inner}}{D_{outer}}\right)}^{-0.17},$$ where Pr_w is the Prandtl number based on the wall temperature. [9]

Nusselt Number Parameterization By Tabulated Data

When Heat transfer parameterizationis Tabulated data - Colburn factor vs. Reynolds number, the average Nusselt number is

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

where J_M is the Colburn-Chilton factor.

When Heat transfer parameterization is Tabulated data - Nusselt number vs. Reynolds number & Prandtl number, the block interpolates the Nusselt number from the three-dimensional array of average Nusselt numbers as a function of both the average Reynolds number and average Prandtl number,

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

Pipe Effects

The block lets you include dynamic compressibility and fluid inertia effects. Turning on each of these effects can improve model fidelity at the cost of increased equation complexity and potentially increased simulation cost:

• When you disable dynamic compressibility, the liquid is assumed to spend negligible time in the pipe volume. Therefore, there is no accumulation of mass in the pipe, and mass inflow equals mass outflow. This is the simplest option. It is appropriate when the liquid mass in the pipe is a negligible fraction of the total liquid mass in the system.

• When you enable dynamic compressibility, an imbalance of mass inflow and mass outflow can cause liquid to accumulate or diminish in the pipe. As a result, pressure in the pipe volume can rise and fall dynamically, which provides some compliance to the system and modulates rapid pressure changes. This is the default option.

• If dynamic compressibility is enabled, you can also turn on fluid inertia. This effect results in additional flow resistance, besides the resistance due to friction. This additional resistance is proportional to the rate of change of mass flow rate. Accounting for fluid inertia slows down rapid changes in flow rate but can also cause the flow rate to overshoot and oscillate. This option is appropriate in a very long pipe. Turn on fluid inertia and connect multiple pipe segments in series to model the propagation of pressure waves along the pipe, such as in the water hammer phenomenon.

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:

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

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

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:

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

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

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:

  $$f={\left\{-1.8{\log }_{10}\left[\frac{6.9}{\mathrm{Re}}+{\left(\frac{\epsilon }{3.7D_h}\right)}^{1.11}\right]\right\}}^{-2},$$

  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:

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

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

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 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_A is the pressure at a port A.

• p_B is the pressure at a port B.

• Δp_f is the pressure differential due to viscous friction, 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.

• ρ_I is the internal fluid density, which is measured at each pipe segment. If fluid dynamic compressibility is not modeled, this is:

  $$p_{\text{I}}=\frac{p_{\text{A}}+p_{\text{B}}}{2}.$$

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_I is the thermal liquid density at internal node I. Each pipe segment has an internal node.

• $$\dot{V}$$ 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_I is the thermal liquid pressure at the internal node I.

• $$\dot{T}$$_I is the rate of change of the thermal liquid temperature at the internal node I.

• β_I is the thermal liquid bulk modulus.

• α_I is the liquid thermal expansion coefficient.

Energy Balance

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

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

where:

• ϕ_A is the energy flow rate at port A.

• ϕ_B is the energy flow rate at port B.

• Q_H is the heat transfer through the pipe wall.

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

$$\dot{E}={\rho }_0{c}_{p_I}V\frac{d{T}_I}{dt},$$

where:

• c_pI is the fluid specific heat at the internal node of the block.

• V is the pipe volume.

• ρ₀ is the constant fluid density. The block calculates this value from the Nominal liquid temperature and Nominal liquid pressure parameters.

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

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

Examples

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• Open Model

Electric Vehicle Thermal Management

Model the thermal management system of a battery electric vehicle.

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Pressure Loss and Mass Flow Rate in a Thermal Liquid Pipe

How changes in pipe and fluid attributes, such as friction and elevation change, impact the pressure loss through a pipe and how enabling dynamic compressibility impacts the mass flow rate. These effects are especially important in long, narrow pipes because they ensure the flow pressure is strong enough to overcome the pressure loss. This example uses the Simscape™ Fluids™ Pipe (TL) block.

• Open Live Script

Ports

Input

EL — Port elevation difference, m

physical signal

Variable elevation differential between port A and B, specified as a physical signal.

Dependencies

To enable this port, set Elevation gain specification to Variable.

Programmatic Use

Port: EL

G — Gravitational acceleration, m/s^2

physical signal

Variable gravitational acceleration, specified as a physical signal.

Dependencies

To enable this port, set Gravitational acceleration specification to Variable.

Programmatic Use

Port: G

Conserving

A — Pipe opening

thermal liquid

Liquid entry or exit port to the pipe.

Programmatic Use

Port: A

B — Pipe opening

thermal liquid

Liquid entry or exit port to the pipe.

Programmatic Use

Port: B

H — Pipe wall

thermal

Pipe wall temperature.

Dependencies

To enable this port, set Cross-sectional geometry to either Circular, Rectangular, Elliptical, Isosceles triangular, or Custom.

Programmatic Use

Port: H

Hi — Inner pipe wall

thermal

Temperature of the inner pipe wall.

Dependencies

To enable this port, set Cross-sectional geometry to Annular and Heat transfer walls to Inner wall or Inner and outer walls.

Programmatic Use

Port: Hi

Ho — Outer pipe wall

thermal

Temperature of the outer pipe wall.

Dependencies

To enable this port, set Cross-sectional geometry to Annular and Heat transfer walls to Outer wall or Inner and outer walls.

Programmatic Use

Port: Ho

Parameters

Configuration

Enable dynamic compressibility — Whether to model fluid dynamic compressibility

on (default) | off

Whether to model the dynamic compressibility of the liquid. Dynamic compressibility affects the transient response of the system at small timescales. If you select this parameter, the pressure responds dynamically based on the accumulation of fluid mass in the volume. If you clear this parameter, the volume does not accumulate mass and the pressure response is instantaneous. Clearing this setting can improve simulation performance, but may have a negative effect on the simulation robustness. Only clear this setting for small fluid volumes or models with long simulation times.

Programmatic Use

Parameter:	dynamic_compressibility
Values:	"true" | "false"

Enable fluid inertia — Whether to model fluid acceleration

off (default) | on

Whether to account for acceleration in the mass flow rate due to the mass of the fluid.

Dependencies

To enable this parameter, select Enable dynamic compressibility.

Programmatic Use

Parameter:	inertia
Values:	"true" | "false"

Number of segments — Pipe discretization

1 (default) | positive unitless scalar

Number of pipe divisions. Each division represents an individual segment over which pressure is calculated, depending on the pipe inlet pressure, fluid compressibility, and wall flexibility, if applicable. The fluid volume in each segment remains fixed.

Dependencies

To enable this parameter, select Enable dynamic compressibility.

Programmatic Use

Parameter: num_segments

Pipe total length — Total pipe length

5 m (default) | positive scalar in units of length

Total pipe length across all pipe segments.

Programmatic Use

Parameter: pipe_length

Cross-sectional geometry — Pipe geometry

Circular (default) | Annular | Rectangular | Elliptical | Isosceles triangular | Custom

Cross-sectional pipe geometry. A nominal hydraulic diameter and nominal cross-sectional area is calculated based on the cross-sectional geometry.

Programmatic Use

Parameter:

cross_section_geometry

Values:

"fluids.thermal_liquid.pipes_fittings.enum.CrossSectionGeometry.Circular" | "fluids.thermal_liquid.pipes_fittings.enum.CrossSectionGeometry.Annular" | "fluids.thermal_liquid.pipes_fittings.enum.CrossSectionGeometry.Rectangular" | "fluids.thermal_liquid.pipes_fittings.enum.CrossSectionGeometry.Elliptical" | "fluids.thermal_liquid.pipes_fittings.enum.CrossSectionGeometry.Triangular" | "fluids.thermal_liquid.pipes_fittings.enum.CrossSectionGeometry.Custom"

Pipe diameter — Pipe diameter

0.1 m (default) | positive scalar

Diameter for circular cross-sectional pipes.

Dependencies

To enable this parameter, set Cross-sectional geometry to Circular.

Programmatic Use

Parameter: pipe_diameter

Parameter: diameter_bend

Pipe inner diameter — Pipe inner diameter

0.05 m (default) | positive scalar

Inner diameter for annular pipe flow, or flow between two concentric pipes.

Dependencies

To enable this parameter, set Cross-sectional geometry to Annular.

Programmatic Use

Parameter: pipe_inner_diameter

Pipe outer diameter — Pipe outer diameter

0.1 m (default) | positive scalar

Outer diameter for annular pipe flow, or flow between two concentric pipes.

Dependencies

To enable this parameter, set Cross-sectional geometry to Annular.

Programmatic Use

Parameter: pipe_outer_diameter

Pipe width — Rectangular pipe width

0.1 m (default) | positive scalar

Width of rectangular pipe.

Dependencies

To enable this parameter, set Cross-sectional geometry to Rectangular.

Programmatic Use

Parameter: pipe_width

Pipe height — Rectangular pipe height

0.1 m (default) | positive scalar

Height of rectangular pipe.

Dependencies

To enable this parameter, set Cross-sectional geometry to Rectangular.

Programmatic Use

Parameter: pipe_height

Pipe major axis — Elliptical pipe major axis

0.1 m (default) | positive scalar

Major axis for elliptical pipes.

Dependencies

To enable this parameter, set Cross-sectional geometry to Elliptical.

Programmatic Use

Parameter: pipe_major_axis

Pipe minor axis — Elliptical pipe minor axis

0.05 m (default) | positive scalar

Minor axis for elliptical pipes.

Dependencies

To enable this parameter, set Cross-sectional geometry to Elliptical.

Programmatic Use

Parameter: pipe_minor_axis

Pipe side length — Triangular pipe side length

0.1 m (default) | positive scalar

Length of the two equal sides of isosceles-triangular pipes.

Dependencies

To enable this parameter, set Cross-sectional geometry to Isosceles triangular.

Programmatic Use

Parameter: pipe_side

Pipe vertex angle — Triangular pipe vertex angle

30 deg (default) | positive scalar

Vertex angle for triangular pipes. The value must be less than 180 degrees.

Dependencies

To enable this parameter, set Cross-sectional geometry to Isosceles triangular.

Programmatic Use

Parameter: pipe_angle

Cross-sectional area — Cross-sectional pipe area without deformation

0.01 m^2 (default) | positive scalar

Cross-sectional area of the pipe without deformations.

Dependencies

To enable this parameter, set Cross-sectional geometry to Custom.

Programmatic Use

Parameter: area

Hydraulic diameter — Effective diameter of noncircular pipes

0.1128 m (default) | positive scalar in units of length

Effective diameter used in heat transfer, momentum balance, and pipe flexibility equations. For noncircular pipes, the hydraulic diameter is the effective diameter of the fluid in the pipe. For circular pipes, the hydraulic diameter and pipe diameter are the same.

Dependencies

To enable this parameter, either:

• Clear the Enable dynamic compressibility check box and set Cross-sectional geometry to Custom.

• Select Enable dynamic compressibility, set the Pipe wall specification parameter to Rigid and set Cross-sectional geometry to Custom.

Programmatic Use

Parameter: Dh

Heat transfer walls — Wall at which model heat transfer

Inner wall (default) | Outer wall | Inner and outer walls

Wall at which to model heat transfer when the pipe cross section is annular.

Dependencies

To enable this parameter, set Cross-sectional geometry to Annular.

Programmatic Use

Parameter:	annular_heat_transfer_walls
Values:	"fluids.thermal_liquid.pipes_fittings.enum.AnnularPipeWalls.AnnularInner" | "fluids.thermal_liquid.pipes_fittings.enum.AnnularPipeWalls.AnnularOuter" | "fluids.thermal_liquid.pipes_fittings.enum.AnnularPipeWalls.AnnularInnerAndOuter"

Elevation gain specification — Set pipe elevation property

Constant (default) | Variable

Set the pipe elevation as either Constant or Variable. Selecting Variable exposes the physical signal port EL.

Programmatic Use

Parameter:	elevation_spec
Values:	"foundation.enum.constant_variable.constant" | "foundation.enum.constant_variable.variable"

Elevation gain from port A to port B — Change in elevation from port A to port B

0 m (default) | positive scalar in units of length

Elevation differential for constant-elevation pipes.

Dependencies

To enable this parameter, set Elevation gain specification to Constant.

Programmatic Use

Parameter: elevation_gain

Gravitational acceleration specification — Acceleration specification

Constant (default) | Variable

Whether the gravitational acceleration is constant or variable.

Programmatic Use

Parameter:	gravity_spec
Values:	"foundation.enum.constant_variable.constant" | "foundation.enum.constant_variable.variable"

Gravitational acceleration — Acceleration due to gravity at mean pipe elevation

9.81 m/s^2 (default) | scalar

Constant of the gravitational acceleration at the mean elevation of the pipe.

Dependencies

To enable this parameter, set Gravitational acceleration specification to Constant.

Programmatic Use

Parameter: g

Viscous Friction

Viscous friction parameterization — Friction model

Haaland correlation (default) | Nominal pressure drop vs. nominal mass flow rate | Tabulated data - Darcy friction factor vs. Reynolds number

Parameterization of pressure losses due to wall friction. Both analytical and tabular formulations are available.

Programmatic Use

Parameter:	pressure_loss_spec
Values:	"fluids.thermal_liquid.pipes_fittings.enum.PressureLossSpec.Haaland" | "fluids.thermal_liquid.pipes_fittings.enum.PressureLossSpec.Nominal" | "fluids.thermal_liquid.pipes_fittings.enum.PressureLossSpec.Tabulated"

Local resistances specification — Method for quantifying pressure losses in the Haaland correlation

Aggregate equivalent length (default) | Local loss coefficient

Method for quantifying pressure losses due to pipe nonuniformities.

Dependencies

To enable this parameter, set Viscous friction parameterization to Haaland correlation.

Programmatic Use

Parameter:	local_pressure_loss_spec
Values:	"fluids.thermal_liquid.pipes_fittings.enum.LocalLossSpec.AggregateLength" | "fluids.thermal_liquid.pipes_fittings.enum.LocalLossSpec.LossCoefficient"

Total local loss coefficient — Defines loss coefficient over the pipe

0.1 (default) | positive scalar

Loss coefficient associated with each pipe nonuniformity. You can input a single loss coefficient or the sum of all loss coefficients along the pipe.

Dependencies

To enable this parameter, set Viscous friction parameterization to Haaland correlation and Local resistance specifications to Local loss coefficient.

Programmatic Use

Parameter: loss_coefficient

Aggregate equivalent length of local resistances — Minor pressure loss in the pipe expressed as a length

1 m (default) | positive scalar

Length of pipe that would produce the equivalent hydraulic losses as would a pipe with bends, area changes, or other nonuniformities. The effective length of the pipe is the sum of the Pipe length and the Aggregate equivalent length of local resistances.

Dependencies

To enable this parameter, set Viscous friction parameterization to Haaland correlation and Local resistances specification to Aggregate equivalent length.

Programmatic Use

Parameter: length_add

Surface roughness specification — Pipe material for roughness specification

Commercially smooth brass, lead, copper, or plastic pipe: 1.52 um (default) | Steel and wrought iron : 46 um | Galvanized iron or steel : 152 um | Cast iron : 259 um | Custom

Absolute surface roughness based on pipe material. The provided values are ASHRAE standard roughness values. You can also input your own value by setting Surface roughness specification to Custom.

Dependencies

To enable this parameter, set Viscous friction parameterization to Haaland correlation.

Programmatic Use

Parameter:	roughness_spec
Values:	"fluids.thermal_liquid.pipes_fittings.enum.RoughnessSpec.SteelWroughtIron" | "fluids.thermal_liquid.pipes_fittings.enum.RoughnessSpec.GalvanizedSteelIron" | "fluids.thermal_liquid.pipes_fittings.enum.RoughnessSpec.CastIron" | "fluids.thermal_liquid.pipes_fittings.enum.RoughnessSpec.Custom" | "fluids.thermal_liquid.pipes_fittings.enum.RoughnessSpec.CommercialPipe"

Internal surface absolute roughness — Pipe wall absolute roughness

1.5e-5 m (default) | positive scalar in units of length

Pipe wall absolute roughness. This parameter is used to determine the Darcy friction factor, which contributes to pressure loss in the pipe.

Dependencies

To enable this parameter, set Viscous friction parameterization to Haaland correlation and Surface roughness specification Custom.

Programmatic Use

Parameter: roughness

Laminar friction constant for Darcy friction factor — Friction constant for laminar flows

64 (default) | positive scalar

Friction constant for laminar flows. The Darcy friction factor captures the contribution of wall friction in pressure loss calculations. If Cross-sectional geometry is not set to Custom, this parameter is internally set to 64.

Dependencies

To enable this parameter, set Cross-sectional geometry to Custom.

Programmatic Use

Parameter: shape_factor

Laminar flow upper Reynolds number limit — Reynolds number below which the flow is laminar

2e+3 (default) | positive scalar

Reynolds number below which the flow is laminar. Above this threshold, the flow transitions to turbulent, reaching the turbulent regime at the Turbulent flow lower Reynolds number limit setting.

Programmatic Use

Parameter: Re_lam

Turbulent flow lower Reynolds number limit — Reynolds number above which the flow is turbulent

4e+3 (default) | positive scalar

Reynolds number above which the flow is turbulent. Below this threshold, the flow gradually transitions to laminar, reaching the laminar regime at the Laminar flow upper Reynolds number limit setting.

Programmatic Use

Parameter: Re_tur

Nominal mass flow rate — Pipe mass flow rate

[0.1 1] kg/s (default) | scalar | vector

Pipe nominal mass flow rate used to calculate the pressure loss coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the Nominal pressure drop parameter. When this parameter is supplied as a vector, the scalar value K_p is determined as a least-squares fit of the vector elements.

Dependencies

To enable this parameter, set Viscous friction parameterization to Nominal pressure drop vs. nominal mass flow rate.

Programmatic Use

Parameter: mdot_nominal

Nominal pressure drop — Pressure drop over the pipe

[0.001 0.01] MPa (default) | scalar | vector

Pipe nominal pressure drop used to calculate the pressure loss coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the Nominal mass flow rate parameter. When this parameter is supplied as a vector, the scalar value K_p is determined as a least-squares fit of the vector elements.

Dependencies

To enable this parameter, set Viscous friction parameterization to Nominal pressure drop vs. nominal mass flow rate.

Programmatic Use

Parameter: p_diff_nominal

Mass flow rate threshold for flow reversal — Threshold below which numerical smoothing is applied

1e-6 kg/s (default) | positive scalar in units of mass/time

Mass flow rate threshold for reversed flow. A transition region is defined around 0 kg/s between the positive and negative values of the mass flow rate threshold. Within this transition region, numerical smoothing is applied to the flow response. The threshold value must be greater than 0.

Dependencies

To enable this parameter, set Viscous friction parameterization to Nominal pressure drop vs. nominal mass flow rate.

Programmatic Use

Parameter: mdot_threshold

Reynolds number vector for turbulent Darcy friction factor — Reynolds numbers at which to tabulate the Darcy friction factor

[ 400 1000 1.5e+3 3e+3 4e+3 6e+3 1e+4 2e+4 4e+4 6e+4 1e+5 1e+8 ] (default) | vector of positive numbers

Vector of Reynolds numbers for the tabular parameterization of the Darcy friction factor. The vector elements form an independent axis with the Darcy friction factor vector parameter. The vector elements must be listed in ascending order and must be greater than 0.

Dependencies

To enable this parameter, set Viscous friction parameterization to Tabulated data - Darcy friction factor vs. Reynolds number.

Programmatic Use

Parameter: Re_friction_TLU

Darcy friction factor vector — Darcy friction factors at the tabulated Reynolds numbers

[ 0.264 0.112 0.07099999999999999 0.0417 0.0387 0.0268 0.025 0.0232 0.0226 0.022 0.0214 0.0214 ] (default) | vector of positive numbers

Vector of Darcy friction factors for the tabular parameterization of the Darcy friction factor. The vector elements must correspond one-to-one with the elements in the Reynolds number vector for turbulent Darcy friction factor parameter, and must be unique and greater than or equal to 0.

Dependencies

To enable this parameter, set Viscous friction parameterization to Tabulated data - Darcy friction factor vs. Reynolds number.

Programmatic Use

Parameter: friction_factor_TLU

Pipe Wall

Pipe wall specification — Pipe wall flexibility

Rigid (default) | Flexible

Wall flexibility of the pipe. This parameter is independent of pipe cross-sectional geometry. The Flexible setting preserves the initial pipe shape and applies equal expansion of the cross-sectional area. The Flexible setting may not be accurate for non-circular cross-sectional geometry under high deformation.

Dependencies

To enable this parameter, select Enable dynamic compressibility.

Programmatic Use

Parameter:	wall_spec
Values:	"fluids.thermal_liquid.pipes_fittings.enum.wall_spec.rigid" | "fluids.thermal_liquid.pipes_fittings.enum.wall_spec.flexible"

Volumetric expansion specification — Linear expansion correlation

Cross-sectional area vs. pressure (default) | Cross-sectional area vs. pressure - Tabulated | Hydraulic diameter vs. pressure | Based on material properties

Linear expansion correlation. The settings correlate the new cross-sectional area or hydraulic diameter to the pipe pressure.

Dependencies

To enable this parameter, select Enable dynamic compressibility and set Pipe wall specification to Flexible.

Programmatic Use

Parameter:	vol_expansion_spec
Values:	"fluids.thermal_liquid.pipes_fittings.enum.VolumeExpansionSpec.Area" | "fluids.thermal_liquid.pipes_fittings.enum.VolumeExpansionSpec.AreaTLU" | "fluids.thermal_liquid.pipes_fittings.enum.VolumeExpansionSpec.Diameter" | "fluids.thermal_liquid.pipes_fittings.enum.VolumeExpansionSpec.Material"

Static gauge pressure to cross-sectional area gain — Coefficient for area-dependent pipe deformation

1e-6 m^2/MPa (default) | positive scalar

Coefficient for calculating pipe deformation. The block multiplies the value of this parameter by the pressure differential between the segment pressure and atmospheric pressure.

Dependencies

To enable this parameter, select Enable dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Cross-sectional area vs. pressure.

Programmatic Use

Parameter: area_pressure_gain

Static gauge pressure vector — Vector of gauge pressures

[.1,1] MPa (default) | vector

Vector that contains the gauge pressures. The block uses this vector in a table lookup to calculate the pipe cross-sectional area. The vector entries must be strictly positive and monotonically increasing and the vector must be the same length as the Cross sectional area gain vector parameter.

Dependencies

To enable this parameter, select Enable dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Cross-sectional area vs. pressure - Tabulated.

Programmatic Use

Parameter: static_pressure_TLU

Cross sectional area gain vector — Vector of pipe cross-sectional areas

[1e-05, 1.1e-05] m^2 (default) | vector

Vector that contains the pipe cross-sectional areas. The block uses this vector in a table lookup to calculate the pipe cross sectional-area at other pressures. The vector entries must be strictly positive and monotonically increasing and the vector must be the same length as the Static gauge pressure vector parameter.

Dependencies

To enable this parameter, select Enable dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Cross-sectional area vs. pressure - Tabulated.

Programmatic Use

Parameter: area_pressure_gain_TLU

Static gauge pressure to hydraulic diameter gain — Coefficient for diameter-dependent pipe deformation

1e-6 m/MPa (default) | positive scalar

Coefficient for calculating the pipe deformation. The block multiplies the value of this parameter by the pressure differential between the segment pressure and atmospheric pressure.

Dependencies

To enable this parameter, select Enable dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Hydraulic diameter vs. pressure.

Programmatic Use

Parameter: diameter_pressure_gain

Material behavior — Method to specify material behavior

Linear Elastic (default) | Multilinear Elastic

Method the block uses to calculate the material behavior.

Dependencies

To enable this parameter, select Enable dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Based on material properties.

Programmatic Use

Parameter:	material_behavior_spec
Values:	"fluids.thermal_liquid.pipes_fittings.enum.PipeMaterialBehavior.Linear" | "fluids.thermal_liquid.pipes_fittings.enum.PipeMaterialBehavior.Multilinear"

Pipe wall thickness — Width of pipe wall

0.05 m (default) | positive scalar

Thickness of the pipe wall. The block uses this value to calculate stress.

Dependencies

To enable this parameter, select Enable dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Based on material properties.

Programmatic Use

Parameter: wall_thickness

Young's modulus — Young's modulus of pipe wall material

69 GPa (default) | positive scalar

Young's modulus of the material that makes up the pipe wall.

Dependencies

To enable this parameter, select Enable dynamic compressibility, and set Pipe wall specification to Flexible, Volumetric expansion specification to Based on material properties, and Material behavior to Linear Elastic.

Programmatic Use

Parameter: elastic_modulus

Poisson's ratio — Poisson's ratio of pipe wall material

0.33 (default) | positive scalar

Poisson's ratio of the material that makes up the pipe wall.

Dependencies

To enable this parameter, select Enable dynamic compressibility, set Pipe wall specification to Flexible, and set Volumetric expansion specification to Based on material properties.

Programmatic Use

Parameter: poisson_ratio

Stress vector — Stress vector of pipe wall material

[276, 310] MPa (default) | vector

Vector containing the stress values for the material that makes up the pipe wall.

Dependencies

To enable this parameter, select Enable dynamic compressibility and set Pipe wall specification to Flexible, Volumetric expansion specification to Based on material properties, and Material behavior to Multilinear Elastic.

Programmatic Use

Parameter: stress_TLU

Strain vector — Strain vector of pipe wall material

[.004, .02] (default) | vector

Vector containing the strain values for the material that makes up the pipe wall.

Dependencies

To enable this parameter, select Enable dynamic compressibility and set Pipe wall specification to Flexible, Volumetric expansion specification to Based on material properties, and Material behavior to Multilinear Elastic.

Programmatic Use

Parameter: strain_TLU

Check if stress exceeds allowable level — Notification when stress is above specified max

None (default) | Warning | Error

Whether the block does nothing, generates a warning, or generates an error when the stress is above the maximum stress specified by the Maximum allowable stress parameter.

Dependencies

To enable this parameter, select Enable dynamic compressibility and set Pipe wall specification to Flexible, Volumetric expansion specification to Based on material properties, and Material behavior to Multilinear Elastic.

Programmatic Use

Parameter:	stress_assert_action
Values:	"simscape.enum.assert.action.none" | "simscape.enum.assert.action.warn" | "simscape.enum.assert.action.error"

Maximum allowable stress — Maximum allowed stress on pipe wall

400 MPa (default) | positive scalar

Maximum stress the block allows on the pipe wall. Control what the block does if the stress exceeds this value with the Check if stress exceeds allowable level parameter.

Dependencies

To enable this parameter, select Enable dynamic compressibility and set Pipe wall specification to Flexible, Volumetric expansion specification to Based on material properties, Material behavior to Multilinear Elastic and Check if stress exceeds allowable level to Warning or Error.

Programmatic Use

Parameter: stress_max

Volumetric expansion time constant — Pipe deformation time constant

0.01 s (default) | positive scalar

Time required for the wall to reach steady-state after pipe deformation. This parameter impacts the dynamic change in pipe volume.

Dependencies

To enable this parameter, select Enable dynamic compressibility and set Pipe wall specification to Flexible.

Programmatic Use

Parameter: wall_time_constant

Enable pipe thermal expansion — Whether to model thermal expansion

off (default) | on

Whether to account for expansion in the pipe due to temperature changes.

Dependencies

To enable this parameter, select Enable dynamic compressibility and set Pipe wall specification to Flexible.

Programmatic Use

Parameter:	thermal_expansion
Values:	"true" | "false"

Coefficient of thermal expansion — Pipe coefficient of thermal expansion

24 um/(deltaK*m) (default) | positive scalar

Coefficient of linear thermal expansion for the pipe material. This value is the fractional change in size per degree change in temperature at a constant pressure.

Dependencies

To enable this parameter, select Enable dynamic compressibility, set Pipe wall specification to Flexible, and select Enable pipe thermal expansion.

Programmatic Use

Parameter: alpha_L_wall

Thermal expansion reference temperature — Reference temperature for pipe thermal expansion

293.15 K (default) | positive scalar

Reference temperature the block uses when calculating the thermal expansion of the pipe.

Dependencies

To enable this parameter, select Enable dynamic compressibility, set Pipe wall specification to Flexible, and select Enable pipe thermal expansion.

Programmatic Use

Parameter: thermal_expansion_Tref

Environment pressure specification — Method to specify pressure outside pipe

Atmospheric pressure (default) | Specified pressure

Method to specify the pressure of the environment outside of the pipe. To use atmospheric pressure, set this parameter to Atmospheric pressure. To specify a custom pressure, set this parameter to Specified pressure.

Dependencies

To enable this parameter, select Enable dynamic compressibility and set Pipe wall specification to Flexible.

Programmatic Use

Parameter:	environment_spec
Values:	"foundation.enum.pressure_spec.atmospheric" | "foundation.enum.pressure_spec.specified"

Environment pressure — Pressure outside pipe

0.101325 MPa (default) | positive scalar

Pressure in the environment surrounding the pipe.

Dependencies

To enable this parameter, select Enable dynamic compressibility, set Pipe wall specification to Flexible, and set Environment pressure specification to Specified pressure.

Programmatic Use

Parameter: environment_pressure

Heat Transfer

Heat transfer parameterization — Method by which to capture the convective heat transfer with the pipe wall

Gnielinski correlationNominal temperature differential vs. nominal mass flow rate (default) | Dittus-Boelter correlation - Nusselt = a * Re^b * Pr^c | Tabulated data - Colburn factor vs. Reynolds number | Tabulated data - Nusselt number vs. Reynolds number & Prandtl number

Method of calculating the heat transfer coefficient between the fluid and the pipe wall. Analytical and tabulated data parameterizations are available.

Programmatic Use

Parameter:	heat_transfer_spec
Values:	"fluids.thermal_liquid.pipes_fittings.enum.HeatTransferSpec.Gnielinski" | "fluids.thermal_liquid.pipes_fittings.enum.HeatTransferSpec.Nominal" | "fluids.thermal_liquid.pipes_fittings.enum.HeatTransferSpec.DittusBoelter" | "fluids.thermal_liquid.pipes_fittings.enum.HeatTransferSpec.TabulatedColburn" | "fluids.thermal_liquid.pipes_fittings.enum.HeatTransferSpec.TabulatedNusselt"

Nusselt number for laminar flow heat transfer — Nusselt number to use in the heat transfer calculations for laminar flows

3.66 (default)

Ratio of convective to conductive heat transfer in the laminar flow regime. The fluid Nusselt number influences the heat transfer rate.

Dependencies

To enable this parameter, set Cross-sectional geometry to Custom and set Heat transfer parameterization to either:

• Gnielinski correlation.

• Nominal temperature differential vs. nominal mass flow rate.

• Dittus-Boelter correlation - Nusselt = a * Re^b * Pr^c.

Programmatic Use

Parameter: Nu_lam

Nominal mass flow rate — Pipe mass flow rate

[0.1 1] kg/s (default) | scalar | vector

Pipe nominal mass flow rate used to calculate the heat transfer coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the Nominal inflow temperature parameter. When this parameter is supplied as a vector, the scalar value h_p is determined as a least-squares fit of the vector elements.

Dependencies

To enable this parameter, set Heat transfer parameterization to Nominal temperature differential vs. nominal mass flow rate.

Programmatic Use

Parameter: mdot_nominal_heat

Nominal inflow temperature — Pipe inlet temperature

[293.15 293.15] K (default) | scalar | vector

Nominal fluid inlet temperature used to calculate the heat transfer coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the Nominal mass flow rate parameter. When this parameter is supplied as a vector, the scalar value h is determined as a least-squares fit of the vector elements.

Dependencies

To enable this parameter, set Heat transfer parameterization to Nominal temperature differential vs. nominal mass flow rate.

Programmatic Use

Parameter: T_in_nominal

Nominal outflow temperature — Pipe outlet temperature

[300 300] K (default) | scalar | vector

Nominal fluid outlet temperature used to calculate the heat transfer coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the Nominal mass flow rate parameter. When this parameter is supplied as a vector, the scalar value h is determined as a least-squares fit of the vector elements.

Dependencies

To enable this parameter, set Heat transfer parameterization to Nominal temperature differential vs. nominal mass flow rate.

Programmatic Use

Parameter: T_out_nominal

Nominal inflow pressure — Pipe inlet pressure

[0.101325 0.101325] MPa (default) | scalar | vector

Nominal fluid inlet pressure used to calculate the heat transfer coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the Nominal mass flow rate parameter. When this parameter is supplied as a vector, the scalar value h is determined as a least-squares fit of the vector elements.

Dependencies

To enable this parameter, set Heat transfer parameterization to Nominal temperature differential vs. nominal mass flow rate.

Programmatic Use

Parameter: p_in_nominal_heat

Nominal wall temperature — Pipe wall temperature

[303.15 303.15] K (default) | scalar | vector

Pipe wall temperature used to calculate the heat transfer coefficient, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the Nominal mass flow rate parameter. When this parameter is supplied as a vector, the scalar value h is determined as a least-squares fit of the vector elements.

Dependencies

To enable this temperature, set Heat transfer parameterization to Nominal temperature differential vs. nominal mass flow rate and any selection of parameters except setting Cross-sectional geometry to Annular and Heat transfer walls to Inner and outer walls.

Programmatic Use

Parameter: T_wall_nominal

Nominal effective wall temperature — Nominal effective temperature of pipe wall

[303.15 303.15] K (default) | scalar | vector

Nominal effective pipe wall temperature that the block uses to calculate the heat transfer coefficient. Calculate the value of this parameter as weighted average temperature across the outer and inner walls

$$T_{effective}=\frac{U_i{A}_i{T}_i+U_o{A}_o{T}_o}{U_i{A}_i+U_o{A}_o}$$

where U, A, and T are heat transfer coefficient, surface area, and wall temperature, respectively. The subscripts i and o denote the inner and outer walls.

The value of this parameter must be greater than 0 and have the same number of elements as the Nominal mass flow rate parameter.

Dependencies

To enable this temperature, set Cross-sectional geometry to Annular, Heat transfer walls to Inner and outer walls, and Heat transfer parameterization to Nominal temperature differential vs. nominal mass flow rate.

Programmatic Use

Parameter: T_wall_eff_nominal

Coefficient a — Empirical constant a of the Dittus-Boelter correlation

0.023 (default) | positive scalar

Empirical constant a to use in the Dittus-Boelter correlation. The correlation relates the Nusselt number in turbulent flows to the heat transfer coefficient.

Dependencies

To enable this parameter, set Heat transfer parameterization to Dittus-Boelter correlation - Nusselt = a * Re^b * Pr^c.

Programmatic Use

Parameter: a_coeff

Exponent b — Empirical constant b of the Dittus-Boelter correlation

0.8 (default) | positive scalar

Empirical constant b to use in the Dittus-Boelter correlation. The correlation relates the Nusselt number in turbulent flows to the heat transfer coefficient.

Dependencies

To enable this parameter, set Heat transfer parameterization to Dittus-Boelter correlation - Nusselt = a * Re^b * Pr^c.

Programmatic Use

Parameter: b_Re_power

Exponent c — Empirical constant c of the Dittus-Boelter correlation

0.4 (default) | positive scalar

Empirical constant c to use in the Dittus-Boelter correlation. The correlation relates the Nusselt number in turbulent flows to the heat transfer coefficient. The default value reflects heat transfer to the fluid.

Dependencies

To enable this parameter, set Heat transfer parameterization to Dittus-Boelter correlation - Nusselt = a * Re^b * Pr^c.

Programmatic Use

Parameter: c_Pr_power

Reynolds number vector for Colburn factor — Reynolds numbers at which to tabulate the Colburn factor

[100 150 1000] (default) | vector of positive numbers

Vector of Reynolds numbers for the tabular parameterization of the Colburn factor. The vector elements form an independent axis with the Colburn factor vector parameter. The vector elements must be listed in ascending order and must be greater than 0. This parameter must have the same number of elements as the Colburn factor vector. For reversed flows, or flows from B to A, the same data is applied in the opposite direction.

Dependencies

To enable this parameter, set Heat transfer parameterization to Tabulated data - Colburn factor vs. Reynolds number.

Programmatic Use

Parameter: Re_Colburn_TLU

Colburn factor vector — Colburn factors at which the tabulated Reynolds numbers

[0.019 0.013 0.002] (default) | vector of positive numbers

Vector of Colbrun factors for the tabular parameterization of the Colburn factor. The vector elements form an independent axis with the Reynolds number vector for Colburn factor parameter. This parameter must have the same number of elements as the Reynolds number vector for Colburn factor.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to Tabulated data - Colburn factor vs. Reynolds number.

Programmatic Use

Parameter: Colburn_factor_TLU

Reynolds number vector for Nusselt number — Reynolds number at which to tabulate the Nusselt number

[100 150 100] (default) | vector of positive numbers

Vector of Reynolds numbers for the tabular parameterization of Nusselt number. This vector forms an independent axis with the Prandtl number vector for Nusselt number parameter for the 2-D dependent Nusselt number table, Nu(Re,Pr). The vector elements must be listed in ascending order and must be greater than 0.

Dependencies

To enable this parameter, set Heat transfer parameterization to Tabulated data - Nusselt number vs. Reynolds number & Prandtl number.

Programmatic Use

Parameter: Re_Nu_TLU

Prandtl number vector for Nusselt number — Prandtl numbers at which to tabulate the Nusselt number

[1 10] (default) | vector of positive numbers

Vector of Prandtl numbers for the tabular parameterization of Nusselt number. This vector forms an independent axis with the Reynolds number vector for Nusselt number parameter for the 2-D dependent Nusselt number table, Nu(Re,Pr). The vector elements must be listed in ascending order.

Dependencies

To enable this parameter, set Heat transfer parameterization to Tabulated data - Nusselt number vs. Reynolds number & Prandtl number.

Programmatic Use

Parameter: Pr_Nu_TLU

Nusselt number table, Nu(Re,Pr) — Nusselt numbers at the tabulated Reynolds and Prandtl numbers

[ 3.72 4.21; 3.75 4.44; 4.21 7.15 ] (default) | matrix of positive numbers

M-by-N matrix of Nusselt numbers at the specified Reynolds and Prandtl numbers. Linear interpolation is employed between table elements. M and N are the sizes of the corresponding vectors:

• M is the number of vector elements in the Reynolds number vector for Nusselt number parameter.

• N is the number of vector elements in the Prandtl number vector for Nusselt number parameter.

Dependencies

To enable this parameter, Heat transfer parameterization to Tabulated data - Nusselt number vs. Reynolds number & Prandtl number.

Programmatic Use

Parameter: Nu_TLU

Nusselt number correction factor — Nusselt number correction factor for annular cross section

Gnielinski factor (default) | Petukhov & Roizen factor | None

Correction factor that the block applies to the Nusselt number when the pipe cross section is annular.

Dependencies

To enable this parameter, set:

• Cross-sectional geometry to Annular

• Heat transfer walls to either Outer wall or Inner wall

• Heat transfer parameterization to Gnielinski correlation, Nominal temperature differential vs. nominal mass flow rate, or Dittus-Boelter correlation - Nusselt = a * Re^b * Pr^c

Programmatic Use

Parameter:	annular_correction
Values:	"fluids.thermal_liquid.pipes_fittings.enum.AnnularPipeCorrectionFactor.Gnielinski" | "fluids.thermal_liquid.pipes_fittings.enum.AnnularPipeCorrectionFactor.None" | "fluids.thermal_liquid.pipes_fittings.enum.AnnularPipeCorrectionFactor.Petukhov"

Initial Conditions

Initial liquid temperature — Liquid temperature at start of simulation

293.15 K (default)

Liquid temperature at the beginning of the simulation.

Programmatic Use

Parameter: T0_vec

Parameter: T0

Initial liquid pressure — Pipe pressure at beginning of simulation

0.101325 MPa (default) | positive scalar

Pipe pressure at the beginning of the simulation.

Dependencies

To enable this parameter, set select Enable dynamic compressibility.

Programmatic Use

Parameter: p0_vec

Parameter: p0

Nominal liquid temperature — Temperature at nominal operating conditions

293.15 K (default) | scalar

Liquid temperature at nominal operating conditions. The block uses this value to calculate the nominal density to use in the mass and energy conservation equation when dynamic compressibility is disabled.

Dependencies

To enable this parameter, clear the Enable dynamic compressibility checkbox.

Programmatic Use

Parameter: T_nominal

Nominal liquid pressure — Pressure at nominal operating conditions

0.101325 MPa (default) | scalar

Liquid pressure at nominal operating conditions. The block uses this value to calculate the nominal density to use in the mass and energy conservation equation when dynamic compressibility is disabled.

Dependencies

To enable this parameter, clear the Enable dynamic compressibility checkbox.

Programmatic Use

Parameter: p_nominal

Initial mass flow rate from port A to port B — Initial mass flow rate for inertia calculation

0 kg/s (default) | scalar

Initial mass flow rate for pipes with simulated fluid inertia.

Dependencies

To enable this parameter, select Enable dynamic compressibility and Enable fluid inertia.

Programmatic Use

Parameter: mdot0

Algorithms

When the flow is laminar and Heat transfer parameterization is Gnielinski correlation, Nominal temperature differential vs. nominal mass flow rate, or Dittus-Boelter correlation - Nusselt = a * Re^b * Pr^c, the block determines the Nusselt number by using a lookup table and Nusselt number data from [1], [6], and [7]. When the block uses a lookup table, it uses linear interpolation and nearest extrapolation to determine the Nusselt number.

The data that the block uses depends on the Cross-sectional geometry setting:

• 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.

  • If Heat transfer walls is Inner wall:

    $$\frac{D_{inner}}{D_{outer}}$$	Nusselt number
    0.02	32.337
    0.05	17.46
    0.1	11.56
    0.25	7.371
    0.5	5.738
    1	4.861

  • If Heat transfer walls is Outer wall:

    $$\frac{D_{inner}}{D_{outer}}$$	Nusselt number
    0	3.657
    0.02	3.993
    0.05	4.056
    0.1	4.113
    0.25	4.232
    0.5	4.429
    1	4.861

  • If Heat transfer walls is Inner and outer walls:

    $$\frac{D_{inner}}{D_{outer}}$$	Nusselt Number Inner Wall	Nusselt Number Outer Wall
    0.0001	2355.3	2.779
    0.001	322.254	2.823
    0.01	50.454	2.908
    0.02	30.179	2.948
    0.04	18.614	2.999
    0.05	16.058	3.019
    0.06	14.28	3.036
    0.08	11.943	3.068
    0.1	10.459	3.095
    0.15	8.342	3.157
    0.2	7.197	3.213
    0.25	6.471	3.267
    0.3	5.966	3.319
    0.4	5.305	3.421
    0.5	4.889	3.52
    0.6	4.602	3.619
    0.7	4.391	3.715
    0.8	4.23	3.811
    0.9	4.103	3.906
    1	4	4

• When Cross-sectional geometry is Rectangular, the block calculates the Nusselt number from tabulated data.

  $$\frac{\min \left(h,w\right)}{\max \left(h,w\right)}$$	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.

  $$\frac{b_{min}}{a_{maj}}$$	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.

  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.

References

[1] Budynas R. G. Nisbett J. K. and 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., and Thomas A. Butler. Review of proposed failure criteria for ductile materials. No. NUREG/CR-3644; LA-10007-MS. Los Alamos National Lab., NM (USA), 1984.

[4] Hencky, Heinrich. "Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen." ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 4, no. 4 (1924): 323-334.

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

[6] Shah, Ramesh K., and Alexander Louis London. Laminar flow forced convection in ducts: a source book for compact heat exchanger analytical data. Academic press, 2014.

[7] Hirbodi, Kamran, Mahmood Yaghoubi, and David Martin Warsinger. "New Nusselt number correlations for developing and fully developed laminar flows in concentric circular annular ducts." International Communications in Heat and Mass Transfer 134 (2022): 105936.

[8] Petukhov, Boris Sergeevich, and Lev Ionovich Roizen. "Generalized relationships for heat transfer in turbulent flow of gas in tubes of annular section." Teplofizika vysokikh temperatur 2, no. 1 (1964): 78-81.

[9] Gnielinski, Volker. "Heat transfer coefficients for turbulent flow in concentric annular ducts." Heat transfer engineering 30, no. 6 (2009): 431-436.

Extended Capabilities

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

Version History

Introduced in R2016a

R2026b: Model annular cross-section heat transfer on either wall

You can now model heat transfer on the inner, outer, or both pipe walls when Cross-sectional geometry is Annular.

R2026b: Specify pipe external pressure

Specify a custom external pressure by setting the Environment pressure specification parameter to Specified pressure.

R2023b: Model a flexible pipe wall or variable gravitational acceleration

Set the Pipe wall specification parameter to Flexible to model flexible pipe walls. This parameterization also accounts for pipe wall deformation due to temperature changes.

Set the Gravitational acceleration specification parameter to Variable to enable port G and provide gravitational acceleration as a physical signal.

R2023b: Model pipe with length shorter than elevation gain

You can now model a pipe with length shorter than the elevation change when Elevation gain specification is Constant. The block issues a warning rather than an error if the value of the Elevation gain from port A to port B parameter is greater than the value of the Pipe length parameter.

R2023a: Model pipe characteristics

You can now use new parameters for the Pipe (TL) block to control pipe attributes. Use the Cross-sectional geometry parameter to specify the cross-sectional pipe shape. Use the Surface roughness specification parameter to use the pipe wall material to determine surface roughness.