Pipe (TL)

Closed conduit for the transport of fluid between thermal liquid components

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  • Pipe (TL) block

Description

The Pipe (TL) block models thermal liquid flow through a pipe. The temperature across the pipe is calculated from the temperature 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. For pipes with varying elevation, the variable elevation variant exposes physical signal port EL. To switch between a constant and variable elevation change, right-click on the block and select Simscape > Block Choicesand choose Variable elevation or Constant elevation.

You can optionally model fluid dynamic compressibility, inertia, and wall flexibility. When these phenomena are modeled, the flow properties are calculated for each number of pipe segments that you specify.

Pipe Flexibility

Flexible walls are modeled by 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:

D˙=DSDτ,

where:

  • DS is the post-deformation, steady-state pipe diameter,

    DS=DN+Kc(ppatm),

    where Kc is the Static pressure-diameter compliance, p is the tube pressure, and patm is the atmospheric pressure.

  • DN is the nominal pipe diameter, or the diameter previous to deformation:

    DN=4Sπ,

    where S is the Nominal cross-sectional area of the pipe.

  • D is the pipe Hydraulic diameter.

  • τ is the Viscoelastic pressure time constant.

Heat Transfer at the Pipe Wall

You can model 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, QConv and conduction, QCond.

Heat transfer due to conduction is:

QCond=kISHD(THTI),

where:

  • D is the Hydraulic diameter if the pipe walls are rigid, and is the pipe steady-state diameter, DS, if the pipe walls are flexible.

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

  • SH is the surface area of the pipe wall.

  • TH is the pipe wall temperature.

  • TI is the fluid temperature, taken at the pipe internal node.

Heat transfer due to convection is:

QConv=cp,Avg|m˙Avg|(THTIn)[1exp(hSHcp,Avg|m˙Avg|)],

where:

  • cp, Avg is the average fluid specific heat.

  • m˙Avg is the average mass flow rate through the pipe.

  • TIn is the fluid inlet port temperature.

  • h is the pipe heat transfer coefficient.

The heat transfer coefficient h is:

h=NukAvgD,

except when parameterizing by Nominal temperature differential vs. nominal mass flow rate, where kAvg 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:

Nu=f8(Re1000)Pr1+12.7( f8)1/2(Pr2/31),

where:

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

    f={1.8log10[6.9Re+(ϵR3.7D)1.11]}-2,

    where εR 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:

Nu=aRebPrc,

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:

Nu=0.023Re0.8Pr0.4.

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:

Nu=JM(Re)RePr1/3.

where JM 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:

Nu=Nu(Re,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=hNDN1.8m˙N0.8m˙Avg0.8D1.8,

where:

  • m˙N is the Nominal mass flow rate.

  • m˙Avg is the average mass flow rate:

    m˙Avg=m˙A+m˙B2.

  • hN is the nominal heat transfer coefficient, which is calculated as:

    hN=m˙Ncp,NSH,Nln(TH,NTIn, NTH,NTOut,N),

    where:

    • SH,N is the nominal wall surface area.

    • TH,N is the Nominal wall temperature.

    • TIn,N is the Nominal inflow temperature.

    • TOut,N is the Nominal outflow temperature.

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

hDk(m˙DSμ)0.8.

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

h=hNm˙N0.8m˙Avg0.8.

Pressure Losses Due to Viscous Friction

There are multiple ways to model the pressure differential over the pipe. The Haaland correlation provides an analytical model for flows through circular pipes with a Darcy friction factor. The Nominal pressure drop vs. nominal mass flow rate and the Tabulated data - Darcy friction factor vs. Reynolds number parameterizations allow you to provide data that the block will use as a look-up table during simulation.

Analytical parameterization

When Viscous friction parameterization is set to Haaland correlation and the flow is turbulent, the pressure loss due to friction at pipe walls is determined by the Darcy-Weisbach equation:

pApI=fm˙A|m˙A|2ρIDS2(L+LAdd2),

where:

  • L is the Pipe length.

  • LE is the Aggregate equivalent length of local resistances, which is the equivalent length of a tube that introduces the same amount of loss as the sum of the losses due to other local resistances in the tube.

The pressure differential between port B and internal node I is:

pBpI=fm˙B|m˙B|2ρIDS2(L+LAdd2),

When the flow is laminar, the pressure loss due to friction is calculated in terms of the Laminar friction constant for Darcy friction factor, λ. The pressure differential between port A and internal node I is:

pApI=λμm˙A2ρD2S(L+LAdd2).

The pressure differential between port B and internal node I is:

pBpI=λμm˙B2ρD2S(L+LAdd2).

For transitional flows, the pressure differential due to viscous friction is a smoothed blend between the values for laminar and turbulent pressure losses.

Parameterization by Tabulated Data

When Viscous friction parameterization is set to Nominal pressure drop vs. nominal mass flow rate, the pressure loss due to viscous friction is calculated over the two pipe halves with the loss coefficient Kp:

Δpf,A=12Kpm˙Am˙A2+m˙Th2

Δpf,B=12Kpm˙Bm˙B2+m˙Th2

where:

  • m˙Th is the Mass flow rate threshold for flow reversal.

  • Kp is a pressure loss coefficient. For flexible pipe walls, the pressure loss coefficient is:

    Kp=pNm˙N2DN,

    where:

    • pN is the Nominal pressure drop.

    • m˙N is the Nominal mass flow rate.

    The pressure loss coefficient is

    Kp=pNm˙N2,

    when the pipe walls are rigid. When the Nominal pressure drop and Nominal mass flow rate parameters are vectors, the value of Kp is determined as a least-squares fit of the vector elements.

When the

When Viscous friction parameterization is set to Tabulated data – Darcy friction factor vs. Reynolds number, the friction factor is interpolated from the tabulated data as a function of the Reynolds number:

f=f(Re).

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:

pApB=Δpf+ρIgΔz,

where:

  • pA is the pressure at a port A.

  • pB is the pressure at a port B.

  • Δpf is the pressure differential due to viscous friction, Δpf,A+Δpf,B.

  • g is Gravitational acceleration.

  • Δz the elevation differential between port A and port B, or zA - zB.

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

    pI=pA+pB2.

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

pApI=Δpf,A+ρIgΔz2.

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

pBpI=Δpf,BρIgΔz2.

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

pApI=Δpf,A+ρIgΔz2+m¨ASL2,

where:

  • 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:

pBpI=Δpf,BρIgΔz2+m¨BSL2,

where

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=2Lfc,

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:

m˙A+m˙B=0,

where:

  • m˙A is the mass flow rate at port A.

  • m˙B is the mass flow rate at port B.

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

m˙A+m˙B=ρIV˙,

where:

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

  • 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:

m˙A+m˙B=ρIV˙+ρIV(p˙IβIαIT˙I),

where:

  • pI is the thermal liquid pressure at the internal node I.

  • T˙I is the rate of change of the thermal liquid temperature at the internal node I.

  • βI is the thermal liquid bulk modulus.

  • α is the liquid thermal expansion coefficient.

Energy Balance

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

E.=ϕA+ϕB+ϕHm˙AvggΔz,

where:

  • ϕA is the energy flow rate at port A.

  • ϕB is the energy flow rate at port B.

  • ϕH is the energy flow rate at port H.

The total energy is defined as:

E=ρIuIV,

where:

  • uI is the fluid specific internal energy at node I.

  • V is the pipe volume.

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

E˙=ρIV(dudpdpdt+dudTdTdt)I.

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

E˙=ρIV(dudpdpdt+dudTdTdt)I+(ρIuI+pI)(dVdt)I.

Ports

Input

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

Conserving

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Liquid entry or exit port to the pipe.

Liquid entry or exit port to the pipe.

***Thermal boundary between the fluid volume and the pipe wall. Use this port to capture heat exchanges of various kinds—for example, conductive, convective, or radiative—between the fluid and the environment external to the pipe (taking into account the thermal resistance of the wall when it is significant).

Parameters

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Configuration

Whether to model any change in fluid density due to fluid compressibility. When Fluid compressibility is set to On, changes due to the mass flow rate into the block are calculated in addition to density changes due to changes in pressure.

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

Dependencies

This parameter is active when the Fluid dynamic compressibility block parameter is set to On.

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

This parameter is active when the Fluid dynamic compressibility block parameter is set to On.

Total pipe length across all pipe segments.

Cross-sectional area of the pipe without deformations.

Specifies pipe walls as rigid or flexible. Flexible walls are modeled by a uniform radial expansion that maintains the original pipe cross-sectional shape.

Dependencies

This parameter is active when the Fluid dynamic compressibility block parameter is set to On.

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

This parameter is active when the Fluid dynamic compressibility block parameter is set to Off or when it is set to On but the Pipe wall specification block parameter is set to Rigid.

Elevation differential for constant-elevation pipes. The elevation gain must be less than or equal to the Pipe total length.

Dependencies

This parameter is exposed in the block dialog box when the block variant is set to Constant elevation. Change the block variant to Variable elevation if necessary, for example to capture the tilting of the pipe in during simulation.

Constant of the gravitational acceleration (g) at the mean elevation of the pipe.

Coefficient of pipe radial deformation due to changes in pressure. This is a material property of the pipe.

Dependencies

This parameter is active when the Pipe wall specification block parameter is set to Flexible.

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

Dependencies

This parameter is active when the Pipe wall specification block parameter is set to Flexible.

Viscous Friction

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

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

Dependencies

This parameter is active when the Viscous friction parameterization block parameter is set to Haaland correlation.

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

Dependencies

This parameter is active when the Viscous friction parameterization block parameter is set to Haaland correlation.

Friction constant for laminar flows. The Darcy friction factor captures the contribution of wall friction in pressure loss calculations.

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.

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.

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 Kp is determined as a least-squares fit of the vector elements.

Dependencies

This parameter is active when the Viscous friction parameterization block parameter is set to Nominal pressure drop vs. nominal mass flow rate.

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 Kp is determined as a least-squares fit of the vector elements.

Dependencies

This parameter is active when the Viscous friction parameterization block parameter is set to Nominal pressure drop vs. nominal mass flow rate.

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

This parameter is active when the Viscous friction parameterization block parameter is set to Nominal pressure drop vs. nominal mass flow rate.

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. For reversed flows, or flows from B to A, the same data is applied in the opposite direction.

Dependencies

This parameter is active when the Viscous friction parameterization block parameter is set to Tabulated data - Darcy friction factor vs. Reynolds number.

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

This parameter is active when the Viscous friction parameterization block parameter is set to Tabulated data - Darcy friction factor vs. Reynolds number.

Heat Transfer

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

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

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to Gnielinski correlation, Nominal temperature differential vs. nominal mass flow rate, or Dittus-Boelter correlation.

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 hp is determined as a least-squares fit of the vector elements.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to Nominal temperature differential vs. nominal mass flow rate.

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

This parameter is active when the Heat transfer parameterization block parameter is set to Nominal temperature differential vs. nominal mass flow rate.

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

This parameter is active when the Heat transfer parameterization block parameter is set to Nominal temperature differential vs. nominal mass flow rate.

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

This parameter is active when the Heat transfer parameterization block parameter is set to Nominal temperature differential vs. nominal mass flow rate.

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

This parameter is active when the Heat transfer parameterization block parameter is set to Nominal temperature differential vs. nominal mass flow rate.

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

This parameter is active when the Heat transfer parameterization block parameter is set to Dittus-Boelter correlation.

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

This parameter is active when the Heat transfer parameterization block parameter is set to Dittus-Boelter correlation.

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

This parameter is active when the Heat transfer parameterization block parameter is set to Dittus-Boelter correlation.

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

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

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.

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. The vector elements must be listed in ascending order and must be greater than 0.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to Tabulated data - Nusselt number vs. Reynolds number & Prandtl number.

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. The vector elements must be listed in ascending order.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to Tabulated data - Nusselt number vs. Reynolds number & Prandtl number.

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

This parameter is active when the Heat transfer parameterization block parameter is set to Tabulated data - Nusselt number vs. Reynolds number & Prandtl number.

Initial Conditions

Liquid temperature at the start of the simulation, specified as a scalar or vector. A vector n elements long defines the liquid temperature for each of n pipe segments. If the vector is two elements long, the temperature along the pipe is linearly distributed between the two element values. If the vector is three or more elements long, the initial temperature in the nth segment is set by the nth element of the vector.

Absolute liquid pressure at the start of the simulation, specified as a scalar or vector. A vector n elements long defines the liquid pressure for each of n pipe segments. If the vector is two elements long, the pressure along the pipe is linearly distributed between the two element values. If the vector is three or more elements long, the initial pressure in the nth segment is set by the nth element of the vector.

Extended Capabilities

C/C++ Code Generation
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