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# Pipe (TL)

Rigid conduit for liquid transport

## Library

Thermal Liquid/Elements

## Description

The Pipe (TL) block represents a pipeline segment with a fixed volume of liquid. The liquid experiences pressure losses and heating due to, respectively, viscous friction and conductive heat transfer with the pipe wall. Viscous friction follows from the Darcy-Weisbach law, while the heat exchange coefficient follows from Nusselt number correlations. Heat transfer can occur in an unsteady manner.

The block includes parameters to account for the dynamic compressibility and inertia of liquid in a pipe. However, by default the block treats liquid flow through the pipe as steady and liquid mass within the pipe as constant. In this mode, the momentum and mass equations of this block are in their steady states. The liquid behaves as if it were incompressible. Pressure waves due to liquid inertia are absent in the pipe.

Depending on the effects you include, the block can function in three configurations: resistive tube, compressible resistive tube, and pipeline segment. The table summarizes the effects present in each configuration.

ConfigurationDynamic CompressibilityFlow InertiaThermal Dynamics
Resistive tubeOffOffOn
Compressible resistive tubeOnOffOn
Pipeline segmentOnOnOn

The configuration to use depends on the relevant effects the model must capture. The pipeline segment configuration provides the greatest accuracy. However, this configuration also increases model complexity, raising the simulation computational cost and challenging the convergence to a numerical solution in rapid transient processes. As the simplest in the list, the resistive tube configuration provides a good starting point in a model. This is the default configuration of the block.

To view the block source code, at the MATLAB® command line enter:

```edit <matlabroot>\toolbox\physmod\simscape\library\m\...
+foundation\+thermal_liquid\+elements\CONFIGURATION.ssc```

Replace <matlabroot> with the output of the matlabroot command and CONFIGURATION with one of the three configuration names:

• resistive_tube

• compressible_resistive_tube

• pipeline_segment

Use this block in the resistive tube configuration when:

• Thermal dynamic effects are important but flow dynamic effects, which have a smaller time scale, are not.

• Liquid mass in the pipe is a negligible fraction of the total liquid mass in the system.

The resistive tube configuration is the recommended starting point for this block, even if fluid dynamic compressibility and flow inertia are important. The simulation results using this configuration provides reasonable initial conditions for more advanced configurations in which fluid dynamic compressibility and flow inertia are important—e.g. compressible resistive tube and pipeline segment configurations.

A typical application of the resistive tube configuration is the thermal analysis of an open-loop circuit linked to reservoirs or of a closed-loop circuit in which the majority of the liquid mass exists outside the pipe.

Use this block in the compressible resistive tube configuration when:

• Thermal dynamic effects are important but flow dynamic effects, which have a smaller time scale, are not.

• Liquid mass in the pipe is not negligible with respect to the total liquid mass in the system

A typical application of the compressible resistive tube configuration include the thermal analysis of a closed-loop circuit, not linked to any reservoir, in which thermal expansion raises liquid pressure—e.g. a car cooling system.

Use this block in the pipeline segment configuration when the characteristic time of the thermal liquid system is close to the liquid compressibility time scale:

where L is the characteristic longitudinal length of the pipe in which waves can develop and a is the speed of sound in the liquid. A typical application of the pipeline segment configuration is the study of the water hammer effect due to fast-shutting valves.

The following equations govern the behavior of liquid in the pipe:

where

 A Cross-sectional area of the pipe V Volume of liquid in the pipe L Length of the pipe pint, pA, pB Pressures of liquid in the pipe, at port A, and at port B Tint, TA, TB Temperatures of liquid in the pipe, at port A, and at port B , Mass flow rates of liquid into the pipe at port A and at port B vA, vB Velocities of liquid into the pipe at ports A and B Fvd,A, Fvd,B Viscous dissipations between the pipe volume center and ports A and B βint Isothermal bulk modulus of liquid in the pipe aint Isobaric coefficient of thermal expansion of liquid in the pipe cint Specific heat of liquid in the pipe uint Internal energy of liquid in the pipe ρint Density of liquid in the pipe ΦA, ΦB, ΦW Thermal fluxes into the pipe at ports A, B, and W

In the resistive tube configuration, the two momentum equations (second and third in the preceding system of equations) combine into a single pressure difference equation relating the liquid pressures at ports A and B. The liquid pressure inside the pipe equals the average of these two liquid pressures. Variations in liquid density are small, making momentum fluxes insignificant with respect to viscous forces.

Liquid velocities at ports A and B follow from the mass flow rates into the pipe through the same ports:

where ρA,u and ρB,u are the upwind mass densities of liquid at ports A and B.

Viscous forces at pipe inlets A and B depend on the flow regime (laminar or turbulent):

where

 DH Hydraulic diameter of the pipe fA, fB Darcy friction factors in the two pipe halves adjacent to pipe inlets A and B KS Shape factor of the pipe Leq Aggregate equivalent length of local pipe resistances ReA, ReB Reynolds numbers at pipe inlets A and B Rel Maximum Reynolds number corresponding to laminar flow in the pipe ReB Minimum Reynolds number corresponding to turbulent flow in the pipe νA,u, νB,u Upwind liquid dynamic viscosities at pipe inlets A and B

The block smooths the transition between laminar and turbulent flow regimes (Ret > Re > Rel) based on the Reynolds number. At pipe inlets A and B, the Reynolds numbers are

where μA,u and μB,u are the upwind liquid dynamic viscosities at pipe inlets A and B. The Darcy friction factor, DH, satisfies the Haaland approximation in the turbulent flow regime:

where r is the roughness of the internal pipe surface.

The following equation governs the convective heat transfer between the pipe wall and the liquid it encloses:

where

 DH Hydraulic diameter of the pipe TW Temperature of the pipe wall (port W) TMTD Mean liquid temperature difference between pipe inlets A and B P Cross-sectional perimeter of the pipe

The cross-sectional perimeter of the pipe, P, follows from the hydraulic diameter:

The heat transfer coefficient follows from empirical correlations involving the Nusselt number. These correlations relate the Nusselt number to powers of Reynolds and Prandtl numbers, the ratio of the pipe hydraulic diameter to its height, and the ratio of the liquid dynamic viscosities at the inlet and wall temperatures. The correlation used depends on the flow regime in the pipe—laminar or turbulent. The block smooths the transition between flow regimes based on the Reynolds number:

where

 Nu Nusselt number Nul Nusselt number at the maximum Reynolds number corresponding to laminar flow Nut Nusselt number at the minimum Reynolds number corresponding to turbulent flow Pr Prandtl number Re Reynolds number al, bl, cl, dl, el Empirical correlation coefficients for Nusselt number computation in the laminar flow regime at, bt, ct, dt, et Empirical correlation coefficients for Nusselt number computation in the turbulent flow regime kInt Thermal conductivity of liquid in the pipe μInt Dynamic viscosity of liquid in the pipe

In the laminar regime, the default coefficient values follow from the Sieder and Tate correlation:

In the turbulent flow regime, the default coefficient values follow from the Colburn correlation:

## Assumptions and Limitations

• Pipe walls are not compliant.

• Flow through the pipe is fully developed.

• Gravitational effects on liquid pressure are negligible.

## Dialog Box and Parameters

### Geometry

Longitudinal length

Enter the longitudinal length of the pipe. This is the length of the pipe along the direction of flow. The default value is 5 m.

Hydraulic diameter

Enter the hydraulic diameter of the pipe. This is the diameter of a cylindrical pipe with the same cross-sectional area. The default value is 0.1128 m.

Cross-sectional area

Enter the cross-sectional area of the pipe. This is the area of the pipe normal to the direction of flow. The default value is 0.01 m^2.

### Viscous Friction

Aggregate equivalent length of local resistances

Enter the combined length of all local resistances present in the pipe. Local resistances include bends, fittings, armatures, and pipe inlets and outlets. The effect of the local resistances is to increase the effective length of the pipe segment. The default value is 1 m.

Shape factor

Enter the shape factor of the pipe. This dimensionless factor encodes the ratio between the height and width of the pipe, correcting for noncircular cross-sectional shapes. The block uses this factor to determine pressure losses in the laminar flow regime. The default value is 64, corresponding to a pipe with circular cross-section.

Internal surface absolute roughness

Enter the absolute roughness of the internal surface of the pipe. This roughness equals the average height of surface defects inside the pipe. The block uses the absolute roughness to determine pressure losses in the turbulent flow regime. The default value is 1.5e-5 m, corresponding to drawn tubing.

Laminar flow upper margin

Enter the Reynolds number above which flow begins to transition from laminar to turbulent. This number equals the maximum Reynolds number corresponding to fully developed laminar flow. The default value is 2000.

Turbulent flow lower margin

Enter the Reynolds number below which flow begins to transition from turbulent to laminar. This number equals to the minimum Reynolds number corresponding to fully developed turbulent flow. The default value is 4000.

### Heat Transfer

Laminar regime Nusselt number correlation coefficients

Enter a vector with the empirical correlation coefficients for convective heat transfer in the laminar flow regime. The coefficients must appear in the order [al bl cl dl el], corresponding to the empirical correlation

The block uses the empirical correlation to determine heat transfer between the liquid and the pipe surface in the laminar regime. The default vector is [1.86 1/3 1/3 1/3 0.14], from the Sieder and Tate correlation in the laminar regime.

Turbulent regime Nusselt number correlation coefficients

Enter a vector with the empirical correlation coefficients for convective heat transfer in the turbulent flow regime. The coefficients must appear in the order [at bt ct dt et], corresponding to the empirical correlation

The block uses the empirical correlation to determine heat transfer between the liquid and the pipe surface in the turbulent regime. The default vector is [0.023 0.8 1/3 0 0], from the Colburn correlation in the turbulent regime.

### Effects and Initial Conditions

Fluid dynamic compressibility

Select whether to account for the dynamic compressibility of the liquid. Dynamic compressibility gives the liquid density a dependence on pressure and temperature, impacting the transient response of the system at small time scales. Selecting On displays the additional parameter Initial fluid pressure in the pipe. The default setting is Off.

Flow inertia

Select whether to account for the flow inertia of the liquid. Flow inertia gives the liquid a resistance to changes in mass flow rate. Selecting On displays the additional parameter Initial mass flow rate oriented from A to B. The default setting is Off.

Initial fluid temperature inside the pipe

Enter the liquid temperature in the pipe at time zero. The default value is 293.15 K.

Initial fluid pressure inside the pipe

Enter the liquid pressure in the pipe at time zero. This parameter appears only when Fluid dynamic compressibility is On. The default value is 1 atm.

Initial mass flow rate oriented from A to B

Enter the mass flow rate from port A to port B at time zero. This parameter is visible only when Flow inertia is On. The default value is 0.1 kg/s.

## Ports

The block has two thermal liquid conserving ports, A and B, and one thermal conserving port, W.