# Pipe (TL)

Rigid conduit for fluid flow in thermal liquid systems

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• Simscape / Foundation Library / Thermal Liquid / Elements

## Description

The Pipe (TL) block represents a pipeline segment with a fixed volume of liquid. The liquid experiences pressure losses due to viscous friction and heat transfer due to convection between the fluid and the pipe wall. Viscous friction follows from the Darcy-Weisbach equation, while the heat exchange coefficient follows from Nusselt number correlations.

### 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 dynamic compressibility is off, 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 dynamic compressibility is on, 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 on, 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.

### Mass Balance

The mass conservation equation for the pipe is

${\stackrel{˙}{m}}_{\text{A}}+{\stackrel{˙}{m}}_{\text{B}}=\left\{\begin{array}{cc}0,& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{dynamic}\text{\hspace{0.17em}}\text{compressibility}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{'off'}\\ V\rho \left(\frac{1}{\beta }\frac{dp}{dt}-\alpha \frac{dT}{dt}\right),& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{dynamic}\text{\hspace{0.17em}}\text{compressibility}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{'on'}\end{array}$

where:

• ${\stackrel{˙}{m}}_{\text{A}}$ and ${\stackrel{˙}{m}}_{\text{B}}$ are the mass flow rates through ports A and B.

• V is the pipe fluid volume.

• ρ is the thermal liquid density in the pipe.

• β is the isothermal bulk modulus in the pipe.

• α is the isobaric thermal expansion coefficient in the pipe.

• p is the thermal liquid pressure in the pipe.

• T is the thermal liquid temperature in the pipe.

### Momentum Balance

The table shows the momentum conservation equations for each half pipe.

 For half pipe adjacent to port A ${p}_{A}-p=\left\{\begin{array}{cc}\Delta {p}_{\text{v,A}},& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{inertia}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{off}\\ \Delta {p}_{\text{v,A}}+\frac{L}{2S}{\stackrel{¨}{m}}_{\text{A}},& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{inertia}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{on}\end{array}$ For half pipe adjacent to port B ${p}_{B}-p=\left\{\begin{array}{cc}\Delta {p}_{\text{v,B}},& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{inertia}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{off}\\ \Delta {p}_{\text{v,B}}+\frac{L}{2S}{\stackrel{¨}{m}}_{\text{B}},& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{inertia}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{on}\end{array}$

In the equations:

• S is the pipe cross-sectional area.

• p, pA, and pB are the liquid pressures in the pipe, at port A and port B.

• Δpv,A and Δpv,B are the viscous friction pressure losses between the pipe volume center and ports A and B.

### Viscous Friction Pressure Losses

The table shows the viscous friction pressure loss equations for each half pipe.

 For half pipe adjacent to port A For half pipe adjacent to port B

In the equations:

• λ is the pipe shape factor.

• ν is the kinematic viscosity of the thermal liquid in the pipe.

• Leq is the aggregate equivalent length of the local pipe resistances.

• D is the hydraulic diameter of the pipe.

• fA and fB are the Darcy friction factors in the pipe halves adjacent to ports A and B.

• ReA and ReB are the Reynolds numbers at ports A and B.

• Rel is the Reynolds number above which the flow transitions to turbulent.

• Ret is the Reynolds number below which the flow transitions to laminar.

The Darcy friction factors follow from the Haaland approximation for the turbulent regime:

$f=\frac{1}{{\left[-1.8{\mathrm{log}}_{10}\left(\frac{6.9}{\mathrm{Re}}+{\left(\frac{1}{3.7}\frac{r}{D}\right)}^{1.11}\right)\right]}^{2}},$

where:

• f is the Darcy friction factor.

• r is the pipe surface roughness.

### Energy Balance

The energy conservation equation for the pipe is

$V\frac{d\left(\rho u\right)}{dt}={\varphi }_{\text{A}}+{\varphi }_{\text{B}}+{Q}_{H},$

where:

• ΦA and ΦB are the total energy flow rates into the pipe through ports A and B.

• QH is the heat flow rate into the pipe through the pipe wall.

### Wall Heat Flow Rate

The heat flow rate between the thermal liquid and the pipe wall is:

${Q}_{H}={Q}_{conv}+\frac{k{S}_{H}}{D}\left({T}_{H}-T\right),$

where:

• QH is the net heat flow rate.

• Qconv is the portion of the heat flow rate attributed to convection at nonzero flow rates.

• k is the thermal conductivity of the thermal liquid in the pipe.

• SH is the surface area of the pipe wall, the product of the pipe perimeter and length.

• TH is the temperature at the pipe wall.

Assuming an exponential temperature distribution along the pipe, the convective heat transfer is

${Q}_{conv}=|{\stackrel{˙}{m}}_{avg}|{c}_{p,avg}\left({T}_{H}-{T}_{in}\right)\left(1-\text{exp}\left(-\frac{h{A}_{H}}{|{\stackrel{˙}{m}}_{avg}|{c}_{p,avg}}\right)\right),$

where:

• ${\stackrel{˙}{m}}_{avg}=\left({\stackrel{˙}{m}}_{A}-{\stackrel{˙}{m}}_{B}\right)/2$ is the average mass flow rate from port A to port B.

• ${c}_{{p}_{avg}}$ is the specific heat evaluated at the average temperature.

• Tin is the inlet temperature depending on flow direction.

The heat transfer coefficient, hcoeff, depends on the Nusselt number:

${h}_{coeff}=Nu\frac{{k}_{avg}}{D},$

where kavg, is the thermal conductivity evaluated at the average temperature. The Nusselt number depends on the flow regime. The Nusselt number in the laminar flow regime is constant and equal to the Nusselt number for laminar flow heat transfer parameter value. The Nusselt number in the turbulent flow regime is computed from the Gnielinski correlation:

$N{u}_{tur}=\frac{\frac{{f}_{avg}}{8}\left({\mathrm{Re}}_{avg}-1000\right){\mathrm{Pr}}_{avg}}{1+12.7\sqrt{\frac{{f}_{avg}}{8}}\left({\mathrm{Pr}}_{avg}^{2/3}-1\right)},$

where favg is the Darcy friction factor at the average Reynolds number, Reavg, and Pravg is the Prandtl number evaluated at the average temperature. The average Reynolds number is computed as:

${\mathrm{Re}}_{avg}=\frac{|{\stackrel{˙}{m}}_{avg}|D}{S{\mu }_{avg}},$

where μavg is the dynamic viscosity evaluated at the average temperature. When the average Reynolds number is between the Laminar flow upper Reynolds number limit and the Turbulent flow lower Reynolds number limit parameter values, the Nusselt number follows a smooth transition between the laminar and turbulent Nusselt number values.

### Assumptions and Limitations

• The pipe wall is rigid.

• The flow is fully developed.

• The effect of gravity is negligible.

## Ports

### Conserving

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Thermal liquid conserving port associated with the inlet or outlet of the pipe. This block has no intrinsic directionality.

Thermal liquid conserving port associated with the inlet or outlet of the pipe. This block has no intrinsic directionality.

Thermal conserving port associated with the temperature of the pipe wall. This temperature may differ from the temperature of the thermal liquid inside the pipe.

## Parameters

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### Geometry

The length of the pipe along the direction of flow.

The internal area of the pipe normal to the direction of the flow.

Diameter of an equivalent cylindrical pipe with the same cross-sectional area.

### Friction and Heat Transfer

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. This length is added to the geometrical pipe length only for friction calculations. The liquid volume inside the pipe depends only on the pipe geometrical length, defined by the Pipe length parameter.

Average depth of all surface defects on the internal surface of the pipe, which affects the pressure loss in the turbulent flow regime.

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 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.

Dimensionless factor that encodes the effect of pipe cross-sectional geometry on the viscous friction losses in the laminar flow regime. Typical values are 64 for a circular cross section, 57 for a square cross section, 62 for a rectangular cross section with an aspect ratio of 2, and 96 for a thin annular cross section [1].

Ratio of convective to conductive heat transfer in the laminar flow regime. Its value depends on the pipe cross-sectional geometry and pipe wall thermal boundary conditions, such as constant temperature or constant heat flux. Typical value is 3.66, for a circular cross section with constant wall temperature [2].

### Effects and Initial Conditions

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.

Select whether to account for the flow inertia of the liquid. Flow inertia gives the liquid a resistance to changes in mass flow rate.

#### Dependencies

Enabled when the Fluid dynamic compressibility parameter is set to On.

Liquid pressure in the pipe at the start of simulation.

#### Dependencies

Enabled when the Fluid dynamic compressibility parameter is set to On.

Liquid temperature in the pipe at the start of simulation.

Mass flow rate from port A to port B at time zero.

#### Dependencies

Enabled when the Fluid inertia parameter is set to On.

## References

[1] White, F. M., Fluid Mechanics. 7th Ed, Section 6.8. McGraw-Hill, 2011.

[2] Cengel, Y. A., Heat and Mass Transfer – A Practical Approach. 3rd Ed, Section 8.5. McGraw-Hill, 2007.

## Version History

Introduced in R2013b