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Pipe with variable liquid volume in an isothermal liquid system

**Library:**Simscape / Fluids / Isothermal Liquid / Pipes & Fittings

The Partially Filled Pipe (IL) block models a pipe with the capacity for varying internal liquid levels. The pipe may also become completely dry during simulation.

In addition to liquid connections at ports **A** and
**B**, port **AL** receives the inlet liquid level
from connected blocks as a physical signal. If the value at **AL** is
greater than zero, **A** is submerged. If the value at
**AL** is less than or equal to zero, the port is exposed. The pipe
liquid level is transmitted as a physical signal to connecting blocks at port
**L**.

Port **A** is always higher than port **B**. If port
**A** becomes exposed, the pipe can be filled through port
**B**. When fluid enters the pipe through port
**B**, the mass flow rate through port **A** is 0
until the pipe is fully filled, at which point $${\dot{m}}_{A}={\dot{m}}_{B}.$$

This block can be used in conjunction with the Tank (IL) block when fluid levels
are expected to fall below the tank inlet. Multiple Partially Filled Pipe (IL)
blocks can also be connected in series or parallel. However, because a partially
filled pipe can only be filled at port **B**, if port
**A** of one block in a parallel configuration becomes exposed,
it may not be possible to refill this pipe if its connection at port
**B** cannot refill the pipe.

The pressure differential over the pipe,
*p*_{A} –
*p*_{B}, comprises losses due to wall
friction and hydrostatic pressure differences between the liquid surface height and
the liquid height at port **A**:

$${p}_{A}-{p}_{B}=\Delta {p}_{loss}+\Delta {p}_{elev}.$$

Friction losses depend on the fluid regime in the pipe. If the flow is laminar, the friction losses are:

$$\Delta {p}_{loss}=\frac{-{\dot{m}}_{B}\upsilon {f}_{s}\widehat{L}}{2{D}_{h}^{2}{A}^{2}},$$

where:

*ν*is the fluid kinematic viscosity.*f*_{S}is the**Laminar friction constant for Darcy friction factor**.$$\widehat{L}$$ is the sum of the pipe length and its

**Aggregate equivalent length of local resistances**, in proportion to the pipe fill level: $$\widehat{L}=\left(L+{L}_{add}\right)\frac{l}{{l}_{\mathrm{max}}}.$$*D*_{h}is the pipe hydraulic diameter. If the pipe cross-section is not circular, the hydraulic diameter is the equivalent circular diameter.

If the flow is turbulent, the friction losses are:

$$\Delta {p}_{loss}=\frac{-{\dot{m}}_{B}\left|{\dot{m}}_{B}\right|f\widehat{L}}{2\rho {D}_{h}^{2}{A}^{2}}.$$

*f* is the Darcy friction factor for turbulent flows, which is
determined by the empirical Haaland correlation:

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

where *ε* is the **Internal surface
absolute roughness**. The Reynolds number is based on the mass flow
rate at port **B** and the pipe hydraulic diameter.

The hydrostatic pressure difference is $$\Delta {p}_{elev}=\rho gl.$$

The flow in the pipe is dictated by the internal fluid level and the conditions at
port **B**. The pipe can be filled or drained at
**B** if the pipe is partially empty. If the pipe is fully
filled, $${\dot{m}}_{A}={\dot{m}}_{B}.$$, and mass is conserved:

$${\dot{m}}_{A}+{\dot{m}}_{B}=0.$$

The mass of fluid in the pipe is determined by the relative fill level of the pipe:

$$M=\rho AL\frac{l}{{l}_{\mathrm{max}}}.$$

This block does not account for dynamic compressibility or fluid inertia, and does not model the dynamics of air (or second liquid) in the pipe.