Isothermal liquid element that models laminar leakage flow for various geometries (2024)

Isothermal liquid element that models laminar leakage flow for various geometries

Since R2020a

Libraries:
Simscape / Foundation Library / Isothermal Liquid / Elements

• Circular — Provide the diameter and longitudinal length.

• Annular — Provide the inner diameter, outer diameter, and longitudinal length.

• Rectangular — Provide the width, height, and longitudinal length.

• Elliptical — Provide the major axis, minor axis, and longitudinal length.

• Equilateral triangular — Provide the length of the triangle side and longitudinal length.

• Custom — Provide the flow resistance value, that is, the ratio of pressure loss to mass flow rate. In this configuration, the block acts as a linear hydraulic resistance.

$${\dot{m}}_{\text{A}}=\{\begin{array}{ll}\frac{K_{geom}}{\nu L}\left(p_A-p_B\right),\hfill & \text{commongeometries}\hfill \\ \frac{\overline{\rho }}{R}\left(p_A-p_B\right),\hfill & \text{custom}\hfill \end{array}$$

$$K_{\text{geom}}=\{\begin{array}{ll}\frac{\pi d^4}{128},\hfill & \text{circular}\hfill \\ \frac{\pi }{128}\left(d_o^4-d_i^4-\frac{{\left(d_o^2-d_i^2\right)}^2}{\log \left(d_o/d_i\right)}\right),\hfill & \text{annular}\hfill \\ \frac{w{h}^3}{12}\left(1-\frac{192h}{w{\pi }^5}\tanh \left(\frac{\pi w}{2h}\right)\right),\hfill & \text{rectangular}\hfill \\ \frac{\pi {\left(ab\right)}^3}{64\left(a^2+b^2\right)},\hfill & \text{elliptical}\hfill \\ l_{side}^4\frac{\sqrt{3}}{320},\hfill & \text{triangular}\hfill \end{array}$$

$$\overline{\rho }=\frac{{\rho }_A+{\rho }_B}{2}$$

• $$\dot{m}$$_A is the mass flow rate through port A.

• p_A and p_B are pressures at ports A and B, respectively.

• K_geom is the coefficient calculated based on the cross-sectional geometry.

• d is the diameter of the circular cross section.

• d_o is the outer diameter of the annular cross section.

• d_i is the inner diameter of the annular cross section.

• w is the width of the rectangular cross section.

• h is the height of the rectangular cross section.

• a is the major axis of the elliptical cross section.

• b is the minor axis of the elliptical cross section.

• l_side is the length of the side of the triangular cross section.

• L is the longitudinal length.

• R is the linear hydraulic resistance for custom geometry.

• ν is the liquid kinematic viscosity at atmospheric pressure, which is a global parameter defined by the Isothermal Liquid Properties (IL) block connected to the circuit.

• $$\overline{\rho }$$ is the average fluid mixture density.

• ρ_A and ρ_B are fluid mixture density values at ports A and B, respectively. Equations used to compute the fluid mixture density depend on the selected isothermal liquid model. For detailed information, see Isothermal Liquid Modeling Options.

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