Inner hydraulic diameter is defined with:

$\mathrm{D\_\_h\_act}\=\mathrm{D\_\_h}$

Flow area is defined with:

$\mathrm{A\_\_act}\=\mathrm{A\_\_cir}$

Geometrical coefficient for laminar flow is defined with:

$\mathrm{Geo\_\_act}\=1$

Geometrical length for Bend to get the values from tables:

$\mathrm{D0\_\_Bend}\=0$

Total loss coefficient:

$\mathrm{zeta\_\_act}\=\left\{\begin{array}{cc}\mathrm{zeta\_\_in}& \mathrm{Use} \mathrm{\ζ} \mathrm{as} \mathrm{input}\=\mathrm{true}\\ \mathrm{zeta}& \mathrm{others}\end{array}\right.$

(*) Reference[1] : page.365-366.

Inner hydraulic diameter is defined with:

$\mathrm{D\_\_h\_act}\=\mathrm{D\_\_h}$

Flow area is defined with:

$\mathrm{A\_\_act}\=\mathrm{A\_\_cir}$

Geometrical coefficient for laminar flow is defined with:

$\mathrm{Geo\_\_act}\=1$

Geometrical length for Bend to get the values from tables:

$\mathrm{D0\_\_Bend}\=\mathrm{D\_\_h}$

Local resistance is defined with:

$\mathrm{zeta\_\_loc}\=\mathrm{A\_\_loc\_Elbow}\cdot \mathrm{C\_\_loc}\cdot \max \left({10}^{-8}\,0.95\cdot \sin {\left(\frac{\mathrm{\θ}}{2}\right)}^2\+2.05\cdot \sin {\left(\frac{\mathrm{\θ}}{2}\right)}^4\right)$

Correction factor C is defined with:

$\mathrm{C\_\_loc}\=1$

Correction factor A is defined with:

$\mathrm{A\_\_loc\_Elbow}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\mathrm{\θ}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_A\_Elbow}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Friction resistance is defined with:

$\mathrm{zeta\_\_fri}\=\mathrm{k\_\_\δ}\cdot \mathrm{k\_\_Re}\cdot \mathrm{zeta\_\_loc}$

Correction factor k_delta (Roughness dependency) is defined with:

$\mathrm{k\_\_\δ}\=\min \left(1.5\,\max \left(1.0\,1.0\+500\cdot \frac{\mathrm{roughness}}{\mathrm{D\_\_h\_act}}\right)\right)$

Correction factor k_Re (Reynolds number dependency) is defined with:

$\mathrm{k\_\_Re}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\mathrm{Re}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_k\_Re\_Elbow}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Total resistance is defined with:

$\mathrm{zeta\_\_act}\=\mathrm{zeta\_\_fri}$

(*) Reference: page.365-366 in [1].

Inner hydraulic diameter is defined with:

$\mathrm{D\_\_h\_act}\=\frac{2}{\frac{1}{\mathrm{a\_\_rec}}\+\frac{1}{\mathrm{b\_\_rec}}}$

Flow area is defined with:

$\mathrm{A\_\_act}\=\mathrm{A\_\_rec}$

Geometrical coefficient for laminar flow is defined with:

$\mathrm{Geo\_\_act}\=\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\frac{\mathrm{b\_\_rec}}{\mathrm{a\_\_rec}}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_geo\_rec}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Geometrical length for Bend to get the values from tables:

$\mathrm{D0\_\_Bend}\=\mathrm{a\_\_rec}$

Local resistance is defined with:

$\mathrm{zeta\_\_loc}\=\mathrm{A\_\_loc\_Elbow}\cdot \mathrm{C\_\_loc}\cdot \max \left({10}^{-8}\,0.95\cdot \sin {\left(\frac{\mathrm{\θ}}{2}\right)}^2\+2.05\cdot \sin {\left(\frac{\mathrm{\θ}}{2}\right)}^4\right)$

Correction factor C is defined with:

$\mathrm{C\_\_loc}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\frac{\mathrm{b\_\_rec}}{\mathrm{a\_\_rec}}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_C\_Elbow}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Correction factor A is defined with:

$\mathrm{A\_\_loc\_Elbow}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\mathrm{\θ}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_A\_Elbow}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Friction resistance is defined with:

$\mathrm{zeta\_\_fri}\=\mathrm{k\_\_\δ}\cdot \mathrm{k\_\_Re}\cdot \mathrm{zeta\_\_loc}$

Correction factor k_delta (Roughness dependency) is defined with:

$\mathrm{k\_\_\δ}\=\min \left(1.5\,\max \left(1.0\,1.0\+500\cdot \frac{\mathrm{roughness}}{\mathrm{D\_\_h\_act}}\right)\right)$

Correction factor k_Re (Reynolds number dependency) is defined with:

$\mathrm{k\_\_Re}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\mathrm{Re}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_k\_Re\_Elbow}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Total resistance is defined with:

$\mathrm{zeta\_\_act}\=\mathrm{zeta\_\_fri}$

The following diagram shows...

(*) Reference[1] : page.357-339.

Inner hydraulic diameter is defined with:

$\mathrm{D\_\_h\_act}\=\mathrm{D\_\_h}$

Flow area is defined with:

$\mathrm{A\_\_act}\=\mathrm{A\_\_cir}$

Geometrical coefficient for laminar flow is defined with:

$\mathrm{Geo\_\_act}\=1$

Geometrical length for Bend to get the values from tables:

$\mathrm{D0\_\_Bend}\=\mathrm{D\_\_h}$

Local resistance is defined with:

$\mathrm{zeta\_\_loc}\=\{\begin{array}{cc}\mathrm{k\_\_Re}\cdot \mathrm{k\_\_\δ}\cdot \mathrm{A1\_\_loc\_Bend}\cdot \mathrm{B\_\_loc\_Bend}\cdot \mathrm{C\_\_loc}& \mathrm{Re}\>10000\\ \frac{\mathrm{A2\_\_loc\_Bend}}{\max \left(3000\,\mathrm{Re}\right)}\+\mathrm{A1\_\_loc\_Bend}\cdot \mathrm{B\_\_loc\_Bend}\cdot \mathrm{C\_\_loc}& \mathrm{Re}\le 10000\end{array}$

Correction factor C is defined with:

$\mathrm{C\_\_loc}\=1$

Correction factor A1 is defined with:

$\mathrm{A1\_\_loc\_Bend}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\mathrm{\θ}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_A1\_Bend}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Correction factor A2 is defined with:

$\mathrm{A2\_\_loc\_Bend}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\frac{\mathrm{R0}}{\mathrm{D0\_\_Bend}}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_A2\_Bend}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Correction factor B is defined with:

$\mathrm{B\_\_loc\_Bend}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\frac{\mathrm{R0}}{\mathrm{D0\_\_Bend}}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_B\_Bend}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Correction factor k_delta (Roughness dependency) is defined with:

$\mathrm{k\_\_\δ}\=\{\begin{array}{cc}\min \left(1.5\,1.0\+0.001\cdot \frac{\mathrm{roughness}}{\mathrm{D\_\_h\_act}}\right)& \frac{\mathrm{R0}}{\mathrm{D0\_\_Bend}}\le 0.55\\ \min \left(2.0\,\max \left(1.0\,\frac{\mathrm{\λ\_\_tur\_roughness}}{\mathrm{\λ\_\_tur\_smooth}}\right)\right)& \frac{\mathrm{R0}}{\mathrm{D0\_\_Bend}}\>0.55\end{array}$

Friction coefficient of smooth pipe for k_Re calculation is defined with:

$\mathrm{\λ\_\_tur\_smooth}\=0.25\cdot {\left(\frac{1}{\mathrm{log10}\left(\frac{5.74}{\max {\left(\mathrm{Re}\,1.0\right)}^{0.9}}\right)}\right)}^2$

Friction coefficient of rough pipe for k_Re calculation is defined with  (Swamee and Jain's approximation[2]):

$\mathrm{\λ\_\_tur\_roughness}\=0.25\cdot {\left(\frac{1}{\mathrm{log10}\left(\frac{\frac{\mathrm{roughness}}{\mathrm{D\_\_h\_act}}}{3.7}\+\frac{5.74}{\max {\left(\mathrm{Re}\,1.0\right)}^{0.9}}\right)}\right)}^2$

Correction factor k_Re (Reynolds number dependency) is defined with:

$\mathrm{k\_\_Re}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\mathrm{Re}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_k\_Re\_Bend}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Friction resistance is defined with:

$\mathrm{zeta\_\_fri}\=\mathrm{\θ}\cdot \mathrm{\λ}\cdot \frac{\mathrm{R0}}{\mathrm{D\_\_h\_act}}$

The friction factor of flow is calculated with:

$\mathrm{\λ}\=\mathrm{`HeatTransfer.Functions.lambda\_Re`}\left(\mathrm{Re}\,\mathrm{roughness}\,\mathrm{D\_\_h\_act}\,\mathrm{Re\_\_CoT}\,\mathrm{IF\_\_speed}\,\mathrm{Geo\_\_act}\right)$

(*) The above function $\mathrm{`HeatTransfer.Functions.lambda\_Re`}$ is to calculated friction factor for Laminar and Turbulent flow.

     Regarding the implementation of the friction factor calculation, see the reference section below.

Total resistance is defined with:

$\mathrm{zeta\_\_act}\=\mathrm{zeta\_\_loc}\+\mathrm{zeta\_\_fri}$

(*) Reference[1]: page.357-339.

Inner hydraulic diameter is defined with:

$\mathrm{D\_\_h\_act}\=\frac{2}{\frac{1}{\mathrm{a\_\_rec}}\+\frac{1}{\mathrm{b\_\_rec}}}$

Flow area is defined with:

$\mathrm{A\_\_act}\=\mathrm{A\_\_rec}$

Geometrical coefficient for laminar flow is defined with:

$\mathrm{Geo\_\_act}\=\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\frac{\mathrm{b\_\_rec}}{\mathrm{a\_\_rec}}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_geo\_rec}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Geometrical length for Bend to get the values from tables:

$\mathrm{D0\_\_bend}\=\mathrm{a\_\_rec}$

Local resistance is defined with:

$\mathrm{zeta\_\_loc}\=\{\begin{array}{cc}\mathrm{k\_\_Re}\cdot \mathrm{k\_\_\δ}\cdot \mathrm{A1\_\_loc\_Bend}\cdot \mathrm{B\_\_loc\_Bend}\cdot \mathrm{C\_\_loc}& \mathrm{Re}\>10000\\ \frac{\mathrm{A2\_\_loc\_Bend}}{\max \left(3000\,\mathrm{Re}\right)}\+\mathrm{A1\_\_loc\_Bend}\cdot \mathrm{B\_\_loc\_Bend}\cdot \mathrm{C\_\_loc}& \mathrm{Re}\le 10000\end{array}$

Correction factor C is defined with:

$\mathrm{C\_\_loc}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\frac{\mathrm{b\_\_rec}}{\mathrm{a\_\_rec}}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_C\_Bend}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Correction factor A1 is defined with:

$\mathrm{A1\_\_loc\_Bend}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\mathrm{\θ}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_A1\_Bend}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Correction factor A2 is defined with:

$\mathrm{A2\_\_loc\_Bend}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\frac{\mathrm{R0}}{\mathrm{D0\_\_Bend}}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_A2\_Bend}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Correction factor B is defined with:

$\mathrm{B\_\_loc\_Bend}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\frac{\mathrm{R0}}{\mathrm{D0\_\_Bend}}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_B\_Bend}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Correction factor k_delta (Roughness dependency) is defined with:

$\mathrm{k\_\_\δ}\=\{\begin{array}{cc}\min \left(1.5\,1.0\+0.001\cdot \frac{\mathrm{roughness}}{\mathrm{D\_\_h\_act}}\right)& \frac{\mathrm{R0}}{\mathrm{D0\_\_Bend}}\le 0.55\\ \min \left(2.0\,\max \left(1.0\,\frac{\mathrm{\λ\_\_tur\_roughness}}{\mathrm{\λ\_\_tur\_smooth}}\right)\right)& \frac{\mathrm{R0}}{\mathrm{D0\_\_Bend}}\>0.55\end{array}$

Friction coefficient of smooth pipe for k_Re calculation is defined with:

$\mathrm{\λ\_\_tur\_smooth}\=0.25\cdot {\left(\frac{1}{\mathrm{log10}\left(\frac{5.74}{\max {\left(\mathrm{Re}\,1.0\right)}^{0.9}}\right)}\right)}^2$

Friction coefficient of rough pipe for k_Re calculation is defined with  (Swamee and Jain's approximation[2]):

$\mathrm{\λ\_\_tur\_roughness}\=0.25\cdot {\left(\frac{1}{\mathrm{log10}\left(\frac{\frac{\mathrm{roughness}}{\mathrm{D\_\_h\_act}}}{3.7}\+\frac{5.74}{\max {\left(\mathrm{Re}\,1.0\right)}^{0.9}}\right)}\right)}^2$

Correction factor k_Re (Reynolds number dependency) is defined with:

$\mathrm{k\_\_Re}\=\`\mathrm{MapleSim.Interpolate1D`}\left(\mathrm{data}\,\mathrm{Re}\right)$

(*) $\mathrm{`MapleSim.Interpolate1D`}$ is the function of Lookup table of 1D.

(*) data is specified with:

     - If data_source = inline, parameter $\mathrm{table\_\_k\_Re\_Bend}$.

     - If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used.

     - If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Friction resistance is defined with:

$\mathrm{zeta\_\_fri}\=\mathrm{\θ}\cdot \mathrm{\λ}\cdot \frac{\mathrm{R0}}{\mathrm{D\_\_h\_act}}$

The friction factor of flow is calculated with:

$\mathrm{\λ}\=\mathrm{`HeatTransfer.Functions.lambda\_Re`}\left(\mathrm{Re}\,\mathrm{roughness}\,\mathrm{D\_\_h\_act}\,\mathrm{Re\_\_CoT}\,\mathrm{IF\_\_speed}\,\mathrm{Geo\_\_act}\right)$

(*) The above function $\mathrm{`HeatTransfer.Functions.lambda\_Re`}$ is to calculated friction factor for Laminar and Turbulent flow.

     Regarding the implementation of the friction factor calculation, see the reference section below.

Total resistance is defined with:

$\mathrm{zeta\_\_act}\=\mathrm{zeta\_\_loc}\+\mathrm{zeta\_\_fri}$

Friction factor of Laminar flow is calculated with:

$\mathrm{\λ\_\_lam}\=\mathrm{Geo\_\_act}\cdot \frac{64}{\mathrm{Re}}$

And, Turbulent flow's friction factor is defined with (Swamee and Jain's approximation[2]):

$\mathrm{\λ\_\_tur}\=\frac{0.25}{\log {\left(\frac{\frac{\mathrm{roughness}}{\mathrm{D\_\_h\_act}}}{3.7}\+\frac{5.74}{{\mathrm{Re}}^{0.9}}\right)}^2}$

Intermittency is defined with:

$\mathrm{\κ}\=\frac{\tanh \left(\frac{\mathrm{IF\_\_speed}\cdot \left(\mathrm{Re}-\mathrm{Re\_\_CoT}\right)}{2}\right)\+1}{2}$

So, the friction factor is calculated with:

$\mathrm{\λ}\=\left(1-\mathrm{\κ}\right)\cdot \mathrm{\λ\_\_lam}\+\mathrm{\κ}\cdot \mathrm{\λ\_\_tur}$

The following plot is Reynolds number vs Friction factor, and $\frac{\mathrm{roughness}}{\mathrm{D\_\_h\_act}}\=0.001$, $\mathrm{IF\_\_speed}\=0.007$, $\mathrm{Re\_\_CoT}\=3500$, $\mathrm{Geo\_\_act}\=1$.

Pressure difference is calculated with Darcy–Weisbach equation:

$\mathrm{dp}\=\frac{1}{2}\cdot \mathrm{zeta\_\_act}\cdot \frac{1}{{\mathrm{A\_\_act}}^2\cdot \{\begin{array}{cc}\mathrm{\ρ\_\_a}& \mathrm{dp}\ge 0\\ \mathrm{\ρ\_\_b}& \mathrm{others}\end{array}}\cdot {\mathrm{mflow}}^2\cdot \mathrm{sign}\left(\mathrm{mflow}\right)$

In theory, Mass flow rate is calculated with Darcy–Weisbach equation:

$\mathrm{mflow}\=\sqrt{\frac{2\cdot {\mathrm{A\_\_act}}^2}{\mathrm{zeta\_\_act}}}\cdot \sqrt{\{\begin{array}{cc}\mathrm{\ρ\_\_a}& \mathrm{dp}\ge 0\\ \mathrm{\ρ\_\_b}& \mathrm{others}\end{array}\cdot \left|\mathrm{dp}\right|}\cdot \mathrm{sign}\left(\mathrm{dp}\right)$

In the Heat Transfer Library, the following equation is used to resolve difficulties of the numerical calculation:

$\mathrm{mflow}\=\sqrt{\frac{2\cdot {\mathrm{A\_\_act}}^2}{\mathrm{zeta\_\_act}}}\cdot \mathrm{`HeatTransfer.Functions.regRoot2`}\left(\mathrm{dp}\,\mathrm{dp\_small}\,\mathrm{\ρ\_\_a}\,\mathrm{\ρ\_\_b}\,\mathrm{true}\,\mathrm{sharpness}\right)$

(*) $\mathrm{`HeatTransfer.Functions.regRoot2`}$ is the same function as $\mathrm{`Modelica.Fluid.Utilities.regRoot2`}$. To check the details of the package and view the original documentation, which includes author and copyright information, click here.

[1] : Idelchik,I.E.: Handbook of hydraulic resistance. Jaico Publishing House, Mumbai, 3rd edition, 2006.

[2] : Swamee P.K., Jain A.K. (1976): Explicit equations for pipe-flow problems. Proc. ASCE, J.Hydraul. Div., 102 (HY5), pp. 657-664.