This is for Crane model [1] [2].

If $\mathrm{d\_\_a}\>\mathrm{d\_\_b}$,

Loss coefficient of Contraction at Inlet (at port_a):

$\mathrm{K\_\_c}\=\{\begin{array}{cc}\frac{0.8\cdot \sin \left(\frac{\mathrm{\θ}}{2}\right)\cdot \left(1-{\mathrm{\β}}^2\right)}{{\mathrm{\β}}^4}& \mathrm{\θ}\le \frac{45}{180}\cdot \mathrm{Pi}\\ \frac{0.5\cdot \left(1-{\mathrm{\β}}^2\right)\cdot \sqrt{\sin \left(\frac{\mathrm{\θ}}{2}\right)}}{{\mathrm{\β}}^4}& \mathrm{otherwise}\end{array}$

Loss coefficient of Enlargement at Outlet (at port_a):

$\mathrm{K\_\_e}\=\{\begin{array}{cc}\frac{2.6\cdot \sin \left(\frac{\mathrm{\theta }}{2}\right)\cdot {\left(1-{\mathrm{\beta }}^2\right)}^2}{{\mathrm{\β}}^4}& \mathrm{\theta }\le \frac{45}{180}\cdot \mathrm{Pi}\\ \frac{{\left(1-{\mathrm{\beta }}^2\right)}^2}{{\mathrm{\β}}^4}& \mathrm{otherwise}\end{array}$

Thus, the total loss coefficient is defined by using the linear approximation for the transition between Contraction and Enlargement:

$K\&equals;\left\{\begin{array}{cc}\mathrm{K\_\_e}& \mathrm{dp}<-\mathrm{dp\_\_transition}\\ \frac{\mathrm{K\_\_c}-\mathrm{K\_\_e}}{2\cdot \mathrm{dp\_\_transition}}\cdot \mathrm{dp}\&plus;\frac{\mathrm{K\_\_c}\&plus;\mathrm{K\_\_e}}{2}& -\mathrm{dp\_\_transition}\le \mathrm{dp}\le \mathrm{dp\_\_transition}\\ \mathrm{K\_\_c}& \mathrm{dp}\&gt;\mathrm{dp\_\_transition}\end{array}\right.$$$

And, if $\mathrm{d\_\_a}\le \mathrm{d\_\_b}$,

Loss coefficient of Enlargement at Inlet (at port_a):

$\mathrm{K\_\_e}\&equals;\&lcub;\begin{array}{cc}2.6\cdot \sin \left(\frac{\mathrm{\theta }}{2}\right)\cdot {\left(1-{\mathrm{\beta }}^2\right)}^2& \mathrm{\theta }\le \frac{45}{180}\cdot \mathrm{Pi}\\ {\left(1-{\mathrm{\beta }}^2\right)}^2& \mathrm{otherwise}\end{array}$

Loss coefficient of Contraction at Inlet (at port_a):

$\mathrm{K\_\_c}\&equals;\&lcub;\begin{array}{cc}0.8\cdot \sin \left(\frac{\mathrm{\&theta;}}{2}\right)\cdot \left(1-{\mathrm{\&beta;}}^2\right)& \mathrm{\&theta;}\le \frac{45}{180}\cdot \mathrm{Pi}\\ 0.5\cdot \left(1-{\mathrm{\&beta;}}^2\right)\cdot \sqrt{\sin \left(\frac{\mathrm{\&theta;}}{2}\right)}& \mathrm{otherwise}\end{array}$

Thus, the total loss coefficient is defined by using the linear approximation for the transition between Contraction and Enlargement:

$K\&equals;\left\{\begin{array}{cc}\mathrm{K\_\_c}& \mathrm{dp}<-\mathrm{dp\_\_transition}\\ \frac{\mathrm{K\_\_e}-\mathrm{K\_\_c}}{2\cdot \mathrm{dp\_\_transition}}\cdot \mathrm{dp}\&plus;\frac{\mathrm{K\_\_e}\&plus;\mathrm{K\_\_c}}{2}& -\mathrm{dp\_\_transition}\le \mathrm{dp}\le \mathrm{dp\_\_transition}\\ \mathrm{K\_\_e}& \mathrm{dp}\&gt;\mathrm{dp\_\_transition}\end{array}\right.$

This is for Hooper model [2].

Reynolds number for Friction factor calculation is defined with:

$\mathrm{Re\_\_target}\&equals;\max \&lpar;\frac{\&lcub;\begin{array}{cc}\mathrm{\&rho;\_\_a}& \mathrm{dp}\ge 0\\ \mathrm{\&rho;\_\_b}& \mathrm{others}\end{array}\cdot \left|\mathrm{v\_\_b}\right|\cdot {\left(\frac{\mathrm{d\_\_b}}{\mathrm{d\_\_a}}\right)}^2\cdot \mathrm{d\_\_a}}{\&lcub;\begin{array}{cc}\mathrm{\&mu;\_\_a}& \mathrm{dp}\ge 0\\ \mathrm{\&mu;\_\_b}& \mathrm{others}\end{array}}\&comma;10\&rpar;$

$\frac{\&DifferentialD;\mathrm{Re}}{\&DifferentialD;t}\&equals;\frac{\left(\mathrm{Re\_\_target}-\mathrm{Re}\right)}{\mathrm{T\_\_const}}$

The friction factor of flow is calculated with:

$\mathrm{\&lambda;}\&equals;\mathrm{`HeatTransfer.Functions.lambda\_Re`}\left(\mathrm{Re}\mathit{\&comma;}\mathrm{roughness}\mathit{\&comma;}\mathrm{d\_\_a}\&comma;\mathrm{Re\_\_CoT}\&comma;\mathrm{IF\_\_speed}\&comma;1\right)$

(*) The above function $\mathrm{`HeatTransfer.Functions.lambda\_Re`}$ is to calculated friction factor for Laminar and Turbulent flow.
The fundamental implementation is based on the following equations. Especially, the equation of Turbulent flow is Swamee and Jain's approximation[1] .

Intermittencies for transition between Laminar and Turbulent flow are defined with:

$\mathrm{\&kappa;\_\_K\_c}\&equals;\frac{\tanh \left(\frac{\mathrm{IF\_\_speed}\cdot \left(\mathrm{Re}-\mathrm{Re\_\_CoT\_c}\right)}{2}\right)\&plus;1}{2}$

$\mathrm{\&kappa;\_\_K\_e}\&equals;\frac{\tanh \left(\frac{\mathrm{IF\_\_speed}\cdot \left(\mathrm{Re}-\mathrm{Re\_\_CoT\_e}\right)}{2}\right)\&plus;1}{2}$

If $\mathrm{d\_\_a}\&gt;\mathrm{d\_\_b}$,

Loss coefficient of Contraction at Inlet (at port_a):

$\mathrm{K\_\_c}\&equals;\left(1-\mathrm{\&kappa;\_\_K\_c}\right)\cdot \mathrm{K\_\_c\_l}\&plus;\mathrm{\&kappa;\_\_K\_c}\cdot \mathrm{K\_\_c\_t}$

Laminar:

$\mathrm{K\_\_c\_l}\&equals;\&lcub;\begin{array}{cc}\left(1.2\&plus;\frac{160}{\mathrm{Re}}\right)\cdot \left(\frac{1}{{\mathrm{\&beta;}}^4}-1\right)\cdot 1.6\cdot \sin \left(\frac{\mathrm{\theta }}{2}\right)& \mathrm{\theta }\le \frac{45}{180}\cdot \mathrm{Pi}\\ \left(1.2\&plus;\frac{160}{\mathrm{Re}}\right)\cdot \left(\frac{1}{{\mathrm{\beta }}^4}-1\right)\cdot \sqrt{\sin \left(\frac{\mathrm{\theta }}{2}\right)}& \mathrm{otherwise}\end{array}$

Turbulent:

$\mathrm{K\_\_c\_t}\&equals;\&lcub;\begin{array}{cc}\left(0.6\&plus;0.48\cdot \mathrm{\&lambda;}\right)\cdot \frac{\left(1-{\mathrm{\&beta;}}^2\right)}{{\mathrm{\&beta;}}^4}\cdot 1.6\cdot \sin \left(\frac{\mathrm{\&theta;}}{2}\right)& \mathrm{\&theta;}\le \frac{45}{180}\cdot \mathrm{Pi}\\ \left(0.6\&plus;0.48\cdot \mathrm{\lambda }\right)\cdot \frac{\left(1-{\mathrm{\beta }}^2\right)}{{\mathrm{\beta }}^4}\cdot \sqrt{\sin \left(\frac{\mathrm{\theta }}{2}\right)}& \mathrm{otherwise}\end{array}$

Loss coefficient of Enlargement at Outlet (at port_a):

$\mathrm{K\_\_e}\&equals;\left(1-\mathrm{\&kappa;\_\_K\_e}\right)\cdot \mathrm{K\_\_e\_l}\&plus;\mathrm{\&kappa;\_\_K\_e}\cdot \mathrm{K\_\_e\_t}$

Laminar:

$\mathrm{K\_\_e\_l}\&equals;\&lcub;\begin{array}{cc}2\cdot \left(1-{\mathrm{\&beta;}}^4\right)\cdot \frac{1}{{\mathrm{\&beta;}}^4}\cdot 2.6\cdot \sin \left(\frac{\mathrm{\theta }}{2}\right)& \mathrm{\theta }\le \frac{45}{180}\cdot \mathrm{Pi}\\ 2\cdot \left(1-{\mathrm{\beta }}^4\right)\cdot \frac{1}{{\mathrm{\beta }}^4}& \mathrm{otherwise}\end{array}$

Turbulent:

$\mathrm{K\_\_e\_t}\&equals;\&lcub;\begin{array}{cc}\left(1.0\&plus;0.8\cdot \mathrm{\&lambda;}\right)\cdot {\left(1-{\mathrm{\beta }}^2\right)}^2\cdot \frac{1}{{\mathrm{\beta }}^4}\cdot 2.6\cdot \sin \left(\frac{\mathrm{\theta }}{2}\right)& \mathrm{\theta }\le \frac{45}{180}\cdot \mathrm{Pi}\\ \left(1.0\&plus;0.8\cdot \mathrm{\lambda }\right)\cdot {\left(1-{\mathrm{\beta }}^2\right)}^2\cdot \frac{1}{{\mathrm{\beta }}^4}& \mathrm{otherwise}\end{array}$

Thus, the total loss coefficient is defined by using the linear approximation for the transition between Contraction and Enlargement:

$K\&equals;\left\{\begin{array}{cc}\mathrm{K\_\_e}& \mathrm{dp}<-\mathrm{dp\_\_transition}\\ \frac{\mathrm{K\_\_c}-\mathrm{K\_\_e}}{2\cdot \mathrm{dp\_\_transition}}\cdot \mathrm{dp}\&plus;\frac{\mathrm{K\_\_c}\&plus;\mathrm{K\_\_e}}{2}& -\mathrm{dp\_\_transition}\le \mathrm{dp}\le \mathrm{dp\_\_transition}\\ \mathrm{K\_\_c}& \mathrm{dp}\&gt;\mathrm{dp\_\_transition}\end{array}\right.$$$

And, if $\mathrm{d\_\_a}\le \mathrm{d\_\_b}$,

Loss coefficient of Enlargement at Inlet (at port_a):

$\mathrm{K\_\_e}\&equals;\left(1-\mathrm{\&kappa;\_\_K\_e}\right)\cdot \mathrm{K\_\_e\_l}\&plus;\mathrm{\&kappa;\_\_K\_e}\cdot \mathrm{K\_\_e\_t}$

Laminar:

$\mathrm{K\_\_e\_l}\&equals;\&lcub;\begin{array}{cc}2\cdot \left(1-{\mathrm{\&beta;}}^4\right)\cdot 2.6\cdot \sin \left(\frac{\mathrm{\theta }}{2}\right)& \mathrm{\theta }\le \frac{45}{180}\cdot \mathrm{Pi}\\ 2\cdot \left(1-{\mathrm{\beta }}^4\right)& \mathrm{otherwise}\end{array}$

Turbulent:

$\mathrm{K\_\_e\_t}\&equals;\&lcub;\begin{array}{cc}\left(1.0\&plus;0.8\cdot \mathrm{\lambda }\right)\cdot {\left(1-{\mathrm{\beta }}^2\right)}^2\cdot 2.6\cdot \sin \left(\frac{\mathrm{\theta }}{2}\right)& \mathrm{\theta }\le \frac{45}{180}\cdot \mathrm{Pi}\\ \left(1.0\&plus;0.8\cdot \mathrm{\lambda }\right)\cdot {\left(1-{\mathrm{\beta }}^2\right)}^2& \mathrm{otherwise}\end{array}$

Loss coefficient of Contraction at Inlet (at port_a):

$\mathrm{K\_\_c}\&equals;\left(1-\mathrm{\&kappa;\_\_K\_c}\right)\cdot \mathrm{K\_\_c\_l}\&plus;\mathrm{\&kappa;\_\_K\_c}\cdot \mathrm{K\_\_c\_t}$

Laminar:

$\mathrm{K\_\_c\_l}\&equals;\&lcub;\begin{array}{cc}\left(1.2\&plus;\frac{160}{\mathrm{Re}}\right)\cdot \left(\frac{1}{{\mathrm{\&beta;}}^4}-1\right)\cdot {\mathrm{\&beta;}}^4\cdot 1.6\cdot \sin \left(\frac{\mathrm{\theta }}{2}\right)& \mathrm{\theta }\le \frac{45}{180}\cdot \mathrm{Pi}\\ \left(1.2\&plus;\frac{160}{\mathrm{Re}}\right)\cdot \left(\frac{1}{{\mathrm{\beta }}^4}-1\right)\cdot {\mathrm{\&beta;}}^4\cdot \sqrt{\sin \left(\frac{\mathrm{\theta }}{2}\right)}& \mathrm{otherwise}\end{array}$

Turbulent:

$\mathrm{K\_\_c\_t}\&equals;\&lcub;\begin{array}{cc}\left(0.6\&plus;0.48\cdot \mathrm{\&lambda;}\right)\cdot \left(1-{\mathrm{\&beta;}}^2\right)\cdot 1.6\cdot \sin \left(\frac{\mathrm{\&theta;}}{2}\right)& \mathrm{\&theta;}\le \frac{45}{180}\cdot \mathrm{Pi}\\ \left(0.6\&plus;0.48\cdot \mathrm{\lambda }\right)\cdot \left(1-{\mathrm{\beta }}^2\right)\cdot \sqrt{\sin \left(\frac{\mathrm{\theta }}{2}\right)}& \mathrm{otherwise}\end{array}$

Thus, the total loss coefficient is defined by using the linear approximation for the transition between Contraction and Enlargement:

$K\&equals;\left\{\begin{array}{cc}\mathrm{K\_\_c}& \mathrm{dp}<-\mathrm{dp\_\_transition}\\ \frac{\mathrm{K\_\_e}-\mathrm{K\_\_c}}{2\cdot \mathrm{dp\_\_transition}}\cdot \mathrm{dp}\&plus;\frac{\mathrm{K\_\_e}\&plus;\mathrm{K\_\_c}}{2}& -\mathrm{dp\_\_transition}\le \mathrm{dp}\le \mathrm{dp\_\_transition}\\ \mathrm{K\_\_e}& \mathrm{dp}\&gt;\mathrm{dp\_\_transition}\end{array}\right.$

(Reference) Detailed implementation of Friction factor calculation

Friction factor of Laminar flow is calculated with: $\mathrm{\λ\_\_lam}\=\frac{64}{\mathrm{Re}}$ And, Turbulent flow's friction factor is defined with (Swamee and Jain's approximation): $\mathrm{\λ\_\_tur}\=\frac{0.25}{\log {\left(\frac{\frac{\mathrm{roughness}}{\mathrm{d\_\_a}}}{3.7}\+\frac{5.74}{{\mathrm{Re}}^{0.9}}\right)}^2}$ 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 $\frac{\mathrm{roughness}}{\mathrm{d\_\_a}}\=0.001$, $\mathrm{IF\_\_speed}\=0.007$, $\mathrm{Re\_\_CoT}\=3500$.

Pressure difference and flow velocities are calculated with the following equations:

$\mathrm{v\_\_a}\&equals;\frac{\mathrm{mflow}}{\&lcub;\begin{array}{cc}\mathrm{\&rho;\_\_a}& \mathrm{dp}\ge 0\\ \mathrm{\&rho;\_\_b}& \mathrm{others}\end{array}\cdot {\left(\frac{\mathrm{d\_\_a}}{2}\right)}^2\cdot \mathrm{\pi }}$

$\mathrm{v\_\_b}\&equals;\frac{\mathrm{mflow}}{\&lcub;\begin{array}{cc}\mathrm{\&rho;\_\_a}& \mathrm{dp}\ge 0\\ \mathrm{\&rho;\_\_b}& \mathrm{others}\end{array}\cdot {\left(\frac{\mathrm{d\_\_b}}{2}\right)}^2\cdot \mathrm{\pi }}$

$\mathrm{dp}\&equals;\frac{K\cdot \&lcub;\begin{array}{cc}\mathrm{\&rho;\_\_a}& \mathrm{dp}\ge 0\\ \mathrm{\&rho;\_\_b}& \mathrm{others}\end{array}\cdot \cdot {\mathrm{v\_\_a}}^2\cdot \mathrm{sign}\left(\mathrm{v\_\_a}\right)}{2}$

In theory, Mass flow rate and flow velocities are calculated with the following equations with Loss coefficient:

$\mathrm{v\_\_a}\&equals;\sqrt{\frac{2}{K}}\cdot \sqrt{\&lcub;\begin{array}{cc}\frac{1}{\mathrm{\&rho;\_\_a}}& \mathrm{dp}\ge 0\\ \frac{1}{\mathrm{\&rho;\_\_b}}& \mathrm{others}\end{array}\cdot \left|\mathrm{dp}\right|}\cdot \mathrm{sign}\left(\mathrm{dp}\right)$

$\mathrm{mflow}\&equals;{\left(\frac{\mathrm{d\_\_a}}{2}\right)}^2\cdot \mathrm{\&pi;}\cdot \&lcub;\begin{array}{cc}\mathrm{\&rho;\_\_a}& \mathrm{dp}\ge 0\\ \mathrm{\&rho;\_\_b}& \mathrm{others}\end{array}\cdot \mathrm{v\_\_a}$

$\mathrm{v\_\_b}\&equals;\frac{\mathrm{mflow}}{\&lcub;\begin{array}{cc}\mathrm{\&rho;\_\_a}& \mathrm{dp}\ge 0\\ \mathrm{\&rho;\_\_b}& \mathrm{others}\end{array}\cdot {\left(\frac{\mathrm{d\_\_b}}{2}\right)}^2\cdot \mathrm{\&pi;}}$

In the Heat Transfer Library, the equation for $\mathrm{v\_\_a}$ is used to resolve difficulties of the numerical calculation:

$\mathrm{v\_\_a}\&equals;\sqrt{\frac{2}{K}}\cdot \mathrm{`HeatTransfer.Functions.regRoot2`}\left(\mathrm{dp}\&comma;\mathrm{dp\_small}\&comma;\frac{1}{\mathrm{\&rho;\_\_a}}\&comma;\frac{1}{\mathrm{\&rho;\_\_b}}\&comma;\mathrm{true}\&comma;\mathrm{sharpness}\right)$

$$(*) $\mathrm{`HeatTransfer.Functions.regRoot2`}$ 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] : Flow of Fluids Through Valves, Fittings, and Pipe, Crane Valves North America, Technical Paper No. 410M. 1979, p A-26

[2] : Ron Darby, Chemical Engineering Fluid Mechanics 2nd edition, Marcel Dekker, 2001

[3] : William B. Hooper, Calculate Head loss caused by change in pipe size, Chemical Engineering November 1988, p 89