The Change component models sudden or gradual changes in the diameter of a pipe and captures both contraction and enlargement within the component.

The ratio of diameters is calculated as

$\beta \=\{\begin{array}{cc}\frac{d_b}{d_a}& d_a\>d_b\\ \frac{d_a}{d_b}& \mathrm{otherwise}\end{array}$

If $d_a\>d_b$

    Loss coefficient of contraction at inlet:

$K_c\=\{\begin{array}{cc}0.8\sin \left(\frac{\mathrm{\theta }}{2}\right)\frac{1-{\mathrm{\beta }}^2}{{\mathrm{\beta }}^4}& \mathrm{\theta }\le \frac{45}{180}\mathrm{\pi }\\ 0.5\left(1-{\mathrm{\beta }}^2\right)\frac{\sqrt{\sin \left(\frac{\mathrm{\theta }}{2}\right)}}{{\mathrm{\beta }}^4}& \mathrm{otherwise}\end{array}$

    Loss coefficient of enlargement at outlet:

$K_e\=\{\begin{array}{cc}2.6\sin \left(\frac{\mathrm{\theta }}{2}\right)\frac{{\left(1-{\mathrm{\beta }}^2\right)}^2}{{\mathrm{\beta }}^4}& \mathrm{\theta }\le \frac{45}{180}\mathrm{\pi }\\ \frac{{\left(1-{\mathrm{\beta }}^2\right)}^2}{{\mathrm{\beta }}^4}& \mathrm{otherwise}\end{array}$

otherwise, $d_a\le d_b$

    Loss coefficient of contraction at inlet:

$K_{{}_c}\=\{\begin{array}{cc}0.8\sin \left(\frac{\mathrm{\theta }}{2}\right)\left(-{\mathrm{\beta }}^2\+1\right)& \mathrm{\theta }\le \frac{45}{180}\mathrm{\pi }\\ 0.5\left(-{\mathrm{\beta }}^2\+1\right)\sqrt{\sin \left(\frac{\mathrm{\theta }}{2}\right)}& \mathrm{otherwise}\end{array}$

    Loss coefficient of enlargement at outlet:

$K_{{}_e}\=\{\begin{array}{cc}2.6\sin \left(\frac{\mathrm{\theta }}{2}\right){\left(-{\mathrm{\beta }}^2\+1\right)}^2& \mathrm{\theta }\le \frac{45}{180}\mathrm{\pi }\\ {\left(-{\mathrm{\beta }}^2\+1\right)}^2& \mathrm{otherwise}\end{array}$

The total loss coefficient is defined by using the linear approximation for the transition between contraction and enlargement:

$K\&equals;\{\begin{array}{cc}{K}_e& \mathrm{dp}<-{\mathrm{dp}}_{{}_{\mathrm{transition}}}\\ \\ \frac{K_{{}_c}-K_{{}_e}}{2{\mathrm{dp}}_{{}_{\mathrm{transition}}}}\mathrm{dp}\&plus;\frac{K_{{}_c}\&plus;K_{{}_e}}{2}& \&comma;-{\mathrm{dp}}_{{}_{\mathrm{transition}}}\le \mathrm{dp}\le {\mathrm{dp}}_{{}_{\mathrm{transition}}}\\ K_{{}_c}& \mathrm{otherwise}\end{array}$

$p\&equals;p_A-p_B\&equals;\frac{\mathrm{\pi }}{4}\mathrm{\rho }\mathrm{\nu }q\frac{{\left(16\frac{q^4}{{\mathrm{\pi }}^2{A}_{\mathrm{cs}}^2{\mathrm{\nu }}^4}\&plus;{\mathrm{Re}}_{\mathrm{Cr}}^4\right)}^{\frac{1}{4}}}{C_d^2{A}_{\mathrm{cs}}\sqrt{\mathrm{\pi }{A}_{\mathrm{cs}}}}$

$q\&equals;q_A\&equals;-q_B$

$v_a\&equals;4\frac{q}{d_a^2}\mathrm{\pi }{v}_b\&equals;4\frac{q}{d_b^2}\mathrm{\pi }$

References

[1] : Flow of Fluids Through Valves, Fittings, and Pipes, Crane Valves North America, Technical Paper No. 410M. 1979, p A-26