The Pressure Reducing 3 Way Valve component models a hydraulic reducing valve as a sharp-edged orifice with the orifice area dependent on the pressure across the valve.

The area varies linearly from $A_{\mathrm{open}}$ to $A_{\mathrm{close}}$ as the pressure varies from $p_{\mathrm{open}}$  to $p_{\mathrm{close}}$ and remains at the endpoints for pressures outside this range. The valve can be used to regulate the flow path and vent the outlet port if needed based on the preset pressure setting. The path A-B runs between a pressure reducing valve which will regulate and close the flow path based on the outlet pressure at port B. The path B-C will vent the flow from the outlet port to port C when the pressure exceeds the preset pressure setting in the valve.

Based on the orifice area, the pressure vs. flow rate relationship is calculated using the formulation in the Orifice component.

	Formulation Approaches
	One of two approaches can be selected for modeling the flow in the device. When the boolean parameter $\mathrm{Use\; constant\; Cd}$ is true, a constant coefficient of discharge ($C_d$) is used, otherwise a variable coefficient of discharge with maximum value ($C_{d\left(\max \right)}$) and a critical flow number (${\mathrm{Crit}}_{\mathrm{no}}$) are used.

	Optional Volumes
	The boolean parameters $\mathrm{Use\; volume\; A}$, $\mathrm{Use\; volume\; B}$, and $\mathrm{Use\; volume\; C}$ when true, add optional volumes $V_A$, $V_B$, and $V_C$ to ports A, B, and C, respectively. See Port Volumes for details. If two orifices or valves are connected, enabling a volume at the common port reduces the stiffness of the system and improves the solvability.

$p_{\mathrm{AB}}\=p_A-p_B{P}_{\mathrm{BC}}\=p_B-p_C$

$\mathbf{Orifice\; Fluid\; Equations}$

$q_{\mathrm{AA}}\=u_1{A}_{{\mathrm{cs}}_1}{\mathrm{\Re }}_1\=u_1\frac{{\mathrm{D}}_{\mathrm{h1}}}{\mathrm{\nu }}$

$\{\begin{array}{cc}{p}_{\mathrm{AB}}\=\frac{\mathrm{\pi }}{4}\frac{\mathrm{\rho }\mathrm{\nu }{q}_{\mathrm{AA}}}{C_d^2{A}_{{\mathrm{cs}}_1}\sqrt{\mathrm{\pi }{A}_{{\mathrm{cs}}_1}}}{\left(\frac{16q_{\mathrm{AA}}^4}{{\mathrm{\pi }}^2{A}_{{\mathrm{cs}}_1}^2{\mathrm{\nu }}^4}\+{\mathrm{Re}}_{\mathrm{Cr}}^4\right)}^{\frac{1}{4}}& \mathrm{Use\; constant\; Cd}\=\mathrm{true}\\ q_{\mathrm{AA}}\=C_{d\left(\max \right)}\tanh \left(4\frac{\sqrt{\frac{A_{{\mathrm{cs}}_1}}{\mathrm{\pi }}\frac{2\left|p_{\mathrm{AB}}\right|}{\mathrm{\rho }}}}{\mathrm{\nu }{\mathrm{Crit}}_{\mathrm{no}}}\right)A_{{\mathrm{cs}}_1}\sqrt{\frac{2\left|p_{\mathrm{AB}}\right|}{\mathrm{\rho }}}\mathrm{sign}\left(p_{\mathrm{AB}}\right)& \mathrm{otherwise}\end{array}$

$q_{\mathrm{CC}}\=u_2{A}_{{\mathrm{cs}}_2}{\mathrm{\Re }}_2\=u_2\frac{{\mathrm{D}}_{\mathrm{h2}}}{\mathrm{\nu }}$

$\{\begin{array}{cc}{p}_{\mathrm{BC}}\=\frac{\mathrm{\pi }}{4}\frac{\mathrm{\rho }\mathrm{\nu }{q}_{\mathrm{CC}}}{C_d^2{A}_{{\mathrm{cs}}_2}\sqrt{\mathrm{\pi }{A}_{{\mathrm{cs}}_2}}}{\left(\frac{16q_{\mathrm{CC}}^4}{{\mathrm{\pi }}^2{A}_{{\mathrm{cs}}_2}^2{\mathrm{\nu }}^4}\+{\mathrm{Re}}_{\mathrm{Cr}}^4\right)}^{\frac{1}{4}}& \mathrm{Use\; constant\; Cd}\=\mathrm{true}\\ q_{\mathrm{CC}}\=C_{d\left(\max \right)}\tanh \left(4\frac{\sqrt{\frac{A_{{\mathrm{cs}}_2}}{\mathrm{\pi }}\frac{2\left|p_{\mathrm{BC}}\right|}{\mathrm{\rho }}}}{\mathrm{\nu }{\mathrm{Crit}}_{\mathrm{no}}}\right)A_{{\mathrm{cs}}_2}\sqrt{\frac{2\left|p_{\mathrm{BC}}\right|}{\mathrm{\rho }}}\mathrm{sign}\left(p_{\mathrm{BC}}\right)& \mathrm{otherwise}\end{array}$

$\mathbf{Orifice\; Area\; Equations}$

$\{\begin{array}{cc}{A}_{{\mathrm{cs}}_1}\=A_i\=A_t& \mathrm{Exact}\\ \left\{A_{{\mathrm{cs}}_1}\=\min \left(A_{\mathrm{open}}\,\max \left(A_{\mathrm{close}}\,A_i\right)\right)\,t_c\frac{\mathrm{d}{A}_i}{\mathrm{d}t}\+A_i\=A_t\right\}& \mathrm{otherwise}\end{array}$

$A_t\&equals;\{\begin{array}{cc}{A}_{\mathrm{open}}& p_B<p_{\mathrm{open}}\\ A_{\mathrm{open}}-\left(A_{\mathrm{open}}-A_{\mathrm{close}}\right)\mathrm{Smooth}\left(S\&comma;\frac{p_B-p_{\mathrm{open}}}{p_{\mathrm{close}}-p_{\mathrm{contrat}}}\right)& p_B<p_{\mathrm{close}}\\ A_{\mathrm{close}}& \mathrm{otherwise}\end{array}$

$A_{{\mathrm{cs}}_2}\&equals;\{\begin{array}{cc}{A}_{\mathrm{close}}& p_B<p_{\mathrm{open}}\\ A_{\mathrm{close}}\&plus;\left(A_{\mathrm{open}}-A_{\mathrm{close}}\right)\mathrm{SmoothTrans}\left(S\&comma;\left(\frac{p_B-\left(p_{\mathrm{close}}\&plus;\mathrm{dp}\right)}{p_{\mathrm{close}}-p_{\mathrm{open}}}\right)\right)& p_B<2p_{\mathrm{close}}\&plus;\mathrm{dp}-p_{\mathrm{open}}\\ A_{\mathrm{open}}& \mathrm{otherwise}\end{array}$

$S\&equals;\{\begin{array}{cc}\mathrm{smoothness}& \mathrm{smoothTransition}\\ 0& \mathrm{otherwise}\end{array}$

$p_{\mathrm{tot}}\&equals;p\&plus;\{\begin{array}{cc}{k}_p{p}_C& \mathrm{dir}\&equals;1\\ -k_p{p}_C& \mathrm{otherwise}\end{array}$

$\mathbf{Optional\; Volume\; Equations}$

$V_{f_A}\&equals;\{\begin{array}{cc}\mathrm{Va}\left(1\&plus;\frac{p_A}{\mathrm{El}}\right)& \mathrm{Use\; volume\; A}\&equals;\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}{q}_{V_A}\&equals;\{\begin{array}{cc}\frac{\mathrm{d}{V}_{f_A}}{\mathrm{d}t}& \mathrm{Use\; volume\; A}\&equals;\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}$

$V_{f_B}\&equals;\{\begin{array}{cc}\mathrm{Vb}\left(1\&plus;\frac{p_B}{\mathrm{El}}\right)& \mathrm{Use\; volume\; B}\&equals;\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}{q}_{V_B}\&equals;\{\begin{array}{cc}\frac{\mathrm{d}{V}_{f_B}}{\mathrm{d}t}& \mathrm{Use\; volume\; B}\&equals;\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}$

$V_{f_C}\&equals;\{\begin{array}{cc}\mathrm{Vb}\left(1\&plus;\frac{p_C}{\mathrm{El}}\right)& \mathrm{Use\; volume\; C}\&equals;\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}{q}_{V_C}\&equals;\{\begin{array}{cc}\frac{\mathrm{d}{V}_{f_C}}{\mathrm{d}t}& \mathrm{Use\; volume\; C}\&equals;\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}$

$q_A\&plus;q_B\&plus;q_C\&equals;q_{V_A}\&plus;q_{V_B}\&plus;q_{V_C}$

$X\&comma;Y\in \left\{A\&comma;B\&comma;C\right\}$