Gerhard Bollrich: "Technische Hydromechanik 1 (eBook)", + Book, Beuth Verlag GmbH, 2019, + 8th Edition, ISBN 978-3-410-29170-1.
+Jonas Tveit Hinna: "Modelica model of transient pipe flow in hydraulic laboratory systems using the method of characteristics", diff --git a/OpenHPL/Waterway/Gate.mo b/OpenHPL/Waterway/Gate.mo new file mode 100644 index 00000000..4c508fe2 --- /dev/null +++ b/OpenHPL/Waterway/Gate.mo @@ -0,0 +1,158 @@ +within OpenHPL.Waterway; +model Gate "Model of a sluice or tainter gate based on [Bollrich2019]" + outer Data data "Using standard class with system parameters"; + extends Icons.Gate; + extends Interfaces.TwoContacts; + import Modelica.Units.Conversions.to_deg; + parameter SI.Height b "Width of the gate" annotation (Dialog(group="Common")); + parameter SI.Length r "Radius of the gate arm" annotation (Dialog(enable=not sluice, group="Radial/Tainter")); + parameter SI.Height h_h "Height of the hinge above gate bottom" annotation (Dialog(group="Radial/Tainter", enable=not sluice)); + SI.Height h_0 "Inlet water level"; + SI.Height h_2 "Outlet water level"; + SI.Height h_2_limit "Limit of free flow"; + SI.Area A = a*b "Area of the physical gate opening"; + SI.MassFlowRate mdot "Mass flow rate"; + SI.VolumeFlowRate Vdot "Volume flow rate through the gate"; + Real mu_A "Discharge coefficient"; + Real psi "Contraction coefficient"; + //Real psi90 "Contraction coefficient for vertical gate"; + Real chi "Back-up coefficient"; + SI.Angle alpha = C.pi/2 - asin((h_h-a)/r) "Edge angle of the gate"; + Real x,y,z; + Real h0_a = h_0/a; + Real h2_a = h_2/a; + Modelica.Blocks.Interfaces.RealInput a "Opening of the gate [m]" annotation (Placement(transformation( + extent={{-20,-20},{20,20}}, + rotation=270, + origin={0,120}))); +equation + mu_A = psi/sqrt(1+psi*a/h_0); + if sluice then + psi = 1 / (1+ 0.64 * sqrt(1-(a/h_0)^2)) "Sluice gate, i.e. alpha = 90 deg"; + else + psi = 1.3 - 0.8 * sqrt(1 - ((to_deg(alpha)-205)/220)^2) "Normally only valid for a/h_0 --> 0"; + end if; + x = (1-2*psi*a/h_0*(1-psi*a/max(C.small,h_2)))^2; + y = x+z-1; + z = (h_2/h_0)^2; + + h_2_limit = a*psi/2*(sqrt(1+16/(psi*(1+psi*a/h_0)) * (h_0/a))-1); + + if h_2 >= h_2_limit then + chi = sqrt((1+psi*a/h_0) * ((1-2*psi*a/h_0 * (1-psi*a/h_2))- + sqrt((1-2*psi*a/h_0*(1-psi*a/max(C.small,h_2)))^2 + (h_2/h_0)^2 - 1))) "Backed-up flow"; + else + chi = 1 "Free flow"; + end if; + + i.mdot+o.mdot = 0 "Mass balance"; + mdot = i.mdot "Inlet direction for mdot"; + Vdot = chi * mu_A * A * sqrt(2*data.g*h_0) "Volume flow rate through the gate"; + mdot = Vdot * data.rho "Mass flow rate through the gate"; + i.p = h_0 * data.g * data.rho + data.p_a "Inlet water pressure"; + o.p = h_2 * data.g * data.rho + data.p_a "Outlet water pressure"; + annotation ( + preferredView="info", + Documentation(info=" +
Implementation
++The calculation of the flow through the gate is approximated for two different regions and is based +on [Bollrich2019]. +Equation numbers and figure numbers given below are in sync with the numbers of [Bollrich2019]. +
+Free flowing
+
+ ![\"TainterGate](\"modelica://OpenHPL/Resources/Images/TainterGate-freeflow.png\")
+ |
+
+The free flow can be calculated with: + +$$Q_A = \\mu_A \\cdot A \\cdot \\sqrt{2g\\cdot h_0} \\tag{8.24} $$ +(valid for gate opening higher than the downstream water level) +
++With +
+- Opening area +
- $$ A = a\\cdot b $$ +
- Discharge coefficient +
- $$ \\mu_A = \\frac{\\psi}{\\sqrt{1+\\frac{\\psi\\cdot a}{h_0}}} \\tag{8.23}$$ +
- Contraction coefficient sluice gate (\\(\\alpha=90^\\circ\\)) +
- $$ \\psi_{90^\\circ}= \\frac{1}{1+0.64\\cdot \\sqrt{1-(a/h_0)^2}} \\tag{8.25}$$ +
- Contraction coefficient radial gate (for \\(a/h_0 \\rightarrow 0\\)) +
- $$ \\psi_0(\\alpha)= 1.3 -0.8\\cdot\\sqrt{1-\\left(\\frac{\\alpha -205^\\circ}{220^\\circ}\\right)^2} \\tag{8.25a}$$ +
- The edge angle \\(\\alpha\\) of the gate +
- $$ \\alpha = \\left( \\frac{\\pi}{2} - \\arcsin(\\frac{h_h-a}{r})\\right) \\cdot \\frac{180^\\circ}{\\pi} $$ +
+With: ++ ++
+- \\(a \\ldots\\) Vertical gate opening
+- \\(h_h \\ldots\\) Height of the hinge above gate bottom\"
+- \\(r \\ldots\\) Radius of the gate arm
+
Backed-up discharge
+
+ ![\"TainterGate](\"modelica://OpenHPL/Resources/Images/TainterGate-backedup.png\")
+ |
+
+$$Q_A = \\chi \\cdot \\mu_A \\cdot A \\cdot \\sqrt{2g\\cdot h_0} \\tag{8.29} $$ + +With +
+-
+
- Back-up coefficient +
- $$ \\chi = \\sqrt{ + \\left( + 1 + \\frac{\\psi\\cdot a}{h_0} + \\right) \\cdot + \\left\\{ + \\left[ + 1 - 2\\cdot\\frac{\\psi\\cdot a}{h_0} \\cdot + \\left( + 1-\\frac{\\psi\\cdot a}{h_2} + \\right) + \\right] + - \\sqrt{ + \\left[ + 1 - 2 \\cdot \\frac{\\psi\\cdot a}{h_0} \\cdot + \\left( + 1-\\frac{\\psi\\cdot a}{h_2} + \\right) + \\right]^2 + + + \\left( + \\frac{h_2}{h_0} + \\right)^2 + - 1 + } + \\right\\} + } \\tag{8.28}$$ +
Boundary between free and backed-up flow
++The boundary of the height of the water level \\(h_2\\) behind the gate from which on the calculation switches to the backed-up flow (8.29) can be derived from: + +$$ \\frac{h_2^*}{a} = \\frac{\\psi}{2} \\cdot \\left( \\sqrt{ 1 + \\frac{16}{\\psi\\cdot\\left(1+\\frac{\\psi\\cdot a}{h_0}\\right)}\\cdot\\frac{h_0}{a}} - 1 \\right) \\tag{8.26}$$ + +So when \\(\\frac{h_2}{a} \\geq \\frac{h_2^*}{a}\\) then we have back-up flow. +
+")); +end Gate; diff --git a/OpenHPL/Waterway/Gate_HR.mo b/OpenHPL/Waterway/Gate_HR.mo new file mode 100644 index 00000000..b0bf4f5b --- /dev/null +++ b/OpenHPL/Waterway/Gate_HR.mo @@ -0,0 +1,92 @@ +within OpenHPL.Waterway; +model Gate_HR "Model of a tainter gate (HEC-RAS)" + outer Data data "Using standard class with system parameters"; + extends Icons.Gate(final sluice); + extends OpenHPL.Interfaces.TwoContacts; + + parameter SI.Height W "Width of the gated spillway" + annotation (Dialog(group="Geometry")); + parameter SI.Height T "Trunnion height (from spillway crest to trunnion pivot point)" + annotation (Dialog(group="Geometry")); + parameter Real Cd = 0.7 "Discharge coefficient (typically ranges from 0.6 - 0.8)" + annotation (Dialog(group="Coefficients")); + parameter Real TE = 0.16 "Trunnion height exponent, typically about 0.16" + annotation (Dialog(group="Coefficients")); + parameter Real BE = 0.72 "Gate opening exponent, typically about 0.72" + annotation (Dialog(group="Coefficients")); + SI.Length H = h_i "Upstream Energy Head above the spillway crest"; + parameter Real HE = 0.62 "Head exponent, typically about 0.62" + annotation (Dialog(group="Coefficients")); + SI.Height h_i "Inlet water level"; + SI.Height h_o "Outlet water level"; + SI.MassFlowRate mdot "Mass flow rate"; + SI.VolumeFlowRate Vdot "Volume flow rate through the gate"; + Real H_ratio = h_o/h_i "Ratio of outlet to inlet height"; + + Modelica.Blocks.Interfaces.RealInput B "Height of gate opening [m]" annotation (Placement(transformation( + extent={{-20,-20},{20,20}}, + rotation=270, + origin={0,120}))); +protected + SI.VolumeFlowRate Vdot_free "Free base volume flow rate through the gate"; + SI.VolumeFlowRate Vdot_partial "Partially submerged base volume flow rate through the gate"; + SI.VolumeFlowRate Vdot_full "Fully submerged base volume flow rate through the gate"; + Real Cdx "Discharge coefficient tuned for a smooth transition between partially and fully submerged"; + +equation + i.mdot+o.mdot = 0 "Mass balance"; + mdot = i.mdot "Inlet direction for mdot"; + Vdot_free = sqrt(2*data.g)* W * T^TE * B^BE * h_i^HE "Free flow condition"; + Vdot_partial = sqrt(2*data.g)* W * T^TE * B^BE * (3*(h_i-h_o))^HE "Partially submerged"; + Vdot_full = sqrt(2*data.g*(h_i-h_o))* W * B "Fully submerged"; + Cdx = Cd * Vdot_partial/Vdot_full; + if H_ratio <= 0.67 then + Vdot=Cd * Vdot_free "Free flow condition"; + elseif H_ratio <= 0.8 then + Vdot= Cd * Vdot_partial "Partially submerged"; + else + Vdot= Cdx * Vdot_full "Fully submerged"; + end if; + mdot = Vdot * data.rho "Mass flow rate through the gate"; + i.p = h_i * data.g * data.rho + data.p_a "Inlet water pressure"; + o.p = h_o * data.g * data.rho + data.p_a "Outlet water pressure"; + annotation ( + preferredView="info", + Documentation(info=" +Background
++The Tainter Gate (also known as Radial Gate) is modelled based on a simplified calculation + as used in the HEC-RAS simulation software. +More specifically the following documentation is taken from the + +HEC-RAS Hydraulic Reference Manual on Radial Gates. +
+![](\"modelica://OpenHPL/Resources/Images/TainterGate.png\"/)
+ Implementation
++The calculation of the flow through the gate is approximated for three different regions: +
+Free flowing
++$$Q = C_d W T^{T_E} B^{B_E} H^{H_E} \\sqrt{2g} \\tag{1} $$ +(valid for gate opening higher than the downstream water level) +
+Partially submerged
+
+$$Q = C_d W T^{T_E} B^{B_E} (3H)^{H_E} \\sqrt{2g} \\tag{2} $$
+(valid for the region where the ratio of downstream water level to upstream water level (H_ratio = h_o/h_i) is between 0.67 and 0.8)
+
Fully submerged
+
+$$Q = C_{dx} A \\sqrt{2gH} \\tag{3}$$
+(otherwise with Cdx being a discharge coefficient tuned for a smooth transition between partially and fully submerged)
+
+Note:
+The use of Cdx is different to the implementaion as done in HEC-RAS. This was done in order to have a smoother transition from the partially to fully submerged region.
+