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Documentation of Tainter Gate implementation after Bollrich2019
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,126 @@ | ||
| within OpenHPL.Waterway; | ||
| model TainterGate3 "Model of a tainter gate based on [Bollrich2019]" | ||
| outer Data data "Using standard class with system parameters"; | ||
| extends Icons.TainterGate; | ||
| extends OpenHPL.Interfaces.ContactPort; | ||
|
|
||
| // parameter SI.Height b_max "Maxium opening of the gate"; | ||
| // parameter SI.Height b_min = 0 "Minimum opening of the gate"; | ||
| parameter SI.Length r "Radius of the gate arm"; | ||
| parameter SI.Height a "Height of the hinge above gate bottom"; | ||
| parameter SI.Height b "Width of the gate"; | ||
| parameter Real Cc_[3] = {1,2,3} "Polinomial factors of contraction coefficient Cc {linear,quadratic,cube}"; | ||
| SI.Height h_i "Inlet water level"; | ||
| SI.Height h_o "Outlet water level"; | ||
| Real alpha = h_o/(Cc*u); | ||
| Real beta = h_i/h_o; | ||
| Real gamma = (h_o/(Cc*u))^2 - (h_o/h_i)^2 "Loss factor"; | ||
| Real psi = (1/beta)^2 - 1 + (gamma^2*(beta-1))/(psi_x1 + psi_x2) "Head-loss parameter"; | ||
| Real psi_ = (1/beta)^2 - 1 + (gamma^2*(beta-1))/(psi_x1 - psi_x2) "Neg Head-loss parameter"; | ||
| Real psi_x1 = gamma*beta-2*(alpha-1); | ||
| Real psi_x2 = sqrt((2*(alpha-1)-gamma*beta)^2 - gamma^2*(beta^2-1)); | ||
|
|
||
| SI.Angle theta = C.pi/2 - asin((a-u)/r) "Flow angle of the gate"; | ||
| Real Cc = Cc_[3]*theta^3 + Cc_[2]*theta^2 + Cc_[1]*theta + 1.002282138151680 "Contraction coefficient"; | ||
| SI.VolumeFlowRate Vdot,Vdot_gamma "Volume flow rate through the gate"; | ||
|
|
||
| Modelica.Blocks.Interfaces.RealInput u "Opening of the gate [m]" annotation (Placement(transformation( | ||
| extent={{-20,-20},{20,20}}, | ||
| rotation=270, | ||
| origin={0,120}))); | ||
| equation | ||
| Vdot = b*h_o * sqrt((2*data.g*max(0,(h_i - h_o)))/(psi + 1 - (h_o/h_i)^2)) "Calculated flow"; | ||
| Vdot_gamma = b*h_o * sqrt((2*data.g*max(0,(h_i - h_o)))/(gamma + 1 - (h_o/h_i)^2)) "Calculated flow"; | ||
| 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 (Documentation(info="<html> | ||
| <h4>Implementation</h4> | ||
| <p> | ||
| The calculation of the flow through the gate is approximated for two different regions and is based | ||
| on <a href=\"modelica://OpenHPL.UsersGuide.References\">[Bollrich2019]</a>. | ||
| Equation numbers oand figure numbers given below are in sync with the numbers of <a href=\"modelica://OpenHPL.UsersGuide.References\">[Bollrich2019]</a>. | ||
| </p> | ||
| <h5>Free flowing</h5> | ||
| <table border=\"0\" cellspacing=\"0\" cellpadding=\"2\"> | ||
| <tr> | ||
| <td> | ||
| <img src=\"modelica://OpenHPL/Resources/Images/TainterGate-freeflow.png\" | ||
| alt=\"TainterGate free flow\"> | ||
| </td> | ||
| </tr> | ||
| <caption align=\"bottom\"><strong>Fig. 8.13:</strong> Free flow through the tainter gate (source: | ||
| <a href=\"modelica://OpenHPL.UsersGuide.References\">[Bollrich2019]</a>, page 376)<caption> | ||
| </table> | ||
| <p> | ||
| The free flow can be calulate 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 | ||
| <dl><dd>Opening area</dd> | ||
| <dt> $$ A = a\\cdot b $$ </dt> | ||
| <dd>Discharge coefficient</dd> | ||
| <dt> $$ \\mu_A = \\frac{\\psi}{\\sqrt{1+\\frac{\\psi\\cdot a}{h_0}}} \\tag{8.23}$$ </dt> | ||
| <dd>Contraction coefficient (for \\(a/h_0 \\rightarrow 0\\))</dd> | ||
| <dt> $$ \\psi_0(\\alpha)= 1.3 -0.8\\cdot\\sqrt{1-\\left(\\frac{\\alpha -205^\\circ}{220^\\circ}\\right)^2} \\tag{8.25a}$$ </dt> | ||
| </dl> | ||
| </p> | ||
| </p> | ||
| <h5>Backed-up discharge</h5> | ||
| <table border=\"0\" cellspacing=\"0\" cellpadding=\"2\"> | ||
| <tr> | ||
| <td> | ||
| <img src=\"modelica://OpenHPL/Resources/Images/TainterGate-backedup.png\" | ||
| alt=\"TainterGate backed-up flow\"> | ||
| </td> | ||
| </tr> | ||
| <caption align=\"bottom\"><strong>Fig. 8.16:</strong> Backed-up flow through the tainter gate (source: | ||
| <a href=\"modelica://OpenHPL.UsersGuide.References\">[Bollrich2019]</a>, page 379)<caption> | ||
| </table> | ||
| <p> | ||
| $$Q_A = \\chi \\cdot \\mu_A \\cdot A \\cdot \\sqrt{2g\\cdot h_0} \\tag{8.29} $$ | ||
| With | ||
| <dl> | ||
| <dd>Back-up factor</dd> | ||
| <dt> $$ \\chi = \\sqrt{ | ||
| \\left( | ||
| 1 + \\frac{\\psi\\cdot a}{h_0} | ||
| \\right) \\cdot | ||
| \\left\\{ | ||
| \\left[ | ||
| 1 - 2\\cdot\\frac{\\psi\\cdot a}{h_0} \\cdot | ||
| \\left( | ||
| 1-\\frac{\\psi\\cdot a}{h_2} | ||
| \\right) | ||
| \\right] | ||
| - \\sqrt{ | ||
| \\left[ | ||
| 1 - 2 \\cdot \\frac{\\psi\\cdot a}{h_0} \\cdot | ||
| \\left( | ||
| 1-\\frac{\\psi\\cdot a}{h_2} | ||
| \\right) | ||
| \\right]^2 | ||
| + | ||
| \\left( | ||
| \\frac{h_2}{h_0} | ||
| \\right)^2 | ||
| - 1 | ||
| } | ||
| \\right\\} | ||
| } \\tag{8.28}$$ </dt> | ||
| </dl> | ||
| </p> | ||
| <h5>Boundary between free and backed-up flow</h5> | ||
| <p> | ||
| 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. | ||
| </p> | ||
| </html>")); | ||
| end TainterGate3; |
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