Bottom and Side Outlet Orifices

Note:  The main difference between an Bottom and Side Outlet orifice at the same offset elevation and the same diameter is the depth at which the flow in the orifice will switch between weir flow and orifice flow.  The Side Outlet orifice has Weir flow until the Orifice is full but the Bottom Orifice has Weir flow until the Critical Height which is usually shorter than the maximum depth of the orifice.

For a circular orifice the Critical Height is:

    Critical Height = Orifice Discharge Coefficient / 0.414 * Orifice Opening / 4

 

 For a rectangular orifice the Critical Height is:

            Critical Height = Orifice Discharge Coefficient / 0.414 * (Orifice Opening*Width) / (2.0*(Orifice Opening+Width))

St. Venant Terms in SWMM 5

Note:  An explanation of the four St. Venant Terms in SWMM 5 and how they change for Force Mains.  The HGL is the water surface elevation in the upstream and downstream nodes of the link.  The HGL for a full link goes from the pipe crown elevation up to the rim elevation of the node + the surcharge depth of the node

             dq2 = Time Step * Awtd * (Head Downstream – Head Upstream) /  Link Length  or

              dq2 = Time Step * Awtd * (HGL) /  Link Length

              Qnew = (Qold – dq2 + dq3 + dq4) / (  1 + dq1)

 when the force main is full dq3 and dq4 are zero and

  Qnew = (Qold – dq2) / (  1 + dq1)

  The dq4 term in dynamic.c uses the area upstream (a1) and area downstream (a2), the midpoint velocity, the sigma factor (a function of the link Froude number), the link  length and the time step or

              dq4 = Time Step * Velocity * Velocity * (a2 – a1) / Link Length * Sigma

 the dq3 term in dynamic.c uses the current midpoint area (a function of the midpoint depth), the sigma factor and the midpoint velocity

              dq3 = 2 * Velocity * ( Amid(current iteration) – Amid (last time step) * Sigma

  dq1 = Time Step * RoughFactor / Rwtd^1.333 * |Velocity|

 

The weighted area (Awtd) is used in the dq2 term of the St. Venant equation:

              dq2 = Time Step * Awtd * (Head Downstream – Head Upstream) /  Link Length

 

Orifice and Weir flow calculations

Note:  Orifice and Weir Flow Computations 

 The orifice flow calculation proceeds as follows:

 1. Initially and whenever the setting (i.e., the fraction opened) changes, flow coefficients for both orifice and weir behavior are computed as follows:

    a. For side orifices:

      Define Hcrit = h/2 where h is the opening height.

    b. For bottom orifices:

      i. For a circular orifice, compute area over length (i.e., circumference) as AL = h /4.

      ii. For a rectangular orifice compute AL = h*w/(2*(h+w)) where w is the opening width.

      iii. Compute Hcrit = Cd*AL/0.414 where Cd is the orifice discharge coefficient.

 At step 1b, the critical head for the bottom orifice, where orifice flow turns into weir flow, is found by equating the result of the orifice equation to that of the weir equation

  Cd*Area*sqrt(2g)*sqrt(Hcrit) = Cw*Length*sqrt(Hcrit)*Hcrit or

  Hcrit = (Cd * Area) / (Cw/sqrt(2g) * Length) The value of Cw/sqrt(2g) for a sharp crested weir is 0.414.

 

   c. Compute the flow coefficients (where A is the area of the opening):

      Corif = A*sqrt(2g)*Cd

      Cweir = A*sqrt(2g)*Cd*sqrt(Hcrit)

 2. During flow routing, compute the degree of inlet submergence (f) and head (H) at the current time step:

    a. Define:

      H1 = upstream head (from node with higher head),

      H2 = downstream head (from node with lower head) ,

      Hcrest = elevation of bottom of opening,

      Hcrown = elevation of top of opening,

      Hmidpt = elevation of midpoint of opening

    b. For side orifices:

      f = min{1.0, (H1 - Hcrest) / (Hcrown - Hcrest)}

      if f < 1.0 then          H = H1 - Hcrest,

      else if H2 < Hmidpt then H = H1 - Hmidpt

      else                     H = H1 - H2

   c. For bottom orifices:

      if H2 > Hcrest then H = H1 - H2

      else                H = H1 - Hcrest

      f = min{1.0, H/Hcrit}

 3. Compute the flow through the orifice (Q):

 

   if f < 1.0 then Q = Cweir*f^1.5

   else            Q = Corif*sqrt(H)

 

4: Villemonte correction

 

   If f < 1.0 and H2 > Hcrest then:

    r = (H2 - Hcrest) / (H1 - Hcrest)

   Q = Q * (1 - r^1.5)^0.385

 

 Weir Flow Computations

 

1.  Weir head calculations

         h1 = Upstream Node Depth + Upstream Invert Elevation

         h2 = Downstream Node Depth + Downstream Invert Elevation

         If h2 is greater than h1 then the flow is reversed and h2 = h1 and h1 = h2

        Weir Crest = Upstream Node Invert Elevation + Weir Offset Depth

        Head = h1 – Weir Crest

  2.  Center Weir flow for Transverse Weirs

  Q = Cw * Weir Length * Head^3/2

  3.  Center Weir flow for Side Flow Weirs

  Weir behaves as a transverse weir under reverse flow

  Q = Cw * Weir Length * Head^3/2

           And under normal flow

  Q = Cw * Weir Length * Head^5/3

  4.  Center Weir flow for V Notch Weirs

  Q = Cw * Weir Slope * Head^5/2