page_title: Dimensionless specific energy diagrams for open-channel flow
section_title: Dimensionless diagram
hierarchical_section_title: Dimensionless specific energy diagrams for open-channel flow / Dimensionless diagram
caption_attribution_description: English: Compare E'y' diagram
context_page_description: Energy conservation is an important concept when analyzing open channel flows. For the purposes of the following analysis, energy is conserved for a fluid in an open channel flow, and head losses due to friction will be neglected. The energy calculated at one location in the flow will be equal to the energy calculated at any other location in the same flow.
  The energy for the flow will have a potential energy component calculated from the depth of water in the flow, a pressure
  component, and a kinetic energy component calculated from the velocity of the flow moving through the channel. This is depicted through the Bernoulli equation:
  where:
  E = energy [=] Length*Force,
  v = velocity [=] Length/Time,
  g = acceleration due to gravity [=] Length/Time²,
  y = depth of water in the flow [=] Length,
  p = pressure [=] Force/Length², and
  = specific gravity of the fluid [=] Force/Length³,
  For two locations in the system with the datum chosen as the bottom of a channel with no slope:
  For an open channel flow the fluid, water, is open to the atmosphere so that the pressure throughout the system can be considered equal to atmospheric pressure.
context_section_description: Another way to evaluate a sluice gate problem is to develop the dimensionless form of this diagram dividing the energy equation by the critical depth and substituting Equation 11:
      
  Equation 15 is the dimensionless form of E-y diagram, which is in similar form to the dimensionless form of M-y diagram if we replace y' with 1 / y' .
  Substituting Equation 6 and Equation 11 into Equation 13 we can determine the following relationship:
  With this relationship we know that values on the dimensionless E'-y' diagram with the same value of y' will have the same Froude number.  In this way we can determine the conditions of flow.
  The minimum value of E' on the dimensionless diagram will be the first derivative of Equation 15 with respect to y' (dE'/dy') equated to 0 for the minimum value:
  Giving y' = 1 or yc = y at the minimum value for E'.  We can determine from this that the lowest E' value will be at the critical depth:
  Considering the initial conditions with the sluice gate example above with y = 4.4 ft and yc = 2.3 ft we can calculate y' and E':
  The dimensionless diagram representation for these conditions would be:
  Knowing the value of y' for the subcritical flow is 3.1, and E' is 3.1, the alternate depth value of y' can be determined by 
  finding the supercritical value where the diagram crosses E' at 2.0.  From the graph, this value can be determined to be 0.60.
  This can also be calculated by solving for the alternate value of y' using Equation 15:
      
  Solving for the alternate depth:
  This is the same value we obtained above using the original specific energy diagram (with dimensions) and specific energy equation!
  For the choke conditions described above with y₁ = 7.1 ft and yc = 2.3 ft we can calculate y’ and E’:
  The dimensionless diagram representation for these conditions would be:
  From the graph, this value can be determined to be 0.43.
  Solving for the alternate value of y' using Equation 15:
      
  Solving for the alternate depth:
  This is the same value we assigned for y₂ previously!
caption: Figure6
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