# Partial Differential Equations

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- Created by Unknown User (reynard.juanir), last modified by Abraham Toribio on Feb 26, 2019

**The EXTRAN Model** uses the momentum equation in the links and a special lumped continuity equation for the nodes. Thus, momentum is conserved in the links and continuity in the nodes. The model utilizes the relation Q = A·V frequently in the solution and the derivation of the differential equations used in the solution.

The basic unsteady flow continuity equation with lateral inflow and with cross-sectional area (A) and flow (Q) as dependent variables is (Yen, 1986; Lai, 1986):

**Equation 1:**

where, q is the lateral inflow. In EXTRAN this term is zero and the inflows enter the network at the nodes. (The notation for the variables used in this section are listed in the table below:

Variable | Description |
---|---|

As | Surface Area of Node |

C# | Courant Number for the Conduit |

D | Current Conduit Depth |

V | Average Conduit Velocity |

A | Conduit Cross-sectional Area |

T | Conduit Flow Width |

g | Gravitational Acceleration |

n | Manning’s Roughness |

R | Hydraulic Radius |

H | Hydraulic Head (z + h) |

h | Water Depth in Conduit |

Q | Conduit Flow |

Qf | Conduit Design Flow |

Hf | Distance between Junction Invert and Junction Crown |

S_{c} | Expansion/Contraction Loss Slope |

S_{e } | Entrance/Exit loss in Junctions |

S_{f } | Friction (Energy) Slope |

w | Under-relaxation Parameter |

L | Distance along the Conduit |

Dt | Time step |

z | Invert Elevation |

y | Water Depth in Conduit |

The conduit momentum equation may be written in several forms depending on the choice of dependent variables. Using dependent variables flow [Q] and hydraulic head [H] the momentum equation is written (Lai, 1986):

**Equation 2: **

Equation 2 is the form of the momentum equation used in EXTRAN. The seven terms in the momentum equation are, respectively:

- Local inertia:
- Convective inertia
- Pressure slope
- Entrance/exit loss (S
_{e}) - Contraction/expansion loss (S
_{c}) - Friction slope (S
_{f}) - Bed slope (S
_{o})

In the EXTRAN Model the bottom slope (So, or ) is incorporated into the gradient of H, with defined as . The revised momentum equation using these definitions of bottom slope can be written as:

**Equation 3**:

This momentum equation must be modified to suit the requirements of EXTRAN. The purpose of these modifications to the continuity and momentum equations used by EXTRAN are:

- Eliminate the need for the term in the continuity equation. EXTRAN uses a mean or center Q in conduits and the term is not strictly valid.
- Eliminate one equation from the solution by creating a combined continuity-momentum equation.
- Linearize by expanding the partial differential.

The possible substitutions are numerous and found by substituting A·V for Q or Q/V for A in the term in Equation 3:

**Equation 4: **

**Equation 5: **

and linearizing the equations,

**Equation 6**:

**Equation 7**:

Equation 7 is the conservative form of the convective inertia term. After substituting equation 7 into equation 3, the equivalent form of the momentum equation follows:

**Equation 8**:

The continuity equation 1 may be manipulated to replace the third term of equation 8 as follows:

**Equation 9**:

Substituting equation 9 into equation 8 to eliminate leads to the following equation:

**Equation 10**:

The friction slope, Sf, is defined by Manning’s equation and is calculated based on the center hydraulic radius, **R** and alternatively the upstream and downstream hydraulic radius, Rup and Rdn, respectively, as follows:

**Equation 11**:

where k = (n/1.49)^{2} for U.S. customary units and n^{2} for metric units. By using the absolute value of the velocity term, Sf is a directional quantity and ensures that the frictional force always opposes the flow.

**Equation 12**:

Equation 11 is used whenever Rup and Rdn vary by greater than 300 percent, and equation 12 otherwise. This is a very important term in the momentum equation because the primary balance to the pressure and bed slope is the friction slope. This slope is calculated two ways to better handle the change from a steep water surface slope to a flatter water surface slope.

Incorporating the formulation for Sf (equation 11) into equation 10 yields:

**Equation 13**:

This is the form of the momentum equation used by EXTRAN and it has the dependent variables Q, A, V, and H. Equation 13 is still a partial differential equation and must be translated into a finite difference form for its solution using EXTRAN. The finite difference formulation is discussed in the next section.

Additional terms are incorporated into the dynamic wave equation for the entrance/exit loss from junctions and the expansion/contraction loss in conduits. The expansion/contraction conduit slope (Sc) is given by (Fread, 1977):

**Equation 14**:

* where*:

Kec is the expansion contraction coefficient, and the expansion/contraction losses are:

**Equation 15**:

Expansion and contraction losses between conduits in the program are calculated using the expression where **A** is the mean of the current conduit and either the conduit immediately upstream or downstream. The squared velocity difference is the difference between the mean velocities of the three conduits.

The entrance/exit conduit slope is given by:

**Equation 16**:

* where*:

Kee is the entrance-exit loss coefficient, and the expansion/contraction losses are:

**Equation 17**:

The entrance and exit head loss is actually calculated in the conduit by adding an additional term to the dynamic flow equation. The velocity at both the upstream and downstream end of the conduit is used to calculate the entrance and exit loss depending on the flow direction.

Equations 14 and 16 are added to the combined continuity-momentum equation to account for these additional losses in head. The losses are also used when the kinematic wave equation is used for supercritical flow. These losses are computed but not listed in the following discussions of the finite difference approximations of the partial differential equations.

__Section Pages__- Partial Differential Equations
- Finite Difference Equations
- Implicit Time Weighting
- Significant Differences between EXTRAN Versions 3, 4 and 5
- Finite Difference Solution
- Convergence in EXTRAN
- Automatic Time Step Selection
- Special Conduit Flow Conditions
- Flow and Head Computation during Surcharge and Flooding
- Ground and Invert Elevations
- Flow Control or Diversion Devices
- Storage Devices, Ponds, and Lakes
- Orifices
- Weirs
- Weirs with Tide Gates
- Pump Stations
- Outfall Structures
- Boundary Conditions
- Initial Conditions
- Pit or Junction Losses
- Oscillations during a Hydraulic Jump
- Irregular Closed Conduits
- Irregular Open Channels
- Kinematic and Diffusion Wave Equations
- Special Finite Difference Approximations
- Conduit Depth Calculations