We are going to start the process of generating a useful numerical
model for the differential equations studied in the last lecture. You
will get
a general background in the various finite volume methods used for
power plant simulation. I will point out several limitations inherent
in averaging required for numerical solutions. The material below is
just an outline of the lecture. For a more detailed discussion
read the Field Equations chapter in the TRACE theory manual available
on ANGEL or here.

- Finite volume is closely related to the process of deriving the
differential equations for flow. You can just run the argument
chain in reverse.

- Consider flow in a pipe, and break it into a finite number of volumes
- cell centers indexed j-1, j, j+1
- cell edges labeled j-1/2, j+1/2
- Integrate the differential equation over volume j and apply Gauss' theorem to convert the integral of the divergence to a surface integral of mass fluxes. The result is that this integral is the difference between the mean mass flows at cell edges j+1/2 and j-1/2.

j-1 |
j-1/2 |
j |
j+1/2 |
j+1 |

- Discuss area averages, average of products
- Recall discussion of results for laminar and turbulent flow
- Stratified flow is not a major issue for the two-fluid model
because of the division of equations. However, an extra force
term is required proportional to the spactial derivative of void
fraction along the horizontal direction.

- Non-uniform distribution of bubbles and drops in flow can cause
problems, as can a liquid flow that is a mix of drops and liquid film
on the wall. This is addressed in codes like COBRA-TF by adding a
second liquid field. In this case one liquid field follows the
wall film and the second follows the droplets. You can imagine
scenarios where more than two liquid fields are needed. Detailed
models of bubbly flow need at least two gas fields if you are going to
naturally model the evolution from bubbly to slug flow.

- The volume integral of the time derivative becomes the time derivative of the volume average. The time derivative must be generated by dividing time into chunks . The equations, that you are likely to use, only deal with information at two points in time for each step forward in time. These are generally referred to as the old time level (stuff you already know as a result of initial conditions or the last step forward in time) and the new time level (stuff you don't know until you solve the coupled set of equations). Old time information is generally marked in an equation with a superscript of "n" and new time information marked with a superscript of "n+1". The approximation for the time derivative of the volume average density is then the difference to the new and old time average densities, divided by the time increment (step) between the two points in time.
- Recall location of variables on the staggered mesh
- Look at the original division of the pipe
- Remember that the cell center is taken to be the location of the average cell values for thermodynamic variables.
- Velocities are evaluated at the cell face
- Mass flux at the cell face is the product of the area of the face (input), velocity at the face (calculated directly), and density at the cell face (not calculated directly since density is a cell center variable).
- Spatial average to obtain density at a cell face
- The obvious choice is the average of cell center values
- This doesn't work well for two phase flow especially with standard choices for time level evaluation.
- A simple but inaccurate average called the donor-cell gives good stability behavior, but is generally inaccurate. An exception to the accuracy problems is evident in the homework.
- When selecting time levels for terms in the cell edge flux, we generally deal with one of three well defined forms.
- The
**Fully Implicit**method evaluates both density and velocity at the new time level. This results in a rather complicated coupled set of non-linear equations for the system variables. At the moment only CATHARE uses this procedure rigourously, although RETRAN is close, and TRACE will have the option within a year.

- The
**Fully Explicit**method evaluates both the density and velocity at the old time level. This produces a simple uncoupled set of equations. However, this method is not used in standard power plant simulation tools because of time step limitations.

- The
**Semi-Implicit**method evaluates the density at the old time level and the velocity at the new time level. It results in a coupled set of equations that is much simpler to solve than those produced by a fully implicit method. This is the standard method for RELAP5 and an optional method (NOSETS=1) in TRACE.

- The Stability Enhancing Two-Step (SETS)
or "Nearly Implicit" (a variation used in RELAP5) methods add second
stabilizing evaluation of each equation (mass, energy, momentum) to the
basic semi-implicit equation set to produce results more like a fully
implicit method with less computational effort (see the TRACE Theory
Manual for more details)