Compatibility Equations
From the characteristic equations (repeated below), we can construct three convenient equations known as the compatibility equations.
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(Positive Wave Characteristic) |
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(Negative Wave Characteristic) |
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(Particle Path Characteristic) |
The only unknown terms in these equations are the pressure, velocity and density at point 4. The other terms are found from a previous time step, or from the initial conditions as determined from the steady state MOC simulation.
Rearranging terms:
and defining six new constants (constant for this time step):
We have our simplest equations yet:
Which can be solved directly for pressure and velocity, then used to directly calculate density.
Note:The compatibility equations allow a solution of all interior points in a pipe. However, the pipe endpoints are solved by applying specific boundary conditions. The boundary condition relationships build on the previous methods, and offer the final needed information to generate a complete solution.
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