Branch Transient Theory
At a branch, there can be multiple pipes. Two additional relationships are needed, the conservation of mass and conservation of energy. The total mass flow in and out of the junction must sum to zero. The branch junction is treated as adiabatic, so the energy must sum to zero as well.
A constant stagnation pressure Po is asserted at the branch since the branch is treated as lossless, meaning there is no stagnation pressure loss. By asserting a constant stagnation pressure the branch can essentially be treated as an assigned pressure boundary, though the "known" stagnation pressure must be iterated on.
Using the stagnation pressure solution the flow rate and density can be solved for with the appropriate compatibility equations as is done for the Assigned Pressure Junction.
The total mass flow in can then be calculated
and the total energy flow is found using
Using the above equations and enforcing energy conservation, the stagnation enthalpy can be found as follows
While the conservation of energy is inherently enforced in the solution using the above equation, conservation of mass is not inherently included. Mass conservation is instead achieved using iteration.
The total mass outflow can be written similar to the mass inflow summation above. The total mass outflow can then be calculated similar to the mass inflow using the compatibility equations, known stagnation pressure solution and known stagnation enthalpy.
An iterative method for stagnation pressure is used to drive the mass inflow and outflow difference to zero.
Sonic Choking Transient Theory
Expansion choking can occur at a branch if the total cross-sectional area of the inflow pipes is less than the total cross-sectional area of the outflow pipes. If sonic choking occurs at the branch junction a different solution method must be enacted. See Transient Sonic Choking for more information.
