At a branch, there can be multiple pipes (Wylie, et al, 1993Wylie, E.B., V.L. Streeter & L. Suo, Fluid Transients in Systems, Prentice Hall, Englewood Hills, New Jersey, 1993., pp. 51-53). An additional relationship is needed, and that relationship is the conservation of mass. The total mass flow in and out of the junction must sum to zero. In addition, the branch has a single pressure solution, P_j, and this solution is common to all in-flowing pipes. The compatibility equation is written for each junction. For the pipes flowing into the branch, the positive equation is used (Compatibility Equations, Equation 1)

where Pj  is the branch junction pressure, as yet unknown.

For the pipes flowing out of the branch, the negative equation is used (Compatibility Equations, Equation 2)

Now, sum all of the pipe flow rates into the junction

(1)

where

(2)

(3)

In general, there may be a known flow into the branch (a flow sink) or into the branch (a flow source). Therefore rewrite Equation 1 as

Now solve for the junction pressure

(4)

With the pressure known, substitute back into the compatibility equations to obtain the flow rate into the branch coming from each connecting pipe.

The applied flow rate may vary with time.

Branch Vapor Cavitation Theory

When the calculated pressure at a Branch junction drops below vapor pressure, a vapor cavity forms at the junction.

Assuming a flow source/sink is specified, the vapor volume calculation is as follows:

where ṁ_B is the branch source/sink term defined as positive flow into the branch. The ṁ_in terms are obtained from the positive compatibility equation, while the ṁ_out terms are obtained from the negative equation.

Similar to a pipe interior node, when the vapor volume is negative, the cavity collapses and the fluid pressure then rises above the vapor pressure.