Loop Equations. Equation (1), which defines the relationship among the number of pipes, primary loops, junction nodes and fixed pressure nodes offers a basis for formulating a set of flow- pressure drop equations to describe a pipe system. In terms of the unknown flow (W expressed in lb/s or kg/s) in each pipe, a number of mass continuity and static pressure loss equations (P expressed in lb/ft2 or Pascals) can be written equaling the number of pipes in the system. For each junction node a continuity relationship equating the flow into the junction (Win) to the flow (Wout) is written as: Here We represents the external inflow or outflow (demand) load at the junction node. For each primary loop the static pressure drop (loss) equation can be written for pipe sections in the loop as follows: ΣPL = ΣPC (ℓ equations) (5) ΣPL = pressure drop (loss) in each pipe (including minor loss) ΣPC = pressure increase due to a compressor If there are no compressors in the loop then the pressure loss equation states that the sum of the pressure losses around the loop equals zero. If there are f fixed pressure nodes, (f - 1) independent pressure loss change equations can be written for paths of pipe sections between any two fixed pressure nodes as follows: △PF = ΣPL – ΣPC (f-1 equations) (6) where △PF is the difference in static pressure between the two fixed pressure nodes. Any connected path of pipes within the pipe system can be chosen between these nodes. When identifying these (f-1) pressure change equations care must be taken to avoid redundant paths. The best method to avoid this difficulty is to either choose all parallel paths starting at a common fixed pressure node (like A-B, A-C, A-D, etc.) or to use a series arrangement where the previous ending node for a path is the starting node for the next path (like A-B, B-C, C-D, etc.). Either of these methods will result in (f-1) equations with no redundancy. The program employs the first scheme. As an additional generalization, Equation (5) can be considered to be a special case of Equation (6) where the difference in static pressure (△PF ) is zero for a path which forms a closed loop. Thus, the static pressure change relationships for pipe networks are expressed by (ℓ + f - 1) path equations of the form given by Equation (6). The junction and path equations constitute a set of p simultaneous nonlinear algebraic equations referred to as loop equations. The desired steady state flow requires the solution of this set of equations for th...
