# Development and Application of a Novel Non-Iterative Balancing Method for Hydronic Systems
## 3 Theory and Calculations  
### Progressive Flow Method Theory  
The studied method was originally developed for air systems in order to overcome the iterative nature of classical methods, in the form of application of the same methods previously mentioned for water systems TAB: the ratio method and step-wise method. The assumptions that led to its validation in air systems are considered still valid in water systems; these consist of constant density and a quadratic relationship between velocity and pressure drop under the assumption of a fully turbulent flow regime (ASHRAE [30]). The lumped-parameters representation of a supply air system (Figure 2) is formally the same as that of a close-loop water system, with the notable difference that in an air system, the return path through the atmosphere has no pressure losses, whereas the hydronic piping is affected by pressure drops due to friction in the return path. In the transposition of the method from air to water, the air diffusers have been replaced by the terminal users' exchangers, the fan replaced by a pump, and the dampers replaced by balancing valves. The main physical parameter considered in the model is the loss factor \(R\) due to friction in pipes and fittings, the overall value of which can be suitably increased by throttling the balancing devices. The relevant quantities considered by the equations are flow rate \(\dot{Q}\) [m\({}^{3}\) s\({}^{-1}\)], pressure drop \(\Delta P\) [Pa], and the head pressure provided by the pump \(\Delta H\) [Pa]. In addition to the shared characteristics, the two systems also have differences in actual installations, the most significant of which for the purposes of this research is the tightness. One of the consequences is that while the performance validation of an air systems requires the measurement of the flow rate exactly at the terminals because of air leakages along the ducts, the tightness in the case of water systems allows cumulative measurements to be carried out; according to mass conservation, the branch flow rate is equal to the sum of the flow rates of terminals belonging to that branch, and the flow rate at the pump is equal to the sum of the branches' flow rates. This fact does not lead to any optimization in balancing procedures when using the standard iterative methods or CM, which require flow meters at each terminal. In contrast, in the case of the PFM, the tightness is fully exploited in order to obtain a significant improvement: as described below, flow measurements can be carried out at a single point (e.g., at the pump) while measurements at the terminals can be replaced with an easier to do acquisition of the pressure difference across the user or, in some cases, with the measurement of temperature difference ([PERSON] 2014) [31]. In any case, the analysis of the theory behind the method does not distinguish the way in which the flow rate is measured. This distinction is addressed in terms of further optimization, when the application procedure of the method is presented. Figure 2: Equivalence of graphical representation air vs. water system: both can be represented by the same lumped-parameters model. The theoretical framework of the method was developed considering the lumped-parameters model previously shown in the lower part of Figure 2.  
The relationship between the pressure drop \(\Delta P\) [Pa] on a straight section (major/distributed losses) as well as across fittings (minor/local losses) and the volume flow rate \(\dot{Q}\) [m\({}^{3}\) s\({}^{-1}\)] is assumed to be  
\[\Delta P=R\cdot\dot{Q}^{2} \tag{1}\]  
where \(R\) [Pa-s\({}^{2}\cdot\)m\({}^{-6}\)] is the loss factor, whose overall value for major losses depends on piping geometry, (length and diameter), fluid properties (density), and friction factor \(f\); for minor losses, \(R\) depends on the geometry of fittings and components (i.e., curves, tees, exchangers, balancing valves) and can be expressed by means of density, average fluid velocity, and the geometry- and size-dependent loss coefficient \(k\)[30].