# Development and Application of a Novel Non-Iterative Balancing Method for Hydronic Systems
## 3 Theory and Calculations
### Progressive Flow Method Theory
In the following, the value of \(f\) is considered constant within the typical operating range characterized by a fully turbulent flow, so \(R\) is assumed to be non-dependent on the flow rate value and dependent only on the actual setting of the balancing devices. In case of series and/or parallel arrangement of the components, the relationship (1) can be used appropriately to describe any portion of a system by means of an equivalent loss factor. * For the series configuration of two components or paths whose loss factors are indicated respectively as \(R_{a}\) and \(R_{b}\), the following applies: \[R_{S(a,b)}=R_{a}+R_{b}\] (2)
* For the parallel configuration, the following applies: \[R_{P(a,b)}=\frac{1}{\left(\frac{1}{R_{a}}+\frac{1}{R_{b}}\right)+\frac{2}{ \sqrt{R_{a}\cdot R_{b}}}}\] (3)  
These relationships can be combined to create a model through which balancing methods can be tested and subsequently verified experimentally. From the perspective of how to identify the final balanced configuration, a method is proven to be non-iterative when the parameters whose values are to be assigned are introduced progressively one by one so that the value of every new considered parameter depends only on (i) the design values and (ii) values already determined as a result of a previous calculation or measurement. In this way, at each step, an unknown quantity can be calculated immediately by means of an explicit relation or, in the case of application to a practical procedure, by means of a single calibration/measurement operation that does not involve any further modification of what has been set previously. On a procedural level, this means that the system is operated in such a way that in each current operation, a measured flow rate depends on the position of a single valve, so that its correct position can be easily found by adjusting it until the desired flow rate value is reached. This requires a logic of bottom-to-top engagement of the circuits and the observance at each step of a system flow condition that maintains unchanged the flow rates and settings adjusted in previous steps. So, in order to guarantee the progressive involvement of the parameters to be regulated and avoid any back effect of the new balancing actions, the following conditions are necessary:  
* the procedure starts from the farthest terminal by giving a clear criterion to set the adjustment of its valve position;
* the operator moves backwards and just one terminal at a time is involved;
* a specific control is introduced in order to keep downward (already adjusted) quantities (flow rates and valve positions) unchanged. The introduction of such a control is shared by the CM and PFM, with the difference that the PFM provides that the mentioned control can also take place in real time by meansof a control loop acting on the portion of the system upstream of the point currently being balanced, in order to keep the downstream flow conditions automatically unchanged. Returning to the consideration of the representation of a generic branch (Figure 2), each terminal path (i.e., a radiator, a fancoil unit, or an air-handling unit's coil) is represented as a parallel \(i^{th}\) shunt from the main path, whose \(R_{i}\) value can vary from a minimum value (at valve fully open) to an infinite value according to the throttle position of the balancing valve. As it is shown in Figure 3, and referring to the split \(i\)-point, three sections can be identified: the upstream part of the system on the left, the \(i\)-terminal path, and the downstream D part on the right. The upstream part of the system can be conceptually replaced by an equivalent pump providing the required flow rate \(\left(\dot{Q}_{D}+\dot{Q}_{i}\right)\) at the given head pressure (\(H_{P}\)).