# Development and Application of a Novel Non-Iterative Balancing Method for Hydronic Systems
## 3 Theory and Calculations
### Management of Cases Where the Farthest Terminal Is Not the Most Disadvantaged  
The configuration can be examined with reference to Figure 6, where two different paths are represented branching off from \((i-1)\) points: the first corresponds to the \((i-1)^{th}\) branch which has to be activated; the second, on the right in Figure 6, consists of the downstream part of the system, represented by the equivalent loss factor (\(R_{D(i-1)}\)), which is assumed to have already been balanced.  
According to the general theory of balancing in case of ramification, a choking action in the favored path is required, and this implies the availability of a balancing device on it. In cases where the favored circuit is downstream, the requirement can be satisfied by the presence of an adjustable line loss (\(R_{Line\;i-(i-1)}\)) whose throttling action affects all downstream terminals equally and does not affect their pre-existing mutual balancing.  
Analytically, in cases where the \((i-1)^{th}\) path is disadvantaged, the proper value of the additional line resistance \(R_{Line\;i-(i-1)}\) depends on the new path loss \(R_{(i-1)100}\) (the resistance of the \((i-1)^{th}\) path with the valve \(BV_{i-1}\) in 100% open condition), on the equivalent resistance of the downstream part already balanced \(R_{D(i-1)}\), and on the flow rate values \(Q_{i-1}\) and \(Q_{D}\):  
\[H_{P}=R_{(i-1)100}\times{Q_{i-1}}^{2}=\Big{(}R_{Line\;i-(i-1)}+R_{D(i-1)} \Big{)}\times{Q_{D}}^{2}\]  
\[R_{Line\;i-(i-1)}=R_{(i-1)100}\bigg{(}\frac{Q_{i-1}}{Q_{Di}}\Big{)}^{2}-R_{D( i-1)} \tag{10}\]  
In terms of system equipment, the requirements are shown in Figure 7.  
Figure 6: Insertion of a line-balancing device downstream of the disadvantaged terminal, lumped-parameters model.  
Figure 7: Insertion of a line-balancing device downstream of the disadvantaged terminal, system equipment representation.  
In practice, while the automatic control loop (or an assistant operator) acts on the system (i.e., on the pump), maintaining unchanged the downstream quantities, the operator starts to open the valve \(BV_{i-1}\) and the \((i-1)^{th}\) terminal begins to be flushed, causing a decrease in the downstream flow rate, which is assumed to be immediately compensated by the upstream control. When the terminal valve is fully open, if the flow rate \(Q_{i-1}\) still does not reach the nominal value, this means that the \((i-1)^{th}\) terminal is actually the MD. Then, the operator moves to act on the line valve \(BV_{\mathit{Ilie}\,\dot{c}(i-1)}\) and begins to close it, and the flow deviation combined with the loop compensation increases \(Q_{i-1}\) up to the desired value.  
As mentioned before, the flow rate measurement required for this operation can be performed directly on the \((i-1)^{th}\) terminal if a local flow meter is available; in case it is not, it is possible to measure the total flow rate coming from the pump, whose target value corresponds to the value of the previous downstream flow rate (unchanged) increased by the nominal value of the newly added terminal.  
In any case, the line resistance can be introduced by installing a simple device capable of introducing a modulable pressure drop without any measuring features. Since it is assumed that the pipeline designer knows which terminal will be disadvantaged with respect to the part of the system that remains downstream, the design can provide for the presence of appropriate line balancing valves, allowing the complete PFM procedure to be carried out.