# Development and Application of a Novel Non-Iterative Balancing Method for Hydronic Systems
## 3 Theory and Calculations
### Progressive Flow Method Theory
The downstream part can also be represented by a single equivalent circuit in parallel with the \(i^{th}\) path, characterized by the loss factor \(R_{D}\) and supplied at the same pressure (\(H_{P}\)); at the very start of the procedure, the \(i^{th}\) terminal itself is the farthest and so, the downward loss factor \(R_{D}\) value is assumed to be infinite. The \(i^{th}\) parallel path is characterized by the loss factor \(R_{i}\), which is univocally related to the valve position and whose value can be analytically determined by knowing both pressure and flow rate:  
\[R_{i}=\frac{H_{P}}{\dot{Q}^{2}_{i}} \tag{4}\]  
When the \(i^{th}\) terminal is not the farthest, \(H_{P}\) depends on downstream conditions:  
\[H_{P}=R_{D}\cdot\dot{Q}^{2}_{D} \tag{5}\]  
where \(\dot{Q}_{D}\) is the sum of all the flow rates at the terminals in the downstream portion. So,  
\[R_{i}=R_{D}\frac{\dot{Q}^{2}_{D}}{\dot{Q}^{2}_{i}} \tag{6}\]  
The procedure has been developed in such a way that \(R_{i}\) has to be defined only when the downstream resistance values are already defined, so that \(R_{D}\) is known and \(R_{i}\) can easily be calculated. The analytical step of calculating the value of \(R_{i}\) corresponds, in the practical procedure, to the adjustment of the balancing device which allows the flow rate value \(\dot{Q}_{i}\) to be reached while the simultaneous maintenance of the flow rate value \(\dot{Q}_{D}\) is kept. Figure 3: Representation of a hydronic distribution system at the generic shunt point. According to the previous description, the procedure to be followed when \(R_{D}\) is known is clear; however, the first step still needs to be defined. At the very start of the procedure, and referring to the last terminal \((i=n)\), the situation is quite different (Figure 4), and Equation (6) can be written as  
\[R_{\rm n}=\frac{\rm H_{P}}{\hat{Q}_{n}^{2}} \tag{7}\]  
\(H_{P}\) cannot be calculated using (5) because there is no downstream section already calculated, and for any value of \(H_{P}\) there is a correspondent value of \(R_{n}\) giving the designed \(\hat{Q}_{n}\) flow rate value; so, one additional criterion has to be considered. This criterion can be found via applying the minimum energy requirement; at the design flow rate, the loss factor has to be as low as possible, so the additional information needed is given by consideration of the maximum opening of the valve as the first step of the PFM procedure. Back to the general case in procedure, notice that the calculation of \(R_{D}\) and \(H_{P}\) must be updated at each step by simply by adding the new branch in parallel to the pre-existing downstream equivalent branch as well as the additional line resistance \(R_{i-1,i}\) (Figure 5). Figure 2.0 Lumped-parameters representation of a hydronic distribution system at the farthest terminal. Figure 2.1 Lumped-parameters representation of the downstream part of the circuit; for clarity, the real configuration on the left and its equivalent configuration on the right are enclosed within the dashed rectangular shapes. Referring for generality to the \((i-1)^{th}\) node, as shown in Figure 5 and according to (2) and (3), the following applies:  
Figure 2.2  
Then, applying Equation (6) with updated values, it is possible to calculate the balancing value as  
Figure 2.3  
Every time the operator moves to the next branch, the setting of only one new valve position is involved. The action required is the adjusting of the \((i-1)^{th}\) valve in order to obtain the desired flow rate value \(\dot{\mathcal{Q}}_{i-1}\), and the measurement-driven adjustment allows detection of its final value without any iteration. In the meantime, it is necessary to increase the upstream flow rate to flush the new opening path, keeping unchanged all flow rates on the downstream side as well as the pressure at the \(i^{th}\) shunt nodes. If this is the case, it makes no difference whether the \(\dot{\mathcal{Q}}_{i-1}\) value is measured directly at the terminal or deduced in terms of the increased flow rate provided by the pump.