# Development and Application of a Novel Non-Iterative Balancing Method for Hydronic Systems
## Appendix F Application of the Ratio Method (RM) Procedure
### Application of the Progressive Flow Method (PFM) Procedure  
The PFM method was then applied according to the proposed procedure as shown in Figure 11. Since there are only two terminals for each branch, it is irrelevant whether they are connected to a zone manifold or, as in the case tested, derived from a tee; in both cases they will be referred to the same pressure. To be able to determine which terminal is the unfavorable without reading of terminal probes (actually available in the test rig but not supposed to be used by PFM procedure), the procedure indicates to run the pump at a constant speed, open one terminal at a time, measure the flow rate via the rotameter at the pump, and sort the users according to the ratio of the measured flow rate vs the nominal flow rate.  
In Figure 11--Testing, the flow measurement taken at the terminals using the MBV probes (not supposed to be used with the PFM) is shown for the sole purpose of verifying that the starting configuration with all valves fully open is the same for both methods tested. The method then proceeds as per procedure (Figure 12 Steps from 2 to 9), for each zone (with all other zones closed acting on branch valves), the reference terminal is fully opened and brought directly to its nominal flow rate (as measured by the rotameter at the pump) by acting on the inverter and/or the bypass branch; then the reference loop pressure control is activated, and the other terminal is progressively opened until the two terminals' summed flow rate is measured by the rotameter at the pump according to the concept of flow rate progressivity.  
Afterwards, the operator moves on to the next branch and so on; unlike the RM procedure, in this case there is no iteration, and each terminal can be brought exactly to its nominal flow rate according to the resolution of the instrument. For terminals balancing, the rotameter activated is the smaller, more suitable in terms of range and resolution.  
Once the balancing of the terminals branch by branch is complete, the operator moves on to the balancing of branches (Figure 12).  
All branches are closed except for the most disadvantaged one (branch A), which is brought up to nominal flow rate, the reference control is activated, and branches C, B, and D are progressively opened, again without iteration. This time the rotameter considered is the one with the higher range, which is more suitable in terms of flow values and resolution for balancing branches.  
Finally, adjusting is not required since the right pump speed is intrinsically achieved by the reference control loop along the procedure of branches balancing.  
The final result is shown in Figure A12--step 12, the values reported are measured in terms of the progressive increase in flow rate at the rotameter during the progression of the procedure and incorporate the 10% tolerance level accepted at each step within the limits of the instrument's resolution. The number of measuring actions related to the virtual moving pressure gauge from one terminal to another is only required by the setting of the reference control, and in the current case it counts one each for the four branches and one for the final balancing of branches, so N\({}_{\text{meas}}\)= 1 \(\times\) 4 + 1 = 5. Regarding the number of balancing operations, the procedure requires one operation per terminal and then one per branch, so N\({}_{\text{bal}}\) = 8 + 4 = 12.