Hi. I am using the Hardy Cross method to simulate the gas flow inside a multiple hearth furnace (as shown in figure A). Figure B is a representation of the furnace's body as a pipe network. The segment A-AH represents the actual hearths, while segments B-AI and C-AJ represent the exhaust ducts of the furnace. Segments A-B, A-C, D-E, and so on, represent the small ducts from which the gases flow from the hearths into the ducts. Each of these small ducts has a manual, discrete valve, with 11 positions (0 to 100% opening). I am currently implementing the method in an Excel spreadsheet, since the company I'm working for wants it linked to other process spreadsheets used by them. The point of the project is not to give an exact result, but only an approximation to the behavior of the gas flow inside the furnace while varying each of the valves' positions. I have a position selector for each of the valves in my spreadsheet, which will vary the downstream head loss from the valve accordingly, and thus, flowrate. When a valve is completely closed, the flowrate through its duct is naturally 0, which will allow for open loops in my network. Figure C shows an example of this; in this case, the D-E duct is closed, there's no gas flow through it, so the loop becomes "broken." My question is: will the Hardy Cross method still converge to a solution if there's broken loops? Is it possible for the sum of all head losses inside a broken loop to be zero? Additionally, I'm open to suggestions for different methods I could use to model the gas flow inside the furnace. Thank you for your help.