Hardy Cross method with open loops

[Pencil_lob]

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 hEarth's, 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 hEarth's 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.

 

[Achy47]

Hello,

Thank you for sharing your project with us. The Hardy Cross method is a commonly used technique for solving pipe network problems, so it is a good choice for simulating gas flow inside a multiple hearth furnace. However, as you have pointed out, the method may not work properly if there are open or broken loops in the network.

In general, the Hardy Cross method assumes that the network is fully connected, with no open or broken loops. This means that all the flowrates and head losses in the network are interdependent and can be solved simultaneously. When there are open or broken loops, the flowrates and head losses in those loops are not connected to the rest of the network, and the method may not converge to a solution.

To answer your question, it is possible for the sum of all head losses in a broken loop to be zero, but it is not guaranteed. It depends on the specific conditions and geometry of your furnace. If the head losses in the broken loop happen to cancel each other out, then the method may still converge to a solution. However, if the head losses do not cancel out, the method may fail to converge.

One possible solution to this issue is to manually close the valves in the broken loops, so that they do not affect the rest of the network. This way, you can still use the Hardy Cross method to solve the rest of the network and get an approximation of the gas flow inside the furnace. Another option is to use a different method, such as the Newton-Raphson method, which can handle open and broken loops more effectively.

I hope this helps. Good luck with your project!