Other than Hardy Cross method, linear theory can be used when we encounter a situation where multiple supply nodes exist in the looping pipe network. Prior to the iterative analysis, we need to establish flow loop. One key difference between the methods is the need of pseudo loop connecting the supply node with known head. Another key difference is that we are free to make any assumption on the flow rate and direction for the first iteration of analysis.
By using this method, we can solve for the flow rate through all the pipes simultaneously. In other words, we need matching number of equations, and they can be established using two approaches. The first approach is based on the continuity of flow principle. Focusing on the nodes with unknown head, we create equation from the correlation of pipe flows based on our assumption. The second approach requires us to implement an equation suit for this method on each flow loop, including the pseudo loop. Rather than using the assumed flow rate, imaginary pipe should only carry flow in terms of head difference.
A common way to solve multiple simultaneous equations is by using matrices. With the right tools, the analysis process can be simplified. Another iteration of analysis is needed if the output from the current calculation does not converge with the input. After convergence is achieved, we are good to determine the flow supplied by each reservoir.
The following introduces the linear theory to analyse looping pipe network. Watch the video above for full details.
The expression for head loss and R are determined. Initial pipe flow rate is assumed before conducting linear theory method.
Four of the seven required equations are constructed using principle of continuity of flow.
Three of the seven required equations are constructed out of each flow loop using equation specific for this method.
Arrangement of seven simultaneous equation in matrix form to ease the analysis.
Iteration 1
Processing of iteration 1 output for upcoming iteration
Iteration 2
Iteration 3
Iteration 4
Iteration 5
Iteration 6
Iteration 7
Iteration 8
Output from Linear Theory method
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