Gradient method may be used to solve any pipe network, regardless of whether it is looped or not. This method solves for the head of nodes and flow rate in pipes at the same time. Therefore, this method demands more rigorous effort during analysis.
Same as linear theory, this method we are free to make any assumption on the flow rate and direction for the first iteration of analysis. There are two ways to identify the simultaneous equations that help us to solve for the unknowns. The first way is by implementing 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 way demands us to implement an equation suit for this method for each pipe. This equation involves head of connected nodes and flow rate in pipe.
Matrices are commonly used to solve for these equations. 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 gradient method 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 gradient method.
Four of the eleven required equations are constructed using principle of continuity of flow.
Equation for each pipe which involves head of start and end nodes, and the flow rate.
Seven of the eleven required equations are constructed out of each flow loop using equation specific for this method.
Simplification of simultaneous equations
Arrangement of simultaneous equations in matrix form
Iteration 1
Processing of iteration 1 output for upcoming iteration
Iteration 2
Iteration 3
Iteration 4
Iteration 5
Output from gradient method
0 Comments