1. Computers in Civil Engineering 53:081 Spring 2003 Lecture #10 Pipe Network as a Systems of Non-Linear Equations

2. Network of Pipes (e.g. municipal water supply network) Node Pipe

3. The Problem Determine the distribution of discharge among the various pipes of the network – and in so doing, ensure that the pressures (heads) meet the requirements of domestic and industrial use (including fire hydrants). Engineer’s design problem: satisfy the design requirements while minimizing the cost of the pipes (as small as possible using standard sizes)

4. For a Node Q = volumetric flowrate, e.g. m 3 /s Q D - known outflow demand Continuity Equation (conservation of mass) or (What goes in must come out – no storage at node)

5. For a Pipe u - upstream d - downstream Headloss from Darcy-Weisbach Equation (conservation of energy)

6. Darcy-Weisbach Equation f - friction factor (from Moody diagram) L - length of pipe D - diameter of pipe Q - discharge in pipe A - cross section area of pipe g - gravitational acceleration Q | Q |- ensures head loss is in the flow direction

7. Mathematical Formulation Network of k nodes and m pipes Equations k - continuity equations m - head loss equations total: k+m equations Unknowns k - unknown heads (at nodes: ) m - unknown discharges (for pipes: ) total: k+m unknowns

8. Mathematical Formulation Problem –Set up the equations –Calculate the partial derivatives –Formulate the linear system to be solved Solution System of k+m nonlinear equations with k+m unknowns (recall Q|Q| appears, like Q 2 ) Newton-Raphson Iterative Method (see Lecture #8) Steps

9. Step 1(a): Setting up the Equations known pipe parameters in pipe j Where:

10. Step 1(b): Setting up the Equations While it is in principle possible to solve the equations, numerical solvers don’t “know” about Q, and h, but deal with x 1, x 2, x 3,… Thus rename the variables Q i, and h j as x 1, x 2, x 3,… (let n=m+k)

11. Review: Solving System of Non-Linear Equations f(x 1,x 2,…,x n )=0 Jacobian Function values at i -th iteration values at i -th iteration Newton-Raphson values at next iteration (unknown) See the textbook!

12. Step 2: Calculate the Derivatives Solving the system using Newton-Raphson requires computation of the Jacobian: This seems complicated, but for our problem it really is not.

13. Equation Derivatives Node equation derivatives Pipe equation derivatives give

14. Step 3: Formulate Linear System Jacobian Function values at i -th iteration values at i -th iteration values at next iteration (unknown) This is a linear system. (See the text)

15. Options for Solution l l Design your own iterative procedure using a linear solver l l Design a Matlab function to solve

16. Example Q d =0.8 m 3 /s A B C PipeD [m] f L [m] AC0.12.023120 AB0.10.025 40 BC20.10.025 60 BC30.10.025 80

17. Example (continued) Caution!!! To obtain a specific (unique) solution replace one of the node continuity equations with an imposed head

18. Next: Worked Example Read the textbook!