Problem Network Flow Scott Jewett BIEN 301 January 30, 2007

Problem Diagram Horizontal Pipe Network A C B D 2 ft 3 /s D=6 in D=8 in D=3 in D=9 in 4000 ft 3000 ft f =.025 P A = 120 psi T= 20°C

Required 1.Determine the flow rate and direction in all the pipes 2.Determine the pressures at points B, C, and D.

Assumptions Liquid Incompressible Steady Viscous

Assumptions (cont.) Flow directions Loop directions A C B D Q ac Q cd Q bd Q ab Q bc L1L1 L2L2

Nodal Equations Solve nodal equations Flow out - Flow in = 0 Node A: Node C: Node B: A C B D Q ac Q cd Q bd Q ab Q bc L1L1 L2L2 2ft 3 /s

Use equation 6.10 to obtain head loss as a function of flow rate for each pipe Head loss

Obtain five equations relating flow rate to head loss

Loop Equations Set up loop equations: Sum of head losses around loop = 0 Loop 1: Loop 2: If the flow is opposite the loop, then the head loss is negative. A C B D Q ac Q cd Q bd Q ab Q cb L1L1 L2L2

System of equations Five equations, Five unknowns

Solution Solve using Mathcad or similar tool Q ab = ft 3 /s Q ac =.813 ft 3 /s Q cb =.99 ft 3 /s Q cd = ft 3 /s Q bd =.197 ft 3 /s

Pressure Solution Equation 6.8 relates pressure to head loss h f = ( P a - P b )/(ρg)

Pressure solution P b = P a - ρgh f(ab) P b = 120 psi - ρg(19.116*(Q ab ) 2 ) P b = 108 psi P c = P b - ρgh f(cb) P c = 102 psi P d = P c - ρgh f(cd) P d = 74 psi

Biomedical Application Blood flow –Your body consists of blood vessels with varying: Diameter Friction Height –All of these affect flow rate and pressure.

Questions?