Fluid flows in pipe networks can be analyzed in a manner similar to that used for electric
Question:
Fluid flows in pipe networks can be analyzed in a manner similar to that used for electric resistance networks. Figure P6 shows a network with three pipes. The volume flow rates in the pipes are q1, q2, and q3. The pressures at the pipe ends are pa, pb, and pc. The pressure at the junction is p1. Under certain conditions, the pressure– flow rate relation in a pipe has the same form as the voltage-current relation in a resistor. Thus, for the three pipes, we have
where the Ri are the pipe resistances. From conservation of mass,
q1 = q2 + q3.
a. Set up these equations in a matrix form Ax = b suitable for solving for the three flow rates q1, q2, and q3 and the pressure p1, given the values of pressures pa, pb, and pc and the values of resistances R1, R2, and R3.
Find the expressions for A and b.
b. Use MATLAB to solve the matrix equations obtained in part a for the case where pa = 4320 lb/ft2, pb = 3600 lb/ft2, and pc = 2880 lb/ft2.
These correspond to 30, 25, and 20 psi, respectively (1 psi = 1 lb/in2, and atmospheric pressure is 14.7 psi). Use the resistance values R1 = 10 000, R2 = 14 000 lb sec/ft5. These values correspond to fuel oil owing through pipes 2 ft long, with 2- and 1.4-in. diameters, respectively. The units of the answers are ft3/sec for the flow rates and lb/ft2 for pressure.
Figure P6
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