Fluid flows in pipe networks can be analyzed in a

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 p_a, p_b, and p_c. 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 R_i 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 p_a, p_b, and p_c 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 p_a = 4320 lb/ft², p_b = 3600 lb/ft², and p_c = 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