Fluid flows in pipe networks can be analyzed in a manner

Chapter , Problem 7.60

(choose chapter or problem)

Fluid flows in pipe networks can be analyzed in a manner similar to that used for electric resistance networks. Figure P7.60a shows a network with three pipes, which is analogous to the electrical network shown in part (b) of the figure. The volume flow rates in the pipes are \(q_{1}, q_{2}\), and \(q_{3}\). The pressures at the pipe ends are \(p_{a}, p_{b}\), and \(p_{c}\). The pressure at the junction is \(p_{1}\).

a. Assuming that the linear resistance relation applies, we have

\(q_{1}=\frac{1}{R_{1}}\left(p_{a}-p_{1}\right)\)

Obtain the equations for \(q_{2}\) and \(q_{3}\).

b. Note that conservation of mass gives \(q_{1}=q_{2}+q_{3}\). Set up the equations in a matrix form Aq = b suitable for solving for the three flow rates \(q_{1}, q_{2}\), and \(q_{3}\), and the pressure \(p_{1}\), given the values of the pressures \(p_{a}, p_{b}\), and \(p_{c}\), and the values of the resistances \(R_{1}, R_{2}\), and \(R_{3}\). Find the expressions for matrix A and the column vector b.

c. Use MATLAB to solve the matrix equations obtained in part (b) for the case: \(p_{a}=30 \ \mathrm{psi}, p_{b}=25 \ \mathrm{psi}\), and \(p_{c}=20 \ \mathrm{psi}\). Use the resistance values \(R_{1}=10,000, R_{2}=R_{3}=14,000 \ 1 /(\mathrm{ft}-\mathrm{sec})\). These values correspond to fuel oil flowing through pipes 2 ft long, with 2 in. and 1.4 in. diameters, respectively. The units of the answers should be \(\mathrm{ft}^{3} / \mathrm{sec}\) for the flow rates, and \(\mathrm{lb} / \mathrm{ft}^{2}\) for pressure.