Chapter 1, Problem 20E

The finite element equation for the uniaxial bar can be used for other types of engineering problems if a proper analogy is applied. For example, consider the piping network shown in the figure. Each section of the network can be modeled using an FE. If the flow is laminar and steady, we can write the equations for a single pipe element as:

$\begin{array}{l}{q}_i=K\left(P_i-P_i\right)\\ q_j=K\left(P_j-P_j\right)\end{array}$ where $q_i$ and $q_j$ are fluid flow at nodes i and j, respectively; $p_i$ and $p_j$ are fluid pressure at nodes i and j, respectively; and K is

$K=\frac{\pi D^4}{128\mu L}$ where D is the diameter of the pipe, p is the viscosity, and L is the length of the pipe. The fluid flow is considered positive away from the node. The viscosity of the fluid is $9\times {10}^{-\text{4}}\text{Pa}•\text{s}$.

a. Write the element matrix equation for the flow in the pipe element.

b. The net flow rates into nodes 1 and 2 are 10 and 15 m3/s, respectively. The pressures at the nodes 6, 7, and 8 are all zero. The net flow rate into the nodes 3, 4, and 5 are all zero. What is the outflow rate for elements 4, 6, and 7?