Problem: Suppose water is flowing through a 0.4 m*m pipe (1) with a velocity of 25 m/s and a static pressure of 940 kPa. The pipe splits into two branches, one that is 0.18 m*m (3) and one that is 0.28 m*m (2). At section (3), the static pressure is measured to be 570 kPa, and the velocity is 30 m/s; at section (2), the static pressure is 1140 kPa, and the velocity is unknown. Determine the amount of available power lost in this horizontal y-connection.
Given: we know the areas and pressures at each location, the velocity at two locations, and the fluid properties.
Find: amount of available power lost
Assumptions: steady, incompressible, uniform properties and pressure across the pipe cross-section, horizontal pipe, no pumps/turbines, constant average velocity
After making our assumptions and drawing our control volume, we must apply the continuity equation:
For steady, incompressible flow with three surfaces where flow is crossing the CS:
Since (1) is an inlet, (2) and (3) are outlets, and the velocities are constant averages, and dividing out density:
Now that we have determined the velocity at (2), we can apply the first law:
For steady, incompressible flow with no pumps or turbines, and three surfaces where flow crosses the CS:
Since (1) is an inlet, (2) and (3) are exits, and we have uniform properties and constant average velocities:
Consider the terms:
Since the y is horizontal, there is no difference in z1, z2 or z3:
The terms in the parentheses above is the same as those in expression (I), which we know is equal to zero; therefore, (II) is equal to zero.
Our losses for this problem are defined by:
So we end up with: