Water flows steadily through the elbow as shown in the image below. The flow is driven by a pump which generates a very high pressure difference between points 1 and 2. The vertical distance between points 1 and 2 is small.

What is the simplest way to approach this problem and still obtain reasonable accuracy, if you wanted to calculate the force at location 1 that is holding the elbow in place?

Question

Water flows steadily through the elbow as shown in the image below. The flow is driven by a pump which generates a very high pressure difference between points 1 and 2. The vertical distance between points 1 and 2 is small.

What is the simplest way to approach this problem and still obtain reasonable accuracy, if you wanted to calculate the force at location 1 that is holding the elbow in place?

BlackTom AI · Accepted Answer

The problem asks for a reasonable, simple approach to estimate the force at location 1 that holds the elbow in place, given a high pressure difference between points 1 and 2 with a steady flow and a small vertical separation.
Option 1: 'Using a form of conservation of momentum that assumes a steady flow and non-uniform velocities at the inlet and outlet, and including both body and surface forces.' While it is possible to account for non-uniform velocities, the phrase 'non-uniform velocities at the inlet and outlet' adds complexity that is not necessary for a first-order estimate. Including both body and surface forces can be more complete, but this option overcomplicates the simplest approach when velocities are close to uniform across the cross-sections for a quick calculation.
Option 2: 'Using a form of conservation of momentum that assumes a steady flow and uniform velocities at the inlet and outlet, and including only surface forces.' This aligns with the common first-order method: apply the steady-flow momentum equation to a control volume around the elbow, assume uniform (or nearly uniform) inlet and outlet velocities to simplify the momentum flux terms, and include surface forces (pressure forces on the CV surfaces and the wall reaction). Since gravity and other body forces are often negligible in pipe-flow problems at modest elevations, this yields a practical, reasonable estimate of the reaction force at location 1.
Option 3: 'You would only need to use conservation of mass to solve this problem as the flow is steady.' Mass conservation ( continuity) tells you about how flow rate is maintained, not about forces or the wall reaction. To determine the force at the wall, you must relate momentum flux changes to external forces; thus mass conservation alone is insufficient for the requested force calculation.
Option 4: 'Using the form of conservation of momentum equation that includes the unsteady term and also including both surface and body forces.' The problem states the flow is steady, so including the unsteady term would add unnecessary complexity and may misrepresent the situation. In a steady-state approximation, the unsteady (time-derivative) term is neglected.
In summary, the most appropriate simple approach for a reasonable first-order estimate is to use the steady-flow momentum balance with assumed uniform inlet/outlet velocities and account for surface forces, avoiding extra unsteady terms or more complex velocity profiles. This mirrors the standard practice for quick force estimates on pipe fittings under a large pressure difference, while keeping the analysis tractable and insightful.

AMME2261 AMME9261 (ND) MCQ6-Conservation of Momentum

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