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I'm having a discussion about the pressure in a flexible line. The concern is there is a rigid connector joining two flexible lines of the same diameter but the connecting piece is $40$ % smaller than the two transfer lines. The pump output is variable and at maximum flow rates, the pressure preceding the rigid connector may prove to exceed the capacity of the flexible line, not to mention added stress on the pump itself. I'm told if the pump output is within the tolerance for the pipework, then there is no inherent danger of failure of the flexible pipe, but it is still inefficient at least.

I have expressed concern but been told it's not a problem, yet there is observable dilation of the flexible line prior to the connector. I have little knowledge or qualification in this field so I find myself unable to present a cogent argument backed up with empirical data. If someone could show me a formula to calculate the disparity in the pressure on either side of the connector or how the restriction could cause peaks in pressure prior to the connection it could prove useful.

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You need the pump "head" curve, an estimate of the frictional losses in the system (the friction "head"), and the elevations of the components. If this is a liquid with little heating effects, you apply the mechanical energy balance including a "loss" term for the friction head. With this information you can calculate the pressures at the various points in the system. The pump will operate at a point on its head curve dependent on the flow rate which is dependent on the rest of the system (elevations and friction losses). This is a standard problem in fluid dynamics. In engineering, the "head" is the pressure divided by the product of density and the acceleration of gravity.

I guess that the pump is centrifugal not positive displacement? The Pump Handbook textbook is an excellent reference.

A couple of comments on your question. (1) You say at maximum flow the pressure may be too high; typically the pressure decreases with flow due to the characteristics of the pump head curve and the increase in friction losses. (2) Also, as the pump delivers more flow it requires more power and at "runout" would trip the circuit breaker (assuming electric driven pump). (For very large pumps for some systems, cavitation is of concern as the pump runs out on its head curve.)

For commercial systems with high pressures, the ASME boiler and pressure vessel code requires safety valves to prevent over pressurizing components.

(If the fluid is a gas and/or heating is significant, you need to apply the first law of thermodynamics, along with conservation of mass and momentum; not a simple problem.)

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