Energy in Fluid Mechanics

This guest blog post is part of a series written by Edward J. Farmer, PE, ISA Fellow and author of the new ISA book Detecting Leaks in Pipelines..

A common issue in a lot of pipeline work is ensuring the physical line data and operating data are consistent. This establishes confidence in the information about a project or situation, helps discern if a hypothetical situation can exist, or suggests that a broader view of a situation is appropriate. It also reminds the analyst of all the factors that pertain to a flow situation on a pipeline.

The Bernoulli equation looks at energy at selected locations along a pipeline. The analyst is free to choose these locations but must be sensitive to observability. Ends are always a good place to start. Often, the highest elevation point will be interesting. In some situations, the lowest points can be interesting. Points of delivery from the pipeline or injection into it may be interesting. Usually, work begins with some pipeline data and some operating data from specific sites along the line. Start analysis with those and ensure the core data the study will be based on is valid and consistent with the other known issues first.

For reasons that become apparent with some experience, Bernoulli and his follower, Euler, normally use a surrogate parameter for energy. This parameter is the “head” at the subject locations, reported in a length unit such as m for meters. Reported data is normally in typical engineering units such as velocity and pressure. Converting between these and head is fairly easy, albeit a bit tedious and obscure for newcomers. To efficiently summarize, the common transformations are:

Using the SI system:

• The head of a defined point is determined by its height relative to the project’s elevation datum essentially H = h – d where H is the head in meters, h is the elevation of the point, and d is the elevation of the datum.
• The head of a column of fluid of height y and density ρ is Hy = y * ρ / ρw meters, where ρw is the density of water at standard conditions, and y is the height of the fluid column above the point.
• The head due to pressure is Hp = P / ( ρ*g) meters where g is the gravitational constant.
• The head due to flow velocity is Hv = V² / (2*g) meters where V is the flow velocity in m/s.

Bernoulli’s (and Euler’s) development of these concepts was based on the idea of an isentropic pipeline, one in which the energy in the fluid itself, is constant. This presumes, for example, a constant temperature. Work since then introduces an internal energy term:

• Head due to internal specific energy is e/g where e is calculated in calories per kilogram from specific heat and temperature. This term is not commonly used albeit the concept is often involved via calculations of the fluid in accordance with its thermodynamic properties.

Essentially, the Bernoulli equation develops energy at the points for which the terms are calculated. The difference in energy between those points goes to the mechanical friction involved in moving the fluid. Fundamentally, the change in energy between location 1 and location 2 is dE = E₂ – E₁ m²/s². This converts to a head difference of dH = dE/g meters.

The commonly used Darcy formula for friction flow loss, in head terms, is:

• h_l = f * L * v² / (2D) meters where f is the appropriate friction factor, L is the line length, and D is the diameter of flow.

The head loss between points along the pipeline should match the computed head loss between them. When it doesn’t there is incentive to understand why.

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There are usually at least three entities involved is obtaining pipeline data for these studies. The engineering department will normally know the characteristics of what was built and its current status. Normally they will have pump or compressor curves, data about the pipe and appetences as installed, and about the fluid as used in the design calculations.

The operators will know the current flows and pressures as well as the characteristics of the fluids in use. The business department will know what came into the system and what came out along with some useful data about energy consumed moving product. Hopefully the data needed to resolve a specific question or inquiry will match across all the sources.

If there are discrepancies, there may be some sort of observability issue with one or more of the involved groups and one or more of the involved places. Fluid mechanics, due to noise and measurement limitations may not always be as precise as some engineering undertakings in which everything is easily known in real-time to decimal places.

While a general Bernoulli analysis is not always adequate for resolving pipeline issues it will quickly, understandably, and simply establish where to look for more information or data. Sometimes the point-oriented concept motivates segmenting the analysis to concentrate on specific parts of the general pipeline.

The nature of the equations makes mathematical analysis, such as comparisons and sensitivity analysis, very straightforward and understandable. More precise analysis may involve continuous monitoring, special equipment, or investigating special situations. Keep an open mind and always thing back toward conditions that would produce or exacerbate the issue motivating the original request.

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