User:AreyouMach-ingme/sandbox

 To Do 5/5/22 

Figure out how EGL reacts to heat transfers? -probably best to leave to someone else

Maybe find another source to corroborate the White textbook (Bakhmeteff was good, maybe add others?)

Find one or two nice open-source picture of an EGL to add to the article

- One that shows general behavior

- One that shows how it reacts to different elements of real flows?

= Energy grade line = In fluid dynamics, the energy grade line (EGL) is a method used to visualize Bernoulli's principle in incompressible fluid flows. The energy grade line displays the total energy of a flow as a height or head above or below an arbitrary datum that varies along a flow's length. The EGL can be expressed in many forms, depending on the conditions of the fluid system and the simplifying assumptions made about the system.

Incompressible flow equation
The energy grade line is a rearrangement of Bernoulli's principle into units of length. This is done by dividing both sides by the fluid density and the acceleration due to gravity, resulting in the two equations below. Each equation is valid only for certain types of flows.

For pipe flows
$$h_{pf} = z + \frac{p}{\rho g} + \frac{V^2}{2g}=z + \frac{p}{\gamma} + \frac{V^2}{2g}$$

where:

$$h_{pf}$$ is the total or energy head of the pipe flow, (not to be confused with specific enthalpy, which also uses a lowercase $$h$$)

$$z$$ is the height above an arbitrarily chosen datum,

$$p$$ is the static pressure of the flow, expressed as either a gauge or an absolute pressure

$$\rho$$ is the density of the fluid,

$$g$$ is the acceleration due to gravity,

$$\gamma$$ is the specific weight, and

$$V$$ is the flow velocity.

The first equation can be grouped into three terms which aid in understanding how the energy grade line changes with flow properties. The first term, $$z$$, represents the elevation head and is the height above an arbitrary datum. The second term, $$p/\rho g$$, represents the pressure head and corresponds to the static pressure of the flow. The third term, $$V^2/(2g)$$, represents the velocity head and corresponds to the kinetic energy of the flow.

For open channel flows
For flows in open channels, the fluid surface pressure is always atmospheric, so the surface gauge pressure is zero. The pressure head then equals zero, and the energy grade line equation becomes:

$$h_{oc} = y + \frac{V^2}{2g}$$

where:

$$h_{oc}$$ is the total or energy head of an open channel flow

$$y$$ is the water depth,

$$g$$ is the acceleration due to gravity, and

$$V$$ is the flow velocity.

Idealized versus real flows
While useful, Bernoulli's principle for incompressible flows is an idealization. The assumptions made to find the equation - constant density in a steady, inviscid flow - are not applicable to many real-life flows.

For idealized flows
In idealized flows, all assumptions made for the simplified form of Bernoulli's principle apply to the equation for total head. In this case, it is assumed that the flow is incompressible (making density constant), steady (making the volume of fluid in the defined control volume constant), and inviscid (making frictional losses zero). Finally, it is assumed that no work is being done on the fluid, no heat is being transferred across the control surfaces, and the flow is irrotational. With these assumptions, we know that the total head remains unchanged across the flow. This leads to the behaviors seen in idealized incompressible flows, where a change in (for example) the pressure head can affect either the elevation, the velocity head, or both.

For various real flows
In real flows, many of the idealizing assumptions made above must be broken to accurately reflect the system to be analyzed. A few of these cases will be explored below.

Viscous flows
Most flows have viscous forces acting on them. An exception is superfluids, which are inviscid. In low-speed, fully-developed pipe flows, friction losses stem from the no-slip condition which approximates fluid velocity at solid-fluid interfaces. Since these frictional forces are applied continuously, the energy grade line will slowly drop over the length of a flow. For a circular pipe the friction losses can be calculated using:

$$h_f=f\frac{LV^2}{2dg}$$

where:

$$h_f$$ is the head loss from pipe wall friction,

$$f$$ is the Darcy friction factor, which is dependent on a variety of other variables,

$$L$$ is the length of pipe to be evaluated,

$$V$$ is the flow velocity,

$$d$$ is the diameter of the pipe, and

$$g$$ is the acceleration due to gravity.

If there is a valve or obstruction in the pipe, the energy grade line drops sharply across the throttling device.

Flows with non-zero work
Many fluid-based systems are used for the generation of work. For example, the hydroelectric dam converts the potential energy of a water reservoir into electrical energy through turbines and electrical generators. Work can be converted to head form by the equation:

$h_s = \frac{\overset{\centerdot}{W_s}}$

where:

$$h_s$$ is the head developed from work by a turbine or pump (also known as the shaft work),

$$\overset{\centerdot}{W}$$ is the rate of work (or power) developed by the element,

$$\overset{\centerdot}{m}$$ is the mass flow rate of the fluid, and

$$g$$ is the acceleration due to gravity.

When a flow encounters an element that develops or extracts work, the energy grade line raises or drops, respectively.

Jeffcott rotor
The Jeffcott rotor (named after Henry Homan Jeffcott), also known as the de Laval rotor in Europe, is a simplified lumped parameter model used to solve these equations. A Jeffcott rotor consists of a flexible, massless, uniform shaft mounted on two rigid bearings with a massive disk located halfway between the supports. The simplest form of the rotor constrains the disk to a plane orthogonal to the axis of rotation. This limits the rotor's response to lateral vibration only. If the disk is perfectly balanced (i.e., its geometric center and center of mass are coincident), then the rotor is analogous to a single-degree-of-freedom undamped oscillator under free vibration. If there is some radial distance between the geometric center and center of mass, then the rotor is unbalanced, which produced a force proportional to the disk's mass, m, the distance between the two centers (eccentricity, ε) and the disk's spin speed, Ω. This produces the following second-order linear ordinary differential equation that describes the radial deflection of the disk from the rotor centerline.

$$m \mathbb{\ddot{r}} + k \mathbb{r} = m \varepsilon \Omega^2 \sin(\Omega t)$$

If we were to graph the radial response, we would see a sine wave with angular frequency $$\omega = \Omega/2\pi$$. The oscillation of the disk about the center of rotation is called 'whirl', and in this case, is highly dependent upon spin speed. Not only does the spin speed influence the amplitude of the forcing function, it can also produce dynamic amplification near the system's natural frequency.

While the one-dimensional Jeffcott rotor is a useful tool for introducing rotordynamic concepts, it is important to note that it is a mathematical idealization that only loosely approximates the behavior of real-world rotors.

Whirl
Whirl is another term for the radial response of a rotating system, and is a key measure of a system's behavior. For a Jeffcott rotor, whirl can be visualized by imagining the rotor's centroid orbiting the rigid supports, as in the diagram at right. If the diagrammed rotor is whirling due to unbalance, then we know that it is undergoing forward synchronous whirl. Forward, since the rotor whirls in the same direction as it spins, and synchronous, since the whirl and spin speeds are identical. Expressed as a simple equation, $$\omega = \Omega$$.

Extending to two dimensions and beyond
While a one-dimensional model is useful to explain the basics, to improve the model fidelity, it is useful to extend our analysis to more dimensions. The easiest first step would be to extend to two dimensions ($$x,y$$) allowing the rotor to truly whirl rather than displace along a radial axis. In this case, we can express the behavior of the disk centroid as either a system of real differential equations, or a single complex equation like so:

$$z = x + jy ; m\ddot{z} + kz = m\varepsilon\Omega^2e^{j\Omega t}$$

I'd like to extend the analysis to 2D and introduce circular vs. elliptical whirl. I would also like to correct the section on critical speeds and the Campbell diagram. Perhaps it could be folded into the 2D section. Before I do any major structural edits of the article, I should really lay out my plan on the talk page and wait a while to get some public comments. In the meantime you can edit your version of the article in here.