13.1 MATHEMATICAL PROPERTIES

---

Fluids do work when we have a differential pressure on a surface. The pressure may be expressed as an absolute value. More correctly we should consider atmospheric pressure (gauge pressure).

When we deal with fluids we approximate them as incompressible.

Fluids observe some basic laws,

13.1.1 Resistance

If fluid flows freely, we say it is without resistance. In reality, every fluid flow experiences some resistance. Even a simple pipe has resistance. Of similar interest is the resistance of a value.

Resistance may also result from valves. Valves usually restrict flow by reducing an area for fluid to flow through.

A simple form of valve is a sliding plunger. The valve below is called a two way valve because it will allow fluid to flow in or out.

Four way valve allow fluid force to be applied in both directions of a cylinder motion.

13.1.2 Capacitance

Fluids are often stored in reservoirs or tanks. In a tank we have little pressure near the top, but at the bottom the mass of the fluid above creates a hydrostatic pressure. Other factors also affect the pressure, such as the shape of the tank, or whether or not the top of the tank is open.

To calculate the pressure we need to integrate over the height of the fluid.

Consider a tank as a capacitor. As fluid is added the height of the fluid rises, and the hydrostatic pressure increases. Hence we can pump fluid into a tank to store energy, and letting fluid out recovers the energy. A very common application of this principle is a municipal water tower. Water is pumped into these tanks. As consumers draw water through the system these tanks provide pressure to the system. When designing these tanks we should be careful to keep the cross section constant (e.g. a cylinder). If the cross section varies then the fluid pressure will not drop at a linear rate and you won't be able to use linear analysis techniques (i.e., Laplace and State Space).

The mathematical equations for a constant cross section tank are,

13.1.3 Power Sources

As with most systems we need power sources. In hydraulics these are pumps that will provide pressure and/or flow to the system.

One type of pump uses a piston.

A geared hydraulic pump is pictured below. Other types use vanes and pistons.

Vane based pumps can be used to create fluid flow. As the pump rotates the vanes move to keep a good seal with the outer pump wall. The displacement on the advance and return sides are unequal (aided by the sliding vanes). The relative displacement across the pump determines the fluid flow.

As with the resistance of valves, these are not linear devices. It is essential that we linearize the devices. To do this we look at the pressure flow curves. (Note: most motors and engines have this problem)