Hydrostatics

Laws of hydrostatics and its application

Pressure by external forces

The pressure within a closed system increases by the impact of an external force. The pressure spreads within this system in all directions equally and has therefore at all points of the system the same value. Example of this would be the hydraulic cylinder

The opposite to the closed system forms the open system, such as a pipeline. Here, the following proportionality applies:
pressure p ~ height h

The force that a fluid exerts on a surface is calculated using Pascal's law:

Law of Pascal:   F = p * A   ⇒  p = F / A          
Units:     1 bar = 10⁵ Pa (Pascal) = 100 kPa;  1 Pa = 1 N/m²

Example: A cylinder with a diameter of d₁ = 15 cm has to hold a mass of 1500 kg on height. How much rises the pressure within the system? Solution: p = 8,327 bar

Hydraulics - Calculation piston force

Work machines, which can be described by the law of Pascal:

Pressure intensifier:

Hydraulics - pressure intensifier symbol

Principle: Two differently sized pistons are mechanically connected by a piston rod. If the piston surface A₁ is pressurized, so the force F₁ on the piston rod acts directly on the small piston area A₂:

Hydraulics pressure intensifier - Function

.          F₁      =    F₂                                
p_e1 * A₁ * η   =  p_e2 * A₂

p_e  operating pressure in bar
A  piston area in cm²
F  piston force in N

Example: By using a pressure intensifier the pressure from P_E1 = 30 bar has to be increased to P_E2 = 250bar. How big must be the diameter d₂ when d₁ = 300 mm? The efficiency is 0.9.   

(Solution: d2 = 98,6 mm)

Hydraulic press

Principle: A force F₁ acts on a pressure piston with a small piston area A₁ - a pressure within the closed system is generated. Since the pressure within a closed system is equal, this pressure acts also on the second piston with a large piston area A_2.

Animation: Hydraulic press

Relation between the first and second piston:

first step:       The pressure in both piston is the same  =>  with the law of Pascal p = F/A you get:

 F₁  =  F₂      respectively:      F₁  =  A₁
.A₁         A₂                               F₂      A₂

Oil quantity: The by the fist piston displaced volume V₁ supplies the volume V₂. Similarly, the work done on both pistons W = F * s is equal. The following applies:

the displaced volume V:

We start with;       V₁ = V₂                                               

.           =>          A₁ * s₁ = A₂ * s₂

the work carried out W:

We start with  W₁ = W₂

.     ⇒          F₁ * s₁ = F₂ * s₂ 

Example:

With the help of a hydraulic press, a load of 80 kN is to be lifted. A force of F₁ = 200 N is applied to the first piston with the diameter of 30 mm. Calculate:

1. The required diameter of the load piston in mm
2. The piston stroke s₂, when the fist piston has a stroke of s₁= 400 mm.