Physical principles

Define pressure and flow rate along with their formulas and International System (SI) units, and explain with an everyday example why high pressure does not necessarily imply a high flow rate.

Express these common pressures in pascals: $1\ \text{bar}$, $6\ \text{bar}$ (typical pneumatic workshop) and $210\ \text{bar}$ (typical industrial hydraulics). Explain why in hydraulics the bar or MPa is used instead of the pascal in practice.

Distinguish between absolute pressure and gauge (relative) pressure: what reference each uses and what a workshop pressure gauge reads when the circuit is depressurised. What is the approximate value of atmospheric pressure in bar?

State Pascal's principle and explain why in a hydraulic press the pressure is the same on both pistons even though the forces are very different. Write the expression $\frac{F_1}{S_1} = \frac{F_2}{S_2}$ indicating what each symbol means.

A force $F_1 = 200\ \text{N}$ is applied over a cross-sectional area $S_1 = 20\ \text{cm}^2$ on the small piston of the hydraulic press in the diagram; the large piston has a cross-sectional area $S_2 = 400\ \text{cm}^2$. Calculate the fluid pressure, the force $F_2$ available at the large piston, and the force multiplication factor.

In a hydraulic press, the small piston has a cross-sectional area $S_1 = 20\ \text{cm}^2$ and the large one a cross-sectional area $S_2 = 400\ \text{cm}^2$. If the small piston moves down $10\ \text{cm}$, applying the conservation of oil volume, calculate how far the large piston moves up and explain how this result relates to force multiplication: where does the energy come from?

A flow rate of $30\ \text{L/min}$ flows through a pipe with a cross-sectional area of $2\ \text{cm}^2$. a) Express the flow rate in $\text{m}^3/\text{s}$. b) Calculate the average fluid velocity applying $Q = S \cdot v$. c) If the pipe narrows to half its cross-sectional area, explain what happens to the velocity.

The diagram shows a constriction in a pipe through which oil flows. a) If the flow rate is $12\ \text{L/min}$ and the wide section area is $4\ \text{cm}^2$, calculate the velocity in the wide section. b) Calculate the velocity in the constriction of area $1\ \text{cm}^2$. c) Using conservation of flow rate, explain how the pressure changes in the narrow section.

A workshop hydraulic jack must lift a load of $1\,500\ \text{kg}$ using a mechanism consisting of a manually operated lever (lever arm ratio $8:1$) and a pump piston with a diameter of $15\ \text{mm}$ connected to the main lifting piston. Take $g = 9{,}8\ \text{m/s}^2$.

1. a) If the maximum force that the operator can comfortably apply at the end of the lever is $120\ \text{N}$ and the overall mechanical-hydraulic efficiency of the system is $85\,\%$, determine the minimum inside diameter that the main lifting cylinder must have.
2. b) Knowing that the useful stroke of the pump piston is $30\ \text{mm}$ per stroke, calculate how many times the operator will need to actuate the lever to lift the load to a height of $15\ \text{cm}$.
3. c) A novice operator suggests that, to halve the number of strokes, it would suffice to double the diameter of the pump piston while keeping the same lever. Refute his proposal on technical grounds, explaining what would happen to the force required from the user and referring to the principle of conservation of mechanical energy.

A hydraulic pump delivers $20\ \text{L/min}$ at $50\ \text{bar}$. a) Express the flow rate in $\text{m}^3/\text{s}$ and the pressure in $\text{Pa}$. b) Calculate the hydraulic power delivered, $P = p \cdot Q$, and express it in $\text{kW}$. c) If the efficiency of the power unit is $80\,\%$, calculate the electrical power consumed.

A commercial single-acting hydraulic cylinder must be selected and verified to drive a scissor lift table responsible for lifting a net mass of $1\,200\ \text{kg}$ ($g = 9{,}8\ \text{m/s}^2$). Due to mechanical constraints of the scissors, at the moment of maximum effort the cylinder must develop a thrust equivalent to $2{,}6$ times the weight of the load. The hydraulic system provides a nominal operating pressure of $80\ \text{bar}$. The manufacturer's catalog offers the following standard bore diameters: $50\ \text{mm}$, $63\ \text{mm}$, $80\ \text{mm}$, and $100\ \text{mm}$.

1. a) Size the cylinder by choosing the smallest commercial diameter that ensures a minimum force safety factor of $1{,}3$.
2. b) If the selected cylinder has a stroke of $400\ \text{mm}$ and the hydraulic power unit supplies a constant flow rate of $9{,}5\ \text{L/min}$, calculate the time the table takes to complete its stroke and the mechanical power developed by the cylinder during the phase of maximum effort.
3. c) If in a redesign of the machine the cylinder works in tension (pulling the load instead of pushing it) with a rod diameter of $36\ \text{mm}$, justify quantitatively whether the cylinder chosen in section a) remains valid with the same safety factor or if it is mandatory to upgrade to the next model up.

A hydraulic power unit feeds a continuous actuator with a constant flow rate of $30\ \text{L/min}$. Between the unit outlet and the actuator, control valves and a pipeline are installed, causing a total parasitic pressure drop of $18\ \text{bar}$. The system has a closed reservoir of $50\ \text{L}$ of hydraulic oil (density $\rho = 860\ \text{kg/m}^3$ and specific heat $c_e = 1{,}9\ \text{kJ/(kg}\cdot\text{K)}$).

1. a) Calculate the power dissipated as heat due to these hydraulic head losses and determine the thermal energy introduced into the oil during an uninterrupted $7\ \text{h}$ shift (expressed in $\text{kWh}$ and in $\text{MJ}$).
2. b) During the morning start-up, assuming temporary insulation of the reservoir during the first $20\ \text{min}$ in which almost no heat is dissipated to the environment, predict the temperature increase that the total mass of oil in the reservoir will undergo.
3. c) The plant manager suggests solving the heating issue by replacing the oil with one having three times the viscosity "so that it does not leak through clearances". Refute his proposal on technical grounds, explaining what real effect the increase in viscosity will have on head losses in the pipes and on the overall efficiency of the system.