Exercises: 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.

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.

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.

Explain why pneumatics takes advantage of the fact that air is compressible (accumulators, damping), whereas hydraulics is based on oil being practically incompressible (rigidity and motion precision). Give one application where air compressibility is an advantage and another where it is a disadvantage.

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?

A piston of $50\ \text{cm}^2$ supports a force of $3\,000\ \text{N}$. Calculate the fluid pressure in $\text{N/cm}^2$ and convert it to $\text{MPa}$ ($1\ \text{MPa} = 100\ \text{N/cm}^2$) and bar ($1\ \text{MPa} = 10\ \text{bar}$).

Explain what a pressure gauge and a flowmeter measure, where each is installed in a circuit, and why no hydraulic circuit should operate without at least one visible pressure gauge at the pump outlet.

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.

Compare pneumatics and hydraulics as power transmission technologies: what fluid each uses, what happens regarding the compressibility of air versus oil, what working pressures are typical, and an application where each is clearly superior.

A hydraulic press has a control piston of $200\ \text{mm}^2$ and a working piston of $20\,000\ \text{mm}^2$. a) Calculate the force multiplication factor. b) What force must be applied to stamp with $60\ \text{kN}$? c) If the small piston travels $300\ \text{mm}$ per stroke, how many strokes are needed for the large one to rise by $12\ \text{mm}$?

A pressure intensifier uses a large piston driven by air at $6\ \text{bar}$ to move a small oil piston. If the ratio of cross-sectional areas is $25:1$, calculate the theoretical maximum oil pressure and explain an application of this equipment (pressure testing, high-pressure injection).

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 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.

Car brakes are hydraulic, whereas heavy truck brakes are pneumatic. Analyse the reasons: what forces are needed in each case, what consequence the compressibility of air has on braking modulation, and what safety systems (dual circuit, ABS) offset the risks of each technology.

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.

In an industrial hydraulic press, the actuator cylinder ($D = 100\ \text{mm}$) must perform a working stroke exerting a useful pressing force of $110\ \text{kN}$ at a constant speed of $0{,}04\ \text{m/s}$. The pump drives hydraulic oil through a pipe of $12\ \text{m}$ with several elbows and a directional solenoid valve. The set pressure at the pump outlet is $160\ \text{bar}$, but the pressure gauge installed right at the cylinder inlet reads $135\ \text{bar}$.

1. a) Determine with justification whether the cylinder is capable of performing the required pressing operation with the actual pressure available at its inlet, and calculate the percentage of pumping power being lost in the pipeline.
2. b) Calculate the power dissipated entirely as heat due to hydraulic head losses during the extension of the piston.
3. c) Knowing that head losses in a pipe strongly depend on the inside diameter according to the Darcy-Weisbach law ($\Delta p \propto v^2 / d \propto 1/d^5$), predict what quantitative impact doubling the inside diameter of the supply hoses would have on the pressure drop and fluid temperature, justifying which two design criteria must be balanced to avoid oversizing the installation.

The hydraulic braking system of a car consists of a pedal with a lever ratio of $4{,}5:1$ that acts on a master piston of $20\ \text{mm}$ diameter. The generated pressure is transmitted through non-deformable brake hoses to the calipers of the four wheels. Each wheel has a floating caliper equipped with a piston of $40\ \text{mm}$ diameter that presses the pads against the disc.

1. a) If the driver exerts a force of $240\ \text{N}$ on the pedal, calculate the hydraulic pressure in the circuit (in $\text{bar}$) and the total normal clamping force exerted at each wheel.
2. b) After a prolonged mountain descent using the brakes intensively, the heat evaporates water absorbed by the brake fluid and forms a vapor bubble of $12\ \text{cm}^3$ in the circuit. Knowing that the maximum possible stroke of the master piston is $35\ \text{mm}$, calculate the volume displaced by this piston and predict whether the driver will experience total loss of braking (pedal bottoming out without braking). Quantitatively justify the difference between the compressibility of a clean liquid and that of a fluid with trapped gas.

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.

In an industrial steel hydraulic pipeline of inside diameter $d = 25\ \text{mm}$ and length $L = 40\ \text{m}$, a flow rate of hydraulic oil (density $\rho = 880\ \text{kg/m}^3$) of $45\ \text{L/min}$ circulates at an operating pressure of $60\ \text{bar}$. The acoustic wave speed in the fluid inside the pipe is $c = 1\,250\ \text{m/s}$. A solenoid valve located at the end of the line closes abruptly in a time $t_c = 0{,}03\ \text{s}$.

1. a) Determine whether the closure is classified as sudden (instantaneous) or slow, comparing $t_c$ with the critical round-trip time of the pressure wave ($T_c = 2L/c$).
2. b) Calculate the fluid velocity in the pipe and determine the maximum theoretical surge pressure caused by the water hammer (Joukowsky's formula: $\Delta p = \rho \cdot c \cdot v$). Predict whether the allowable test pressure of the pipe, set by the manufacturer at $110\ \text{bar}$, will be exceeded.
3. c) Justify two design alternatives to protect the line: calculate the minimum regulated closing time the valve should have to limit the surge pressure to less than $10\ \text{bar}$, or explain the operating principle and optimal location of a hydropneumatic accumulator to cushion this transient.

In the design of a vertical hydraulic bottle jack for lifting heavy vehicles, the system consists of: an annular oil reservoir at atmospheric pressure, a manually operated lever pump piston with a diameter of $14\ \text{mm}$, a suction check valve $V_1$, a discharge check valve $V_2$, a lifting ram with a diameter of $70\ \text{mm}$, and a manual needle valve connected in parallel between the main ram chamber and the reservoir.

1. a) Draw the functional schematic of the hydraulic circuit using standardized symbols and interpret with justification the open/closed state of $V_1$, $V_2$, and the needle valve during: 1) the suction stroke (raising the lever), 2) the discharge stroke (lowering the lever), and 3) the controlled lowering of the load.
2. b) During an inspection, it is found that pumping downward causes the load to rise, but when releasing the lever to begin a new stroke, the load drops immediately and the lever is violently forced upward. Diagnose which of the three valve components has failed and what type of fault it has suffered (stuck open, broken, or dirt on the seat).
3. c) If the jack is supporting a suspended mass of $2\,500\ \text{kg}$ ($g = 9{,}8\ \text{m/s}^2$) and the fault diagnosed in section b) occurs, calculate the instantaneous hydraulic force transmitted to the pump piston and justify why it poses a serious accident risk to the operator.

An oil-hydraulic forming station operates on repetitive cycles of $40\ \text{s}$. During the first $5\ \text{s}$, a cylinder advances rapidly requiring a flow rate of $60\ \text{L/min}$ at $140\ \text{bar}$. During the remaining $35\ \text{s}$ of the cycle, the machine performs clamping and part-changing operations where the flow consumption is zero ($0\ \text{L/min}$), maintaining the rest pressure.

1. a) Justify why the use of a hydropneumatic accumulator dramatically reduces the size and cost of the motor-pump unit compared to a design without an accumulator. Calculate the continuous flow rate that the pump must deliver in the system with an accumulator and the percentage reduction in installed hydraulic pump power.
2. b) The selected accumulator is a bladder type working between a maximum pressure of $140\ \text{bar}$ and a minimum allowable pressure after discharge of $105\ \text{bar}$. If the initial gas precharge when empty is $p_0 = 90\ \text{bar}$ and the discharge of the required $5\ \text{L}$ is rapid (an adiabatic process with $\gamma = 1{,}4$, following $p \cdot V^{1{,}4} = \text{cte}$), calculate the total nominal volume that the accumulator must have.
3. c) During maintenance work, a technician suggests refilling the bladder precharge with oxygen-enriched compressed air to reach the pressure without spending a bottle of nitrogen. Categorically refute their proposal by justifying the risk of thermodynamic explosion (diesel effect) resulting from the cycle's pressures and temperatures, and argue why dry nitrogen is the mandatory fluid.