The following duties are to be performed by rotodynamic pumps driven by electric synchronous motors, speed \(100 \pi / n \mathrm{rad} \cdot \mathrm{s}^{-1}(=50 / n \mathrm{rev} / \mathrm{s})\), where \(n\) is an integer: (a) \(14 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\) of water against \(1.5 \mathrm{~m}\) head; \((b)\) oil (relative density \(0.80\) ) at \(11.3 \mathrm{~L} \cdot \mathrm{s}^{-1}\) against \(70 \mathrm{kPa}\) pressure; \((c)\) water at \(5.25 \mathrm{~L} \cdot \mathrm{s}^{-1}\) against \(5.5 \mathrm{MPa}\). Designs of pumps are available with specific speeds of \(0.20,0.60,1.20,2.83,4.0\) rad. Which design and speed should be used for each duty?

Rotodynamic pumps are a type of dynamic pump where energy is continuously added to the fluid by a rotating impeller. This category includes centrifugal pumps, axial flow pumps, and mixed flow pumps. They are generally used to handle large quantities of fluid and can be applied in various industries. The main principle of their operation involves converting mechanical energy into hydraulic energy, which increases the fluid's pressure and flow rate. These pumps are highly efficient and suitable for scenarios where continual flow of fluid is necessary.

Electric synchronous motors are motors whose rotors turn at the same rate as the oscillating magnetic field produced by the stator. They are known for their constant speed capabilities, which makes them ideal for applications that require precision. Unlike asynchronous motors, synchronous motors don't slip — their rotor speed exactly matches the speed of the rotating magnetic field. This precision is crucial in pump systems where consistent and reliable operation is necessary. Synchronous motors are more efficient at converting electrical energy to mechanical energy, and they are often used in combination with rotodynamic pumps to ensure smooth and consistent fluid flow.

Flow rate is the volume of fluid passing a point per unit time, commonly measured in cubic meters per second \(\text{m}^3/\text{s}\) or liters per second \(\text{L}/\text{s}\). To convert from \(\text{L}/\text{s}\) to \(\text{m}^3/\text{s}\), use the relation:
\[ 1 \text{ L/s} = 1 \times 10^{-3} \text{ m}^3/\text{s} \] For example, a flow rate of 11.3 \( \text{L/s}\) converts to 0.0113 \( \text{m}^3/\text{s}\). This conversion is essential when performing calculations with the specific speed formula, which relies on consistent units.

Pressure is often measured in pascals (Pa), while head is measured in meters of fluid. The conversion between pressure and head is important in pump selection and design. The formula used is: \[ H = \frac{P}{\rho g} \] where:
\(H\) is the head in meters
\(P\) is the pressure in pascals
\( \rho \) is the fluid density in \( \text{kg}/\text{m}^3\)
\( g \) is the acceleration due to gravity (9.81 \( \text{m}/\text{s}^2\).
For instance, to convert 70 kPa pressure for oil with a relative density of 0.80, the head calculation simplifies to: \[ H = \frac{70000}{0.80 \times 9.81} = 7.13 \text{ m} \] Understanding this conversion helps in determining the appropriate pump design and configuration.

Selecting the right pump involves matching the specific speed \(N_s\) to the available pump designs. The specific speed, calculated for different duties using the formula: \[ N_s = N \frac{\big(\textbf{\text{Q}} \big)^{1/2}}{H^{3/4}} \] where \( N \) is the rotational speed, \( Q \) is the flow rate, and \( H \) is the head, shows how a pump will perform under certain conditions. A closer match to the specific speed ensures optimum performance and efficiency:
\* For duty (a): using 1.20 rad for the design.
\* For duty (b): using 0.60 rad for the design.
\* For duty (c): using 0.20 rad for the design.
Proper selection ensures the pump operates efficiently, consumes less energy, and meets the required hydraulic demands.