Centrifugal pump

A consequence of Newton's second law of mechanics is the conservation of the angular momentum (or the “moment of momentum”) which is of fundamental significance to all turbomachines. Accordingly, the change of the angular momentum is equal to the sum of the external moments. Angular momentums

$${\displaystyle \rho Qrcu}$$

at inlet and outlet, an external torque ${\displaystyle M}$ and friction moments due to shear stresses ${\displaystyle M\tau }$ act on an impeller or a diffuser, where:

• ${\displaystyle \rho }$ is the fluid density (kg/m³)
• ${\displaystyle Q}$ is the flow rate (m³/s)
• ${\displaystyle r}$ is the radius
• ${\displaystyle c}$ the absolute velocity vector
• ${\displaystyle u}$ is the peripheral circumferential velocity vector.

Since no pressure forces are created on cylindrical surfaces in the circumferential direction, it is possible to write Eq. (1.10) as:[4]

${\displaystyle \rho Q(c_{2}ur_{2}-c_{1}ur_{1})=M+M_{\tau }}$ (1.13)

Based on Eq. (1.13) Euler developed the head pressure equation created by the impeller see Fig.2.2

${\displaystyle Yth.g=H_{t}=c_{2}u.u_{2}-c_{1}u.u_{1}}$ (1)

${\displaystyle Yth=1/2(u_{2}^{2}-u_{1}^{2}+w_{1}^{2}-w_{2}^{2}+c_{2}^{2}-c_{1}^{2})}$ (2)

In Eq. (2) the sum of 4 front element number call static pressure, the sum of last 2 element number call velocity pressure look carefully on the Fig 2.2 and the detail equation.

• ${\displaystyle H_{t}}$: theoretical head pressure: ${\displaystyle g}$ = between 9.78 and 9.82 m/s² depending on latitude, conventional standard value of exactly 9.80665 m/s² barycentric gravitational acceleration
• ${\displaystyle u_{2}=r_{2}\omega }$: peripheral circumferential velocity vector
• ${\displaystyle u_{1}=r_{1}\omega }$: inlet circumferential velocity vector
• ${\displaystyle \omega =2\pi n}$: angular velocity
• ${\displaystyle w_{1}}$: inlet relative velocity vector
• ${\displaystyle w_{2}}$: outlet relative velocity vector
• ${\displaystyle c_{1}}$ inlet absolute velocity vector
• ${\displaystyle c_{2}}$: outlet absolute velocity vector

The color triangle formed by velocity vectors ${\displaystyle u,c,w}$ is called the velocity triangle. This rule was helpful to detail Eq.(1) become Eq.(2) and wide explained how the pump works.

Fig 2.3 (a) shows the velocity triangle of a forward-curved vane impeller; Fig 2.3 (b) shows the velocity triangle of a radial straight-vane impeller. It illustrates rather clearly energy added to the flow (shown in vector ${\displaystyle c}$) inversely change upon flow rate ${\displaystyle Q}$ (shown in vector ${\displaystyle c_{m}}$).

$${\displaystyle \eta ={\frac {\rho .gQH}{P_{m}}}}$$

where:

• ${\displaystyle P_{m}}$ is the mechanics input power required (in watts)
• ${\displaystyle \rho }$ is the fluid density (kg/m³)
• ${\displaystyle g}$ is the standard acceleration of gravity (9.80665 m/s²)
• ${\displaystyle H}$ is the energy head added to the flow (in metres)
• ${\displaystyle Q}$ is the flow rate (in m³/s)
• ${\displaystyle \eta }$ is the efficiency of the pump plant as a decimal

The head added by the pump (${\displaystyle H}$) is a sum of the static lift, the head loss due to friction and any losses due to valves or pipe bends are all expressed in metres of fluid. Power is more commonly expressed as kilowatts (10³ W, kW) or horsepower. The value for the pump efficiency, ${\displaystyle \eta _{\textrm {pump}}}$, may be stated for the pump itself or as a combined efficiency of the pump and motor system.

In normal conditions, common centrifugal pumps are unable to evacuate the air from an inlet line leading to a fluid level whose geodetic altitude is below that of the pump. Self-priming pumps have to be capable of evacuating air from the pump suction line without any external auxiliary devices.

Centrifugal pumps with an internal suction stage such as water-jet pumps or side-channel pumps are also classified as self-priming pumps.[10] Self-Priming centrifugal pumps were invented in 1935. One of the first companies to market a self-priming centrifugal pump was American Marsh in 1938.

Centrifugal pumps that are not designed with an internal or external self-priming stage can only start to pump the fluid after the pump has initially been primed with the fluid. Sturdier but slower, their impellers are designed to move liquid, which is far denser than air, leaving them unable to operate when air is present.[11] In addition, a suction-side swing check valve or a vent valve must be fitted to prevent any siphon action and ensure that the fluid remains in the casing when the pump has been stopped. In self-priming centrifugal pumps with a separation chamber the fluid pumped and the entrained air bubbles are pumped into the separation chamber by the impeller action.

The air escapes through the pump discharge nozzle whilst the fluid drops back down and is once more entrained by the impeller. The suction line is thus continuously evacuated. The design required for such a self-priming feature has an adverse effect on pump efficiency. Also, the dimensions of the separating chamber are relatively large. For these reasons this solution is only adopted for small pumps, e.g. garden pumps. More frequently used types of self-priming pumps are side-channel and water-ring pumps.

Another type of self-priming pump is a centrifugal pump with two casing chambers and an open impeller. This design is not only used for its self-priming capabilities but also for its degassing effects when pumping two-phase mixtures (air/gas and liquid) for a short time in process engineering or when handling polluted fluids, for example, when draining water from construction pits. This pump type operates without a foot valve and without an evacuation device on the suction side. The pump has to be primed with the fluid to be handled prior to commissioning. Two-phase mixture is pumped until the suction line has been evacuated and the fluid level has been pushed into the front suction intake chamber by atmospheric pressure. During normal pumping operation this pump works like an ordinary centrifugal pump.