Positive Displacement (PD) Pumps

PD pumps force fluid along using volume changes (e.g. bike pumps, the heart)
All PD pumps deliver a periodic flow
They deliver any fluid regardless of viscosity (dynamic pumps struggle with viscous fluids)
They are self priming (will be filled automatically)
They can operate under high pressures (300 atm) but low flow rates (25 m$^3$h$^{-1}$)
flow rate can be only be changed by vary speed or displacement

Dynamics Pumps

add momentum to fluid by fast moving blades or vanes
classified based on direction of flow at exit:
- centrifugal
- axial
- mixed flow
fluid increases momentum while moving through open passages and extra velocity is converted to pressure through exiting it into a diffuser section
provide high flow rates (up to 70000 m$^3$h$^{-1}$) but usually at moderate pressure rises (a few atm)
require priming

Centrifugal Pumps

fluid enters through eye of casing and gets caught in impeller blades
fluid is whirled outwards until it leaves via the expanding area section, known as the diffuser or volute

Blades

backward inclined blades - most common and efficient, intermediate pressure rise, less robust
straight blades - simplest geometry, high pressure rise, less robust
forward inclined blades - more blades but smaller, lowest pressure rise, lowest efficiency, more robust

Integral Analysis of Centrifugal Pumps

Derivation in slides (p. 23-25).

\begin{align} \frac{w_s}{g} - \left(\frac{u_2-1-q}{g}\right) = \left(\frac{p_2}{\rho g} + z_2 + \frac{v_2^2}{2g}\right) -\left(\frac{p_1}{\rho g} + z_1 + \frac{v_1^2}{2g}\right) \ H_s - H_f &= H = H{T,2} - H_{T,1} \nonumber \end{align}

where H_s is supplied head to pump, H_f friction loss head, H is head supplied to fluid, H_{T,1} is total head at inlet, and H_{T,2} is total head at outlet.

Assuming that z_1 \approx z_2, v_1 \approx v_2 (from inlet and outlet diameters are equal) then:

\begin{equation} H \approx \frac{p_2-p_1}{\rho g} \end{equation}

and the power to the fluid (water horsepower) is:

\begin{equation} P_w = \rho QgH \end{equation}

where Q is volumetric flow rate.

Power supplied to the pump (brake horsepower), P = \omega T, lets us find the overall pump efficiency:

\begin{equation} \eta = \frac{P_w}{P} = \frac{\rho QgH}{\omega T} = \eta_h \eta_m \eta_v \end{equation}

where:

\eta_h = 1 - \frac{H_f}{H_s} is hydraulic efficiency
\eta_m = 1- \frac{P_f}{P} is mechanical efficiency
\eta_v = \frac{Q}{Q+Q_L} (where Q_L is loss due to leakage flow) is the volumetric efficiency

Performance

Cavitation

Cavitation is when bubbles form in liquid by sudden pressure drop, followed by their implosion when original pressure is restored. The implosion generates a high pressure wave that can damage nearby solid surfaces.

In a centrifugal pump, the fluid pressure drops at the impeller's eye, where it has the minimum value. If pressure falls below saturation pressure, bubbles appear. Pressure grows as the fluid flows between the blades as the ducts are diverging. Pressure is maximum at the trailing edge of the blades, on their front side. This is where cavitation occurs and causes wear on the blade.

https://www.michael-smith-engineers.co.uk/resources/useful-info/pump-cavitation
https://www.youtube.com/watch?v=g1o5Z9o7b0
https://www.youtube.com/watch?v=eMDAw0TXvUo
https://www.youtube.com/watch?v=1Lbxtjfdat4

Net Positive Suction Head (NPSH)

The following conditions must be satisfied to prevent cavitation:

\begin{equation} H_i - \frac{p_v}{\rho g} > \text{NSPSH} \end{equation}

where H_i = \frac{p_i}{\rho g} + \frac{v_i^2}{2g} is total head at inlet, p_v is saturation pressure at T_i.

It is important that the inlet pressure is as high as possible. To do this, one can reduce frictional losses (e.g. shorter smoother pipes) or install the pump lower down (even below the reservoir) (slides p. 36).

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