3.3 Positive displacement pump theory
Positive displacement pump theory
Positive displacement pump theory explains the theoretical principles of positive displacement pumps and they are basically simple. For each working cycle or each revolution or stroke, a certain quantity of fluid is enclosed and transported from the pump intake to the pump outlet irrespective of counter-pressure. The enclosed volume of fluid depends entirely on the size of the displacement pump cavities or so called displacement, hence the name displacement pump. The maximum attainable pressure head increase is primarily dependent only upon the mechanical strength of the construction and the driving power available. The pressure head increase can be limited by means of a maximum pressure valve (relief valve, safety valve) installed in the system or integrated in the pump.
Ignoring internal leakage, the volume rate of flow delivered becomes:
Qtheor = D * n / 60 (Equ. 3.31)
Qtheor = volume rate of flow (m³/s)
D = displacement (m³ / revolution or stroke)
n = revolution or strokes per minute (1/min)
The general equations (Equ. 3.1 to 3.7) used for determining turbine pump power requirements can also be applied to displacement pumps. For displacement pumps however, it is usual to state the pressure increase p (the difference between the inlet and outlet pressures) instead of delivery head H. Assuming that the kinetic energy and the potential energy at the intake and outlet are the same:
p = ρ * g * H (Equ. 3.32)
where
p = pressure increase (Pa)
ρ = density (kg/m³)
g = acceleration due to gravity 9,806 (m/s²)
H = delivery head (m)
And for power requirement
P = (Q * p) / η (Equ. 3.33)
where
P = power (W)
Q = volume flow (m³/s)
p = pressure increase (Pa)
η = efficiency (1 / %)
Efficiency represents the overall efficiency. It is particularly interesting to note that the volumetric efficiency Ƞv for displacement pumps is determined by:
where
Ƞv = volumetric efficiency (%)
Q = actual delivered volume flow
Qtheor = volume rate of flow equivalent to the displacement
ΔQ = internal leakage (slip)
Since internal leakage (slip) is not dependent on Q then the volumetric efficiency increases with speed (increased r/min or strokes per min).
The presence of entrained and dissolved gas or air in the fluid and similarly, to a certain extent, the onset of cavitation (the formation of vapour bubbles) further reduces the actual volume flow delivered. This relationship is especially accentuated in the case of viscous fluids, which may have large concentrations of entrained or dissolved gas or have insufficient supply pressure feed to completely fill the displacement.
By the introduction of a “capacity loss factor”, f which is equal to or less than unity, it is possible to separately identify the pump’s internal leakage (slip) from such losses which depend upon fluid and the type of pump installation. The intake flow Qin to the pump is thus:
Qin = f * Q = f * Ƞv * Qtheor
where Q is the volume flow for gas free liquid.
The outlet flow measured in units of volume will be less than the intake flow because of the compression of the gas. Very high pressure increases have a similar effect on the compressibility of liquids. The volume of water reduces by approximately 2% when pressurized to 50 MPa.