For both thermodynamic and ‘conventional’ measurement techniques, pump parameters are summarised by the following expression (the Pump Equation):

η.ME.PW = q.ρ.g.H ..........(equation 1)

(S.I. units are used in the following)

The left-hand side of equation (1) is the electrical power (watts, or joules per second) applied to the fluid, after losses in the motor drive and pump: -

η is the pump efficiency (expressed as a fraction)

ME is the motor and drive efficiency (expressed as a fraction)

PW is the electrical power to the motor (watts)

The right-hand side of equation 1 is the energy per second imparted to the fluid, and also has the units of watts (joules per second): -

q is flow rate, in m³/s

ρ is the fluid density, in kg/ m³, and is a function of temperature and pressure

g is the acceleration due to gravity, in m/s²

H is pump total head, in m

In the thermodynamic method, the pump efficiency, η, is determined from changes in enthalpy (internal energy per unit mass), using temperature and pressure probes. The calibration of these probes can be readily checked on-site. The uncertainty in η is primarily due to the uncertainty in differential temperature measurements. This is minimised by Robertson Technology’s CoolTip™ dual sensor temperature probes.

The flow rate, q, is determined from equation (1), rearranged:

q = η.ME.PW / (ρ.g.H)

Thus flow rate can be derived without the need for a separate flow meter.

## THE THERMODYNAMIC METHOD

η = EH / EM for pumps (for turbines, η = EM / EH)

where EH is the hydraulic energy per unit mass of fluid

and EM is the mechanical energy per unit mass of fluid

EH = dp / ρ and EM = a . dp + cp . dt

dp is the differential pressure, and

dt is the differential temperature

These parameters are measured by the temperature and pressure probes.

###### MEASUREMENT OF PUMP EFFICIENCY

cp is the specific heat capacity, a is the isothermal coefficient, and ρ is the fluid density. These are known for the fluid (e.g. ISO 5198 Tables for water). For slurries, the fractions of liquids and solids must also be known. These can be calculated from the slurry and solids densities.

Accurate measurements of both pump efficiency and flow rate can be made, without the need for a conventional flow meter.

The thermodynamic technique requires measurement of only two parameters, temperature and pressure, to determine pump efficiency. Due to the advances we have made in accurate and stable temperature measurement, the uncertainty in pump efficiency measurements is typically less than 1%. Decisions on pump refurbishment or system control can be taken with confidence.

System-related energy savings can be optimised by monitoring of pump combinations.

The equipment is easy to install, with minimum disruption to operations.

Calibration checks can be carried out on-site.

###### ADVANTAGES OF THE THERMODYNAMIC METHOD

On-site constraints often make it difficult to accurately measure pump efficiency under installed conditions by the same method that pump manufacturers traditionally use for works tests. In the ‘traditional’ technique, pump efficiency is calculated from equation (1) as follows:

η = q.ρ.g.H/ ME.PW

This requires measurement of flow rate, head, and power. Of these parameters, all except flow rate are also common to the thermodynamic method. Flow rate is the most difficult to determine accurately. Many pumps do not have accurate, individual flow meters, which are high-cost items, especially for larger diameter pipes, and can be difficult or impossible to install, maintain, and carry out calibration checks on-site. Flow meter accuracy can be dependent on installed pipe lengths prior to and after the measuring device, the pump’s operating point, and other factors, such as the build-up of debris in pipes or on sensors, or cavitation and air entrainment. Often, just the total flow from the station, or from each group of pumps, is measured, so there is no information on how each pump is performing, and what pumps require attention. Pipe installations are sometimes compromised in the interest of minimising civil costs. Conventional flow meters are likely to have an installed uncertainty of 5 to 10%, and this will lead to a corresponding uncertainty in the pump efficiency measurement, a potential error so large that it is impracticable for pump refurbishment or system control decisions.