R

Radial force generally means a force exerted in a radial direction towards the centre or away from the centre. When a body or fluid element with mass moves in a circular path, an equilibrium exists in the radial direction between the outwardly directed centrifugal force (inertia force) und the inwardly directed centripetal force (acceleration force).

In centrifugal pumps, radial force (also radial thrust) is the term used to designate the force acting on the pump rotor. The radial force at the rotor of a centrifugal pump determines the bending load and resulting bending deformation of the shaft and the load on the radial bearings.

The resulting radial force at the rotor of a centrifugal pump comprises the hydraulic radial force generated by hydrodynamic processes, and mechanically influenced components from the effect of an unbalance and (in the case of a non-vertical pump shaft) the weight of the rotor parts themselves.

The hydraulic radial force at the impeller of a centrifugal pump is (in the case of radial and mixed flow impellers) assumed to act on the centre of the impeller width at the outlet. In relation to the fixed reference system of the casing, it is possible to distinguish between non-rotating and rotating (e.g. rotating at the pump's rotational speed, like the unbalance force) and, in terms of time dependence, between steady and non-steady (mainly periodic) force components. The hydraulic radial force at an impeller contains both steady and non-steady components.

A radial impeller is an impeller at which the flow (see Flow rate) leaves the impeller in radial direction, perpendicular to the pump shaft.

Radial thrust in centrifugal pump engineering is a hydraulic radial force in the plane of the impeller, generated by the interaction between the impeller and the pump casing or the diffuser.

Steady radial force

The steady hydraulic radial force (R) in volute casing pumps is generated by the interaction between the impeller and the casing. Its vector changes its magnitude and direction with the Q/Q_opt ratio (q), i.e. the quotient between the flow rate (Q) and the flow rate (Q) at best efficiency point (Q_opt).
The magnitude increases with the density (ρ), the projected impeller outlet area (B · D) and the head (H) where q is constant, and the direction (φ) remains unchanged. See Fig. 1 Radial thrust

Radial thrust is defined as:

R = K · ρ · g · H · D · B
R  Radial force
K  Radial force coefficient See Fig. 2 Radial thrust
ρ  Density of the fluid handled
g  Acceleration due to gravity
H  Head
D  Outside diameter of impeller
B  Impeller outlet width

If the radial force (R) is plotted against the Q/Q_opt ratio (q), a minimum results for volute casing pumps at the design point (q = 1) whereas the radial force increases rapidly in the low-flow range (q < 1) and the overlaod range (q > 1) (see Operating behaviour). The radial force (R) magnitude depends heavily on the specific speed (n_s). 

However, if the impeller is positioned eccentrically in the base circle of the volute, notable de-centring forces also occur at the design point while the radial forces may be smaller at other duty points. Fig. 1 Radial thrust shows an example of a volute casing pump with a specific speed of n_s = 26 rpm and its force vector curves, where radial force vectors for six different Q/Q_opt ratios (q) radiate from the centre (incl. the connecting lines between end points). It also shows the vectors for one centred and four eccentric impeller positions in the pump casing (the direction of point A points to the volute's tongue).

Vectors R_{A, B, C, D} for the best efficiency points (at q = 1) are included on all five force vector curves as an example. At BEP the centred rotor position (curve Z) results in the smallest radial forces. The force minimum of the remaining force vector curves for eccentric impeller positions is larger and also lies outside of the best efficiency point.

In theory, radial forces do not occur with pumps fitted with diffusers because of the symmetrical flow around the impeller's circumference. However, in reality radial forces do occur as a result of inaccuracies in the manufacturing process. The max. value of this radial force can be estimated by applying a radial force coefficient of (K = 0.09). This value applies irrespective of the pump's Q/Q_opt ratio (q). The radial force's direction on pumps fitted with diffusers is arbitrary. See Fig. 2 Radial thrust

The forces acting on impellers fitted in annular casings are lowest in the low-flow range and continually rise towards the overload range. Double volute casings are frequently used (see Pump casing) in order to reduce the forces. See Fig. 6 Volute casing pump

Unsteady radial force

Unsteady and in some cases rotating radial forces may be superimposed onto the steady radial forces. They may have different causes and characteristics. The best-known are the unsteady hydraulic radial forces with a frequency = number of impeller vanes x rotational speed. These radial forces appear to a greater or lesser extent in all types of pumps. 

In particular in diffuser-type pumps (see Diffuser) rotating hydraulic radial forces develop under low flow conditions and when the rotor is in centred position (rotating frequency approx. 1/10th of the pump speed).

If a centrifugal pump'spump casing features a vertical joint relative to the shaft, this is referred to as a radially split design. The opposite is the axially split design.

A rainwater harvesting system collects and uses rainwater to meet some or all of the water requirements for business or general domestic use.

If correctly dimensioned, it enables cost savings to be made by reducing the amount of drinking water used. Furthermore, the rainwater is not drained away into the sewer system; and therefore incurs no sewer charges. Rainwater can be used for many purposes: flushing toilets, in washing machines or for watering gardens. See Fig.1 Rainwater harvesting system

The use of rainwater harvesting systems also reduces the risk of flooding in many sealed areas with sewerage systems, since an extreme increase in water level as a result of sudden heavy rain can be delayed or completely prevented.

Rate of drainage is a term used in land drainage, i.e. dewatering through an underground pipe system. The rate of drainage gives the flow rate per unit area of land to be drained; e.g. for polders (dyked land), the rate of drainage amounts to approximately 0.7 to 0.8 l / s and hectare (10,000 m²).
Maximum values are approximately 2 l/s ha.

Many electric consumers require reactive power to build and dissipate their magnetic fields. It fluctuates periodically between generator and load. The intensity of this energy over time is quantified by the reactive power.

Reactor internal pumps are reactor pumps which are arranged in the reactor pressure vessel, thus enabling coolant circulation without any external pipelines, see Fig. 5 Reactor pump 

The reactor pump which acts as a reactor coolant pump is a centrifugal pump, which circulates the coolant necessary to carry away the decay heat. See Fig. 1 Reactor pump

Depending on the type of reactor plant, it is necessary to distinguish between pressurised, boiling and heavy water reactor pumps, and liquid-metal reactor pumps. Different designs of reactor pumps exist for the different reactor types. 

Pressurised water reactors and heavy water reactors

The high power input of pressurised water reactors and heavy water reactors necessitates the use of reactor pumps with shaft seal and integral bearing arrangement (axial, radial and rolling element bearing) and integrated oil supply, see Fig. 1 Reactor pump, or of close-coupled reactor pumps with drives in the form of conventional electric motors. See Fig. 2 Reactor pump

The shaft seal consists either of several mechanical seals arranged in series, or of a combination of hydrostatic and mechanical seals. 

Depending on the support arrangement and on the type of suspension forces and moments have to be absorbed by the pump casing. This results in a number of different casing designs and wall thicknesses. The reactor pump casing is designed in the form of a spherical or pot-shaped casing, for example. The design pressure is 175 bar, and the corresponding temperature 350 °C. See Fig. 3 Reactor pump

Boiling water reactor 

Two different reactor pump concepts are used for boiling water reactors. In the first concept, for reactor pressure vessels with built-in jet pumps the reactor pumps are welded into the external piping and are designed as driving water pumps, their casing mostly being a double volute (see Volute casing pumps). See Fig. 4 Reactor pump

Two such driving water pumps incorporated in piping loops only handle approximately one third of the entire coolant flow. Instead, they drive the jet pumps which circulate the balance of the coolant flow within the pressure vessel. The driving water pumps are equipped with shaft seals and a conventional electric motor drive.

The second concept involves several reactor pumps accommodated in the reactor pressure vessel as reactor internal pumps. These enable the coolant to circulate without the need for external piping. Whereas the driving water pumps are equipped with shaft seals and conventional electric motor drives, the insert pumps are also built with a wet rotor motor. See Fig. 5 Reactor pump

These electric motors are speed-controlled, enabling the pump flow rate to be adjusted and the reactor power to be controlled. The system pressure (design pressure) amounts to 90 bar, for example, and the design temperature to 300 °C.

Liquid metal cooled reactors

In the case of liquid metal cooled reactors, in which sodium is used as coolant, reactor pumps with a free fluid surface are used in the casing tube between the shaft seal and the pump impeller. 

The free space is filled with an inert gas to prevent any sodium reactions, so that the shaft seal is required to protect against this protective gas and not against the liquid metal. The pump shaft is guided by a liquid-metal lubricated hydrostatic bearing (see plain bearing) situated immediately alongside the impeller. 

The design pressure (system pressure) is 10 bar, and the corresponding temperature 580 °C. See Fig. 6 Reactor pump

Apart from reactor pumps, centrifugal pumps are also required in the auxiliary circuits and safety circuits of the various types of reactor. These include, for example, the volume control system, the decay heat removal system with high-pressure and low-pressure safety feed, the fuel storage pool, the secondary reactor loop and the water treatment system. See Figs. 7 and 8 Reactor pumps

The pumps in these types of circuits must meet specific requirements that usually do not need to be accounted for in centrifugal pump design.

The requirements for centrifugal pumps in auxiliary and safety circuits: 

• Extremely high nozzle forces and moments must be transferred to the foundation (see Pump nozzle load).
• The pressure boundary must permit 100 % volumetric inspection and lend itself to detailed stress analysis.
• Very steep QH characteristic curves (flow rate, head) in order to implement extremely large operating ranges
• Low NPSHR values
• Due to the radioactivity in the fluid handled, any outboard leakage must be prevented (even in the case of temperature shocks).
• To minimise possible radiation exposure, the reactor pump must be easy to service in the event of maintenance inspection.
• Continued operation even in cases where the building is affected by earthquake or a plane crash, to avoid consequences such as flooding of the pump rooms, shutdown of the cooling water supply for mechanical seals and bearings, and a room temperature of over 65 °C with a relative humidity of 100 %

This form of impeller trimming involves the reduction of the impeller vane tips on centrifugal pumps equipped with diffusers.

The impeller shrouds remain untrimmed for reasons of the hydraulic interaction between the impeller and diffuser. To establish the impeller vane diameter reduction necessary, the requirements mentioned under impeller trimming should be referred to. See Fig. 1 Impeller trimming

A reed relay is a relay used to switch a circuit and integrates a reed contact. The reed contacts (reeds) are hermetically sealed in a glass tube which is either evacuated, creating a vacuum, or filled with a protective gas. They also double as the contact spring and magnet armature.

The contact is actuated by an external magnetic field produced by a nearby permanent magnet (reed contact) or generated electrically in an accompanying magnet coil (reed relay). The magnetic field causes the reeds to attract each other, thereby closing the circuit. As soon as the magnetic field is gone or diminishes beyond a certain point, the contact opens again due to the spring effect.

Refinery pumps transport petroleum and its derived products in refineries, petrochemical plants and the chemical industry. They are used in temperature ranges from -120 °C to +450 °C at pressures of about 65 bar.

As the fluids handled are often highly volatile and flammable, the pump components in contact with the fluid handled are always made of ductile materials such as unalloyed steel, chrome steel and, less frequently, also nodular cast iron.

The required NPSHR value is particularly important and governs the selection of the drive speed and type of pump.

Refinery pumps are most commonly single-stage horizontal volute casing pumps in back pull-out design. See Fig. 1 Refinery pump

Depending on the operating conditions, particularly with regard to the suction characteristics refinery pumps can also be designed as horizontal or vertical can-typemultistage pumps or as horizontal double-suction pumps in between-bearings design. See Figs. 2, 3 Refinery pump

Refinery pumps have to comply with specific design codes, such as the well-known codes of the American Petroleum Institute (API 610) and the International Standards Organisation (ISO 13709). The codes which describe refinery pumps in greater detail refer to them as "heavy duty" pumps as they are generally suitable for heavy-duty operation.

They lay down certain design features applying to different operating temperatures, e.g. the arrangement of pump feet and nozzles, the mode of splitting the casing, the sealing elements and the temperature limits for cooling the bearings.

The shaft seals of refinery pumps are usually mechanical seals of various types and arrangements. The mechanical seals and their operating modes are described in API 682 or ISO 187498.

Great emphasis is placed on a sturdy and heavy construction as the piping is almost always hot and exerts considerable forces and moments on the refinery pump and its baseplate. These forces and moments must be absorbed safely without warping the pump or its baseplate (see Pump nozzle load).

The concept of relative velocity (w) is used in turbomachinery design and defines the velocity of a fluid particle in relation to the rotating impeller. Together with the absolute velocity (v) and the circumferential speed (u) it forms the velocity triangle.

Relays are electrical devices/components that are operated via electric current. They are usually electromagnetic, can be switched remotely, and typically have two positions. A control circuit is used for activation and additional circuits can be switched as required.

Relay applications

• Simultaneous control of several circuits by a single control circuit
• Control of a high-power circuit via a low-power control circuit
• Galvanic isolation of the controlling circuit from the circuit to be switched

Normally-open (NO) contacts connect the circuit when the relay is activated; normally-closed (NC) contacts disconnect the circuit when the relay is activated.

The remote control switch is an electrical switchgear operated by mechanical force (also see Relay, Contactor). The force or moment is transferred by a motor, for example, to the operating shaft of the switch.

A remote control system remotely performs open and closed loop control, protection or security-related tasks via a telecommunication network (also see Communications system).

Remote data transmission, or RDT, describes data transmission as it takes place between computers via a medium (telephone network, radio waves, or optics) and a specific protocol. This protocol dictates how data, which is prepared using special hardware, is exchanged.

Renewable energies are regenerative or alternative energies which make use of energy from natural and renewable resources. These include solar power (e.g. photovoltaic pump systems), wind power and bioenergy (e.g. biogas plants) as well as geothermal energy and hydropower (e.g. river turbine).

According to DIN 31051 this term denotes measures that are taken to restore a system's technical resources to their desired condition. These are described in DIN EN 13306:2018-02.

A residual current device is a safety device that disconnects a power circuit whenever it detects an excessive level of earth leakage current (residual current) in the earth conductor (PE). Residual current can be attributed to dangerous contact voltage resulting from an insulation fault. All-pole disconnection takes place within 0.2 seconds.

Residual current devices are available for different rated residual currents. Variants with tripping currents of 30 mA are also used for personal protection. Greater tripping currents primarily provide protection against fires caused by earth leakage current.

Devices with rectifier circuits such as frequency inverters, which in the event of a malfunction can produce DC leakage current, must not be used downstream of standard residual current devices. In this scenario, AC/DC-sensitive residual current devices with higher tripping current are required to prevent unwanted tripping when the drive system is activated.

A resistive sensor measures changes in electrical resistance caused by the application of force, e.g. by mechanical stretching, for example, as for strain gauges (also see Sensor).

The restart inhibit is a facility that prevents a machine from restarting automatically.

The return guide vanes of a multistage radial pump (e.g. boiler feed pump) form a guiding device downstream of the diffuser. The flow is guided radially inwards so that the approach flow of the fluid handled to the next stage is as vortex-free as possible (see Inlet conditions).

Reverse flow occurs if pumping is interrupted, for example, due to a power failure and if the check valve has not yet closed or no such valve has been installed. This may be an intentional measure designed to influence the impact of a pressure surge in certain instances. The pump set must then be able to withstand the "runaway speed" (approx. 140 % the rated speed).

Reverse osmosis (RO) is a physical process used to separate substances. Its purpose is to use a semipermeable membrane to separate dissolved substances and to extract the solvent in a purer form (see Seawater desalination system).

The reverse running speed is the speed that occurs in a centrifugal pump when the fluid flows in reverse direction through the pump at a certain head (i.e. difference in total head between discharge nozzle and suction nozzle).

This situation can arise in systems whose system characteristic curves have high static heads (H_sys,0), but it can also be observed in centrifugal pumps operating in parallel. Following a drive failure and when the discharge line is open, the fluid will reverse its direction through the pump, and the pump rotor will rotate at reverse running speed following this change of flow direction (see Turbine mode). The reverse running speed is generally markedly higher than the normal operating speed and is dependent on the system conditions (in particular the current pressure head) and on the pump's specific speed (n_s). The max. reverse running speed for radial pumps (n_s ≈ 40 rpm) is approximately 25 % higher than the operating speed of the pump, and for axial pumps (ns ≥ 100 rpm) it is up to 100 % higher.

These operating conditions can also occur if a slow-closing shut-off element – rather than a check valve (see Valve) – is used for protection against surge pressures. This allows the back-flowing fluid to run through the centrifugal pump.

If the surge pressure is caused by a power failure of the drive and a backstop has not been installed, the pump shaft will also rotate in reverse. The risk involved for plain bearings and mechanical seals which can only be operated in one direction of rotation must also be considered.

If the back-flowing fluid is near its boiling point (see Vapour pressure) it can evaporate in the pump or in the discharge-side throttle device.

The ratio of reverse running speed when operating with steam and liquids can rise to dangerously high values, as a function of the square root of the ratio of the density of liquid/vapour.

If the motor is switched on when the centrifugal pump is running in the reserve direction, the run-up time of the pump set (see Start-up process) is considerably extended. With asynchronous motors it is advisable to observe the additional temperature rise of the motor under such conditions.

Excessively high reverse running speeds that would lead to damage to the pump set can only be prevented by taking the appropriate measures.

Measures to prevent high reverse running speeds

• Fitting a mechanical backstop on the pump shaft
• Installation of a reliable, automatically closing check valve (e.g. swing check valve) in the piping

The Reynolds number is a dimensionless characteristic coefficient used in fluid mechanics. It describes the ratio of the inertia force to the friction force (see Affinity laws).

A ring-section pump is a pump with several stage casings of the same type in tandem arrangement (see Multistage pump, Boiler feed pump and Pump casing).

A river turbine is based on the innovative concept of directly converting the kinetic energy of the flow in rivers into mechanical work.

Conventional hydro power stations generally make use of the geodetic head difference (partly by means of artificial constructions such as weir systems or dams). The height difference between the upper and lower water level is converted into usable mechanical work in the pump house (in the turbine impeller).

Unlike conventional hydro power stations, river turbines do not require any structure to be built across the water body. Following the principle of wind energy installations, the kinetic energy of the axial flow through the turbine impeller directly drives the impeller. The concept of a river turbine has some substantial advantages over conventional hydro power stations.

Advantages of river turbines

• Simple design
• No hindrance to commercial shipping traffic
• Low impact on aquatic ecology (fish migration)
• Low impact on water body morphology (sediment transport, sedimentation)

As illustrated in the schematic drawing of a river turbine, the coarse screen fitted in the direction of flow prevents the turbine impeller (not shown) from any damage by flotsam. The screen is connected to the inlet nozzle. The generator is fastened inside the turbine housing by means of braces. The outlet nozzle in the outgoing flow increases the volume flow rate inside the river turbine and thus the speed at the rotor level. See Fig. 1 River turbine

This clearly improves the total efficiency of the installation as the hydraulic power (P_h) of the turbine equals the speed to the power of three. P_h is defined as:

The rolling element bearing is used to support shafts and consists of the bearing races or discs and the rolling elements which can be spherical, cylindrical, tapered or barrel-shaped. See Fig. 1 Rolling element bearing

In addition, they are frequently equipped with a cage which prevents contact between adjoining rolling elements. See Fig. 2 Rolling element bearing

Depending on the direction of force, a distinction is made between transverse (radial) bearings, axial (thrust) bearings and designs such as radial deep groove ball bearings which can handle both radial and axial forces. See Fig. 3 Rolling element bearing

Fig. 4 Rolling element bearing shows a typical shaft bearing arrangement for a low-pressure pump with cylindrical roller and angular contact ball bearings.

In radial bearings, the rolling elements run between races or rings (ring bearing), and in axial or thrust bearings they run between discs (disc bearing).

Advantages of rolling element bearings over plain bearings

• Friction coefficient is approx. 25 to 50 % lower
• Thanks to narrower clearance gap, they run more precisely
• Require less space
• Straightforward maintenance
• Straightforward lubricant requirements
• Replaceability assured by extensive standardisation

Disadvantages of rolling element bearings over plain bearings

• Sensitive to shock loads
• More noise during operation
• Generally more expensive than comparable plain bearings
• Wear during standstill (pitting)

The load capacity and service life of rolling element bearings are standardised (DIN 622). A distinction is made between the dynamic and static load capacity. The calculation is preferably carried out in accordance with DIN 622 or in accordance with the rolling element bearing’s manufacturer.

Dynamic load capacity

• (Service life) It defines the number of revolutions or operating hours which the bearing will sustain without symptoms of material fatigue on all bearing components.

Static load capacity

• The term "static load capacity" refers to the static load force that causes a permanent deformation on the rolling element at the point of contact without impairing the bearing's function.

As the rolling velocity increases, so do the centrifugal forces acting on the rolling elements. The bearing's operating temperature increases as a result of higher friction losses; this means that each standard bearing has an upper rotational speed limit. This limit speed can easily be calculated as a function of the bearing type, size, type of lubrication and load; the equations to be used are supplied by bearing manufacturers in their catalogues.

A complete separation of the contact faces by a load-bearing lubricant film does not take place due to the rolling contact motion of the rolling elements. Conventional lubrication with greases or oils of an appropriate consistency is adequate.

The bearing manufacturer usually provides lubrication recommendations which detail the influence of the bearing's operating temperature and the ageing stability of the recommended greases and oils under the prevailing operating conditions, and usually give directions regarding topping up the lubricant fill.

Grease types for rolling element bearing lubrication

• Calcium soap greases are water-repellent and need to be topped up at frequent intervals. The operating temperatures range from -20 °C to 50 °C.
• Sodium soap greases have good lubricating properties, but absorb water; they are washed away if water penetrates the bearing. The operating temperatures range from -30 °C to 110 °C.
• Lithium soap greases are water-repellent and can sustain high loads. The operating temperatures range from -30 °C to 125 °C.
• Multi-complex soap greases are water-repellent; there is no restriction as to their applications, but are more expensive. The operating temperatures range from -25 °C to 150 °C. Examples are barium-calcium or lithium-magnesium-strontium.

Choice of solid lubricants used for special applications

• Graphite is often used in conjunction with other carriers or lubricants. The operating temperatures are below 400 °C, otherwise risk of oxidation.
• Molybdenum disulphide is a commercial lubricant available in powder form or mixed with pastes, greases or oils. The friction coefficient is very low and sinks even lower with increasing load.
  It can be used in powder form for temperatures from -180 °C to 450 °C.

The trend is towards for-life lubrication, meaning that the initial lubricant fill lasts for the bearing's entire service life.

Rotary vane pumps are rotating positive displacement pumps whose displacement element is an eccentrically arranged hollow cylinder which encloses the outer walls of the radially mobile vanes.

A centrifugal pump's rotating assembly encompasses the impeller, the pump shaft and all associated components which rotate with the shaft.

Rotational speed (also called speed, or speed of rotation) can be quantified as the number of revolutions a rotating system makes within a defined period of time. Thee unit used for rotational speed is s^–1 (rev/s); pump speed is generally given in min^–1 (rpm).

The rotating frequency of the pump shaft therefore characterises a pump's rotational speed (n). It should not be confused with specific speed (n_s) and is always defined as a positive figure.

The pump direction of rotation is specified as clockwise or anti-clockwise and is separate to the defined direction of rotation of the impeller, which, when turning to the right with respect to the direction of inflow, is clockwise.

The selection of pump rotational speed is closely related to the characteristics of the pump hydraulic system (circumferential speed, impeller, specific speed), as the overall strength and economic efficiency of the pump and drive system need to be taken into account.

Most pumps operate at rotational speeds between 1000 and 3000 rpm but frequently reach in excess of 6,000 rpm with special gearing and turbine drives.

Larger centrifugal pumps (e.g. cooling water pumps for power stations), however, are typically mated to slow-running electric drives that are very costly. Reduction gears between the drive and pump maintain today's low pump speeds of just 200 rpm.

Rotational speed (n) is proportionate to angular velocity (ω), the latter of which is more conducive to physical calculations and is the quotient of the plane angle and time interval. The unit is rad/s. The rad (radiant) is equal to the plane angle (57.296 degrees), which intersects an arc of 1 m in length as the centre angle of a circle with a 1 m radius.

This is represented with the number 1 in practice. The following relationship exists between rotational speed (n) and angular velocity (ω): 

he rotor is the rotating, moving part of an electric machine. On direct current motors, the term armature can also be used for this part. Its counterpart is the stator.

A rubber-lined pump is a centrifugal pump, whose wetted components are covered with a rubber lining. A distinction must be made between soft rubber lining (multilayer lining, mainly intended as protection against wear) and hard rubber lining (single layer lining, mainly intended as protection against corrosion). See Fig. 1 Rubber-lined pump

In the case of high specific speed centrifugal pumps, the operating limit under low flow conditions is referred to as the rumor limit. See Figs. 9, 10 Cooling water pump and Figs. 3 to 5 Characteristic curves selection chart

The period of time a centrifugal pump needs from the time the drive is switched off or stops due to a failure to the point where the pump shaft stops rotating is referred to as the run-down time. The opposite procedure's duration is termed run-up time.

The run-up time is the period of time during which the drive (see Drive) is started and accelerated up to the operating point of the centrifugal pump. It is calculated from the acceleration torque (T_b.mittel) averaged over the rotational speed (n):

The calculation made with this formula will only produce useful values, however, if the acceleration torque is approximately constant across the entire speed range. Should the TM and TP curves approach each other considerably at certain points, the run-up time must be calculated in sequence using a computational or graphical method. To this end, the speed range is split into sections (Δni) in which constant acceleration torque values (Tbi) are used for calculation. See Fig. 1 Run-up time

The run-up time (ta) is the product of the sum of the individual steps: See Fig. 1 Run-up time

The run-down time (t_down) is calculated in the same way as the run-up time. The only difference is that the acceleration torque (T_bi) is replaced with the starting torque T_Pi = f(n), which produces a load, or deceleration, torque in this scenario: