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# Centrifugal Pump (TL)

Pressure source based on the centrifugal action of a rotating impeller

• Library:
• Simscape / Fluids / Thermal Liquid / Pumps & Motors

## Description

The Centrifugal Pump (TL) block models the pressure rise established across a pump by the centrifugal action of a fanlike rotor, or impeller, coupled to a spiral chamber, or volute. The pump is parameterized by head and brake power, performance metrics that are often graphed in technical datasheets as functions of flow capacity and impeller shaft speed. Effects attributed to impeller geometry, stage count, and volute type, among other pump design elements, are assumed to be reflected in the performance data. No specific pump architecture or flow direction—axial, radial, or mixed—are assumed.

Side view of a typical centrifugal pump

The pump is powered by an external device—a prime mover, often an electrical motor—which spins the impeller shaft (port R) against the pump housing (port C). The impeller accelerates the flow that it receives from the inlet (port A), funneling it through the volute (and, in some cases, a flow diffuser). As it nears the outlet (port B), the flow loses speed, causing a pressure rise, as required by the physics of Bernoulli's principle. The flow can momentarily flip direction, causing port A to function as the outlet and port B as the inlet. However, such conditions are unusual and fall outside of the normal mode of operation.

### Mechanical Orientation

The pump generates flow when the impeller shaft is spun in a particular direction, determined by the setting of the Mechanical orientation block parameter. If the selected orientation is `Positive`, the impeller shaft rotation (at port R) must be positive relative to the pump housing (port C). If the selected orientation is `Negative`, the impeller shaft rotation must be negative. No power transmission occurs when the impeller shaft is spun counter to the direction prescribed; the pump is then idle. The positive direction of flow—that generated during normal operation—is always from port A to port B.

#### Shaft Speed Saturation

The shaft speed obtained from port R relative to port C is redefined in the block calculations to be positive whenever its sign matches that prescribed by the Mechanical orientation block parameter. Its value is also saturated at a lower threshold bound just slightly above zero. The saturation ensures that the shaft speed cannot flip sign, an event that would allow the pump to transmit power when spun counter to its mechanical orientation. The positive threshold speed ensures that singularities due to division by zero cannot occur and thereby cause simulation to fail. The modified shaft speed is:

`$\omega =\left\{\begin{array}{ll}{\omega }_{\text{Th}},\hfill & {\omega }_{\text{In}}ϵ<0\hfill \\ \left(1-\lambda \right){\omega }_{\text{Th}}+\lambda |{\omega }_{\text{In}}|,\hfill & {\omega }_{\text{In}}ϵ<{\omega }_{\text{Th}}\hfill \\ |{\omega }_{\text{In}}|,\hfill & {\omega }_{\text{In}}ϵ\ge {\omega }_{\text{Th}}\hfill \end{array},$`

where ɷ is the shaft speed, with the subscript `Th` denoting the threshold value and the subscript `In` denoting the actual, or input, value; ε is the mechanical orientation of the pump, defined as `+1` if positive and `-1` if negative, and λ determines the width of the transition region, which in turn influences the step size taken by the solver during simulation. Generally, the wider the transition region is, the larger the step size can be, and the faster the simulation can progress. The parameter λ is defined as:

`$\lambda =3{\left(\frac{{\omega }_{\text{In}}}{{\omega }_{\text{Th}}ϵ}\right)}^{2}-2{\left(\frac{{\omega }_{\text{In}}}{{\omega }_{\text{Th}}ϵ}\right)}^{3}.$`

The left graph plots the modified shaft speed against the actual value obtained at port R relative to port C for a pump with positive mechanical orientation. The right graph plots the same speed for a pump with negative mechanical orientation. Region I corresponds to a fully saturated shaft speed, region II to a partly saturated shaft speed, and region III to an unsaturated shaft speed.

### Pump Parameterization

The performance data behind the pump parameterizations is specified in the block in tabulated form. There are two parameterizations: 1-D and 2-D. The 1-D parameterization takes in data on head and brake power, each as a function of flow capacity at some fixed (or reference) fluid density and shaft speed. The 2-D parameterization takes in data on the same variables but now as functions also of shaft speed. The choice of parameterization is set by the `Pump parameterization` block parameter.

The dependence of pump performance on shaft speed is captured in both parameterizations via pump affinity laws—expressions relating the characteristics of similar pumps moving fluids of different densities and with their impellers running at different speeds. The laws state that flow rate must be proportional to shaft speed, head to the square of shaft speed, and brake power to the cube of shaft speed. They are applied here to a single pump, to convert the head and brake power at the specified reference speed into their proper values at the actual shaft speed.

#### Lookup Table Operations

Head and brake power data are extended in the 1-D parameterization to negative flow capacities. The data extension is based on cubic polynomial regression for head (region II of left graph) and on linear regression for brake power (region II of right graph). The extension is capped at the negated value of the upper bound on the tabulated flow capacity (-x in left graph, -y in right graph). No data extension is used in the 2-D parameterization.

Within the tabulated data ranges, head and brake power are determined by linear interpolation of the nearest two breakpoints. Outside of the data ranges (extended, in the 1-D parameterization), they are determined by linear extrapolation of the nearest breakpoint. The extrapolation is limited to positive flow capacity and shaft speed in the 2-D parameterization. Simulation outside of the tabulated data ranges may diminish the accuracy of simulation; where supported, it is intended for the handling of transient dynamics only.

### Performance Characteristics

It is common in technical datasheets to characterize pump performance using as variables the head (a length) and flow capacity (a volumetric flow rate). The pump parameterizations provided in the block are, for this reason, based on these variables. Nevertheless, the thermal liquid domain relies on pressure and mass flow rate as its across and through variables, and the block must therefore convert between the two sets of variables in its calculations.

The pump head given in the datasheets is typically the total dynamic head of the pump. Its value is the sum of the static pressure head, the velocity head, and the elevation head. On the scale of the static pressure head of a typical pump, the velocity and elevation heads are generally very small and their values can be rounded off to zero. With this assumption in place, the conversion between the pump head and the pressure rise from inlet to outlet is expressed as:

`$\Delta H=\frac{\Delta p}{\rho g},$`

where ΔH is the total head of the pump, Δp is the static pressure rise across the pump, and ρ and g are the fluid density and gravitational acceleration, respectively. The conversion between flow capacity and mass flow rate is given by:

`$Q=\frac{\stackrel{˙}{m}}{\rho },$`

where Q is the flow capacity and $\stackrel{˙}{m}$ the mass flow rate through the pump.

#### Flow Capacity

The flow capacity is determined at the simulated operating conditions from the first pump affinity law:

`$\frac{{Q}_{\text{R}}}{Q}=\frac{{\omega }_{\text{R}}}{\omega },$`

where ɷ is the saturated shaft speed. The subscript `R` denotes a reference value—either a reference condition reported by the pump manufacturer or a performance variable (here the flow capacity) obtained for those conditions. Expressed in terms of the instantaneous mass flow rate and fluid density, the reference flow capacity becomes:

`${Q}_{\text{R}}=\frac{{\omega }_{\text{R}}}{\omega }\frac{\stackrel{˙}{m}}{\rho },$`

It is this value of the flow capacity that is used during lookup table operations to determine the reference pump head and brake power. In the 2-D parameterization, the instantaneous shaft speed replaces the reference shaft speed in the lookup table operations and the ratio ${\omega }_{\text{R}}}{\omega }$ reduces to `1`, yielding:

`${Q}_{\text{R}}=\frac{\stackrel{˙}{m}}{\rho }\left(=Q\right),$`

#### Pressure Rise

The pump head is determined at the simulated operating conditions from the second pump affinity law:

`$\frac{\Delta {H}_{\text{R}}}{\Delta H}=\frac{{\omega }_{\text{R}}^{2}}{{\omega }^{2}}.$`

In the 1-D parameterization, the reference pump head (ΔHR) is a tabulated function of the reference flow capacity:

`$\Delta {H}_{\text{R}}=\Delta H\left({Q}_{\text{R}}\right).$`

In the 2-D parameterization, it is a tabulated function of the instantaneous flow capacity and shaft speed:

`$\Delta {H}_{\text{R}}=\Delta H\left(Q,\omega \right).$`

Expressing ΔH as an equivalent pressure rise and rearranging terms yields:

`$\Delta p=\frac{{\omega }^{2}}{{\omega }_{\text{R}}^{2}}\rho g\Delta {H}_{\text{R}}.$`

The shaft speed ratio reduces to `1` in the 2-D parameterization and the calculation becomes:

`$\Delta p=\rho g\Delta {H}_{\text{R}}.$`

#### Shaft Torque

The torque on the impeller shaft is determined at the simulated operating conditions from the third pump affinity law:

`$\frac{{P}_{\text{R}}}{P}=\frac{{\omega }_{\text{R}}^{3}}{{\omega }^{3}}\frac{{\rho }_{\text{R}}}{\rho },$`

where P is brake power. In the 1-D parameterization, the reference brake power (PR) is a tabulated function of the reference flow capacity:

`${P}_{\text{R}}=P\left({Q}_{\text{R}}\right).$`

In the 2-D parameterization, it is a tabulated function of the instantaneous flow capacity and shaft speed:

`${P}_{\text{R}}=P\left(Q,\omega \right).$`

Brake power and shaft torque are related by the expression:

`$P=T\omega ,$`

where T is torque. Expressing the instantaneous brake power in the pump affinity law in terms of the instantaneous shaft torque and rearranging terms yields:

`$T=\frac{{\omega }^{2}}{{\omega }_{\text{R}}^{3}}\frac{\rho }{{\rho }_{\text{R}}}{P}_{\text{R}}.$`

The shaft speed ratio reduces to `1` in the 2-D pump parameterization and the calculation becomes:

`$T=\frac{1}{{\omega }_{\text{R}}}\frac{\rho }{{\rho }_{\text{R}}}{P}_{\text{R}}.$`

### Centrifugal and Displacement Pumps

Centrifugal pumps are a type of rotodynamic pump. In conjunction with displacement pumps, they comprise the majority of pumps currently in use. Rotodynamic pumps work by accelerating the flow in an ``open'' compartment—one never closed off from the ports—and using some of its kinetic energy to generate a pressure rise at the outlet. By contrast, displacement pumps work by trapping a controlled fluid volume in a closed compartment before pushing it through the outlet by the action of a piston, plunger, or other mechanical interface.

The pumps differ in their performance characteristics. The flow rate of a centrifugal pump falls rapidly with a change in head between the ports (curve I in the figure). That of a positive-displacement pump varies little (curve II). These characteristics lend centrifugal pumps to applications requiring near-constant head and positive-displacement pumps to applications requiring near-constant flow rate. Centrifugal pumps most closely resemble constant pressure sources; positive-displacement pumps most closely resemble constant flow rate sources.

## Ports

### Conserving

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Opening through which, during normal operation, fluid is sucked into the pump by the centrifugal action of an impeller spinning relative to a mechanical casing.

Opening through which, during normal operation, fluid is expelled from the pump by the centrifugal action of an impeller spinning relative to a mechanical casing.

Mechanical enclosure of the pump, used as a local ground against which to measure the rotary action of the impeller shaft.

Rotary part of the of the pump whose motion relative to the pump casing serves to generate flow.

## Parameters

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Method by which to capture the mechanical and flow dynamics of the pump. The methods provided differ in the number of independent of variables that they require (and therefore on the complexity of the data that must be specified). Both parameterizations are based on performance data readily available in pump datasheets. Reference the characteristic curves provided by the pump manufacturer for this data.

Placement of the mechanical portion of the pump relative to the flow portion. The mechanical orientation of the pump determines the direction in which the impeller must rotate in order to generate flow. The required direction is positive if the mechanical orientation is `Positive` also; it is negative if the mechanical orientation is `Negative` instead. Flipping the mechanical orientation of the pump is conceptually equivalent to reversing the spiral direction of the impeller vanes.

M-element array with the flow capacity breakpoints at which to specify the performance characteristics of the pump. The array values must be greater than or equal to zero and increase monotonically from left to right. When the 1-D pump parameterization is selected, the array must be equal in size to the Head vector and Brake power vector arrays; when the 2-D pump parameterization is selected, the array must be equal instead to the number of rows in the Head table and Brake power table matrices.

N-element array with the impeller shaft speed breakpoints at which to specify the performance characteristics of the pump. The array values must be greater than or equal to zero and increase monotonically from left to right. The array must be equal in size to the number of columns in the Head table and Brake power table matrices of the 2-D pump parameterization. This parameter is disabled when the 1-D pump parameterization is selected.

M-element array with the pressure heads established from inlet to outlet at the breakpoints defined in the Capacity vector array. The pressure heads are specified at a single impeller shaft speed, which must be obtained from the pump datasheet and entered in the Reference angular velocity block parameter. The Head vector parameter is disabled when the 2-D pump parameterization is selected.

M-by-N matrix with the pressure heads established from inlet to outlet at the breakpoints defined in the Capacity vector and Angular speed vector arrays. Each of the M matrix rows corresponds to one of the M Capacity vector array elements. Each of the N matrix columns corresponds to one of the N Angular speed vector array elements. The Head table parameter is disabled when the 1-D pump parameterization is selected.

M-element array with the brake power levels transmitted by the impeller shaft at the breakpoints defined in the Capacity vector array. The brake power levels are specified at a single impeller shaft speed, which must be obtained from the pump datasheet and entered in the Reference angular velocity block parameter. The Brake power vector parameter is disabled when the 2-D pump parameterization is selected.

M-by-N matrix with the brake power levels transmitted by the impeller shaft at the breakpoints defined in the Capacity vector and Angular speed vector arrays. Each of the M matrix rows corresponds to one of the M Capacity vector array elements. Each of the N matrix columns corresponds to one of the N Angular speed vector array elements. The Head table parameter is disabled when the 1-D pump parameterization is selected.

Fluid density at which the tabulated pump performance data is specified. This density is typically appended to the characteristic curves provided by the pump manufacturer. The pump's brake power is scaled during simulation by a factor equal to the ratio of the actual to reference fluid densities (in accordance with the third pump affinity law).

Impeller shaft speed at which the pump performance data is specified. The reference shaft speed is generally appended to the characteristic curves provided by the pump manufacturer. The tabulated head and brake power are scaled during simulation using the second and third pump affinity laws to obtain their instantaneous values at the actual shaft speed.

Minimum speed required of the impeller shaft relative to the impeller casing in order for the pump to generate flow. The shaft speed is saturated at this value as it falls toward zero, with a smooth transition region provided by a cubic polynomial function. The saturation eliminates from the block calculations any singularities caused by division by a zero shaft speed. The smoothing relaxes the solver step size requirements at near-zero shaft speeds, allowing simulation to progress at a faster rate. The width of the transition region (and the impact on the simulation speed) scale with the magnitude of this parameter.

Area normal to the direction of the flow at each of the ports of the pump. The ports are assumed to be identical in size. For best simulation results, the area specified here should match the flow areas of the components adjacent to the pump. Consider using the Sudden Area Change block if you must connect the pump to a component of different flow area.

#### Variables

Desired mass flow rate into the pump through port A at the start of simulation. This parameter serves as an initial state target, a guide used by Simscape in assembling the initial configuration of the model. How closely the target is met depends on the constraints imposed by the remainder of the model and on the priority level specified.