# Centrifugal Pump (TL)

Pressure source based on the centrifugal action of a rotating impeller

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• Simscape / Fluids / Thermal Liquid / Pumps & Motors

## Description

The Centrifugal Pump (TL) block models rotational conversion of energy from the shaft to the fluid in a thermal liquid network. The pressure differential and mechanical torque are modeled as a function of the pump head and brake power, which depend on pump capacity and are determined by linear interpolation of tabulated data. All formulations are based on the pump affinity laws, which scale pump performance to the ratio of current to reference values of the pump angular velocity and the fluid density. The differences in head due to fluid velocity and elevation change are not modeled.

Centrifugal Pump (TL) Block Schematic

Under nominal operating conditions, the fluid inlet is at port A and the fluid outlet is at port B. While the block supports reversed flows, flow from B to A is outside of the pump normal operating conditions. The pump mechanical reference associated with the pump casing is at port C and the shaft torque and rotational velocity are transferred at port R.

### Analytical Parameterization: Capacity, Head, and Brake Power

The pressure gain over the pump is calculated as a function of the pump affinity laws and the reference pressure differential:

`${p}_{B}-{p}_{A}=\Delta {p}_{ref}{\left(\frac{\omega }{{\omega }_{ref}}\right)}^{2}{\left(\frac{D}{{D}_{ref}}\right)}^{2},$`

where:

• Δpref is the reference pressure gain, which is determined from a quadratic fit of the pump pressure differential between the Maximum head at zero capacity, the Nominal head, and the Maximum capacity at zero head.

• ω is the shaft angular velocity, ωRωC.

• ωref is the Reference shaft speed.

• $\frac{D}{{D}_{ref}}$ is the Impeller diameter scale factor, which can be modified from the default value of 1 if your reference and system impeller diameters differ. This block does not reflect changes in pump efficiency due to pump size.

• ρ is the network fluid density.

The shaft torque is:

`$\tau ={W}_{brake,ref}\frac{{\omega }^{2}}{{\omega }_{ref}^{3}}{\left(\frac{D}{{D}_{ref}}\right)}^{5}.$`

The reference brake power, Wbrake,ref is determined from a linear fit between the Nominal brake power and the Brake power at zero capacity.

The reference capacity is calculated as:

`${q}_{ref}=\frac{\stackrel{˙}{m}}{\rho }\frac{{\omega }_{ref}}{\omega }{\left(\frac{{D}_{ref}}{D}\right)}^{3}.$`

### 1-D Tabulated Data Parameterization: Head and Brake Power as a Function of Capacity

You can model pump performance as a 1-D function of capacity, the volumetric flow rate through the pump. The pressure gain over the pump is based on a reference shaft speed and is a function of the Reference head vector, ΔHref, evaluated at a reference capacity, Qref:

`$\Delta p=\rho g\Delta {H}_{ref}\left({Q}_{ref}\right)\frac{{\omega }^{2}}{{\omega }_{ref}^{2}}{\left(\frac{D}{{D}_{ref}}\right)}^{2},$`

where:

• ω is the shaft angular velocity.

• ρ is the fluid density.

• g is the gravitational constant.

This is derived from the affinity law that relates head and angular velocity:

`$\frac{\Delta {H}_{ref}}{\Delta H}=\frac{{\omega }_{ref}^{2}}{{\omega }^{2}}{\left(\frac{D}{{D}_{ref}}\right)}^{2},$`

The shaft torque is based on the Reference brake power vector, Pref, which is a function of the reference capacity, Qref:

`$T={P}_{ref}\left({Q}_{ref}\right)\frac{{\omega }^{2}}{{\omega }_{ref}^{3}}\frac{\rho }{{\rho }_{ref}}{\left(\frac{D}{{D}_{ref}}\right)}^{5},$`

where ρref is the fluid Reference density.

This is derived from the affinity law that relates brake power and angular velocity:

`$\frac{{P}_{ref}}{P}=\frac{{\omega }_{ref}^{3}}{{\omega }^{3}}\frac{{\rho }_{ref}}{\rho }{\left(\frac{D}{{D}_{ref}}\right)}^{5}.$`

The reference capacity is defined as:

`${Q}_{ref}=\frac{\stackrel{˙}{m}}{\rho }\frac{{\omega }_{ref}}{\omega }{\left(\frac{D}{{D}_{ref}}\right)}^{3},$`

where $\stackrel{˙}{m}$ is the mass flow rate at the pump inlet.

If the simulation is beyond the bounds of the provided table, the pump head is extrapolated linearly and brake power is extrapolated to the nearest point.

### 2-D Tabulated Data Parameterization: Head and Brake Power as a Function of Capacity and Shaft Speed

You can model pump performance as a 2-D function of capacity and shaft angular velocity. The pressure gain over the pump is a function of the Head table, H(Q,w), ΔHref, which is a function of the reference capacity, Qref, and the shaft speed, ω:

`$\Delta p=\rho g\Delta {H}_{ref}\left({Q}_{ref},\omega \right){\left(\frac{D}{{D}_{ref}}\right)}^{2}.$`

The shaft torque is calculated as a function of the Brake power table, Wb(q,w), Pref, which is a function of the reference capacity, Qref, and the shaft speed, ω:

`$T=\frac{{P}_{ref}\left({Q}_{ref},\omega \right)}{{\omega }_{ref}}\frac{\rho }{{\rho }_{ref}}{\left(\frac{D}{{D}_{ref}}\right)}^{5}.$`

The reference capacity is calculated as:

`${Q}_{ref}=\frac{\stackrel{˙}{m}}{\rho }{\left(\frac{{D}_{ref}}{D}\right)}^{3}.$`

If the simulation is beyond the bounds of the provided table, the pump head is extrapolated linearly and brake power is extrapolated to the nearest point.

### Mechanical Orientation

The pump generates power when the shaft at port R rotates in the same direction as the Mechanical orientation setting. Setting this parameter to ```Positive angular velocity of port R relative to port C corresponds to normal pump operation``` means that fluid flows from A to B when R rotates in the positive convention relative to port C. When the shaft rotates counter to the Mechanical orientation setting, torque is generated, but it may not be physically accurate.

### Energy Balance

Mechanical work done by the pump is associated with an energy exchange. The governing energy balance equation is:

`${\varphi }_{A}+{\varphi }_{B}+{P}_{mech}=0,$`

where:

• ΦA is the energy flow rate at port A.

• ΦB is the energy flow rate at port B.

• Pmech is the mechanical power generated due to torque, T, and the pump angular velocity, ω: ${P}_{mech}=T\omega .$

The pump hydraulic power is a function of the pressure difference between pump ports:

`${P}_{hydro}=\Delta p\frac{\stackrel{˙}{m}}{\rho }.$`

## Ports

### Conserving

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Fluid inlet port.

Fluid outlet port.

Casing angular velocity and torque.

Shaft angular velocity and torque.

## Parameters

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Parameterization of the pump head and brake power.

• ```Capacity, head, and brake power at reference shaft speed```: Model pump pressure gain and shaft torque with an analytical formula.

• ```1D tabulated data - head and brake power vs. capacity at reference shaft speed```: Model head and brake power from tabulated data of head and brake power at a given capacity.

• ```2D tabulated data - head and brake power vs. capacity and shaft speed```: Model head and brake power from tabulated data of head and brake power at a given capacity and shaft speed.

Nominal pump volumetric flow rate at a reference shaft angular velocity.

#### Dependencies

To enable this parameter, set Pump parameterization to ```Capacity, head, and brake power at reference shaft speed```.

Nominal pump pressure differential, normalized by gravity and the fluid density, at a reference shaft angular velocity.

#### Dependencies

To enable this parameter, set Pump parameterization to ```Capacity, head, and brake power at reference shaft speed```.

Nominal mechanical shaft power at a reference angular velocity.

#### Dependencies

To enable this parameter, set Pump parameterization to ```Capacity, head, and brake power at reference shaft speed```.

Maximum pump head with no flow at a reference angular velocity. This parameter is used to determine the reference pressure differential over the pump in the analytical parameterization by providing the roots of the quadratic equation for pressure in conjunction with the Nominal capacity, the Nominal head, and the Maximum capacity at zero head parameters.

#### Dependencies

To enable this parameter, set Pump parameterization to ```Capacity, head, and brake power at reference shaft speed```.

Maximum pump power without flow at a reference angular velocity. This parameter is used to determine the reference pump torque in the analytical parameterization by providing the roots of the equation for reference friction torque, in conjunction with the Nominal brake power.

#### Dependencies

To enable this parameter, set Pump parameterization to ```Capacity, head, and brake power at reference shaft speed```.

Maximum fluid load with zero head at a reference angular velocity. This parameter is used to determine the reference pressure differential over the pump in the analytical parameterization by providing the roots of the quadratic equation for pressure in conjunction with the Nominal capacity, the Nominal head, and the Maximum head at zero capacity parameters.

#### Dependencies

To enable this parameter, set Pump parameterization to ```Capacity, head, and brake power at reference shaft speed```.

Reference angular velocity for affinity law calculations. The default value depends on the setting.

#### Dependencies

To enable this parameter, set Pump parameterization to either:

• ```Capacity, head, and brake power at reference shaft speed```.

• ```1D tabulated data - head and brake power vs. capacity at reference shaft speed```.

Vector of volumetric flow rates for the tabular parameterization of pump head or brake power. This parameter corresponds one-to-one with the Reference head vector and Reference brake power vector parameters. The vector elements are listed in ascending order and must be greater than 0.

#### Dependencies

To enable this parameter, set Pump parameterization to ```1D tabulated data - head and brake power vs. capacity at reference shaft speed```.

Vector of pump head values for the 1-D tabular parameterization of pump head and brake power. This parameter corresponds one-to-one with the Reference capacity vector parameter. The vector elements must be listed in descending order and must be greater than or equal to 0.

#### Dependencies

To enable this parameter, set Pump parameterization to ```1D tabulated data - head and brake power vs. capacity at reference shaft speed```.

Vector of pump brake power values for the 1-D tabular parameterization of pump head and brake power. This parameter corresponds to the Reference capacity vector parameter. Vector elements must be greater than 0.

#### Dependencies

To enable this parameter, set Pump parameterization to ```1D tabulated data - head and brake power vs. capacity at reference shaft speed```.

Vector of volumetric flow rates for the tabular parameterization of pump head. This vector forms an independent axis with the Shaft speed vector, w parameter for the 2-D and Brake power table, Wb(q,w) parameters. The vector elements must be listed in ascending order and must be greater than or equal to 0.

#### Dependencies

To enable this parameter, set Pump parameterization to ```2D tabulated data - head and brake power vs. capacity and shaft speed```.

Vector of shaft angular velocity values for the tabular parameterization of pump head. This vector forms an independent axis with the Capacity vector, q parameter for the 2-D and Brake power table, Wb(q,w) parameters. The vector elements must be listed in ascending order and must be greater than 0.

#### Dependencies

To enable this parameter, set Pump parameterization to ```2D tabulated data - head and brake power vs. capacity and shaft speed```.

M-by-N matrix of pump head values at the specified volumetric flow rate and angular velocity. All table elements must be greater than or equal to 0. Linear interpolation is employed between table elements. M and N are the sizes of the corresponding vectors:

• M is the number of vector elements in the Capacity vector, q parameter. All columns of the array must be in strictly descending order. For example, H(1,1) must be the largest value in the column, and H(M,1) must be the smallest.

• N is the number of vector elements in the parameter. All rows must be in strictly ascending order.

#### Dependencies

To enable this parameter, set Pump parameterization to ```2D tabulated data - head and brake power vs. capacity and shaft speed```.

M-by-N matrix of pump brake power values at the specified volumetric flow rate and angular velocity. All values must be greater than 0. Linear interpolation is employed between table elements. M and N are the sizes of the corresponding vectors:

• M is the number of vector elements in the Capacity vector, q parameter.

• N is the number of vector elements in the parameter. All rows must be in strictly ascending order.

#### Dependencies

To enable this parameter, set Pump parameterization to ```2D tabulated data - head and brake power vs. capacity and shaft speed```.

Ratio of model diameter to reference diameter for affinity law calculations. Modify this value if there is a difference between your reference and system impeller diameters, such as when testing pump scaling. For system pumps smaller than the reference pump, use a value less than 1. For system pumps larger than the reference pump, use a value grater than 1. This block does not reflect changes in pump efficiency due to pump size.

Shaft rotational direction for flow from port A to B. This parameter identifies the positive and reverse shaft rotation of R with respect to C.

Fluid density specified in reference or data sheet performance. This parameter is used to scale pump performance between different fluids.

Area at ports A and B.