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Pressure source based on the centrifugal action of a rotating impeller

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

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**.

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:

*Δp*_{ref}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,*ω*_{R}–*ω*_{C}.*ω*_{ref}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, *W*_{brake,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:

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

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**, *ΔH _{ref}*, evaluated at
a reference capacity,

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

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:

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

where *ΔH* is the head.

The shaft torque is based on the **Reference brake power
vector**, *P _{ref}*, which is a
function of the reference capacity,

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

where *ρ _{ref}* is the fluid

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:

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

where $$\dot{m}$$ 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.

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)**, *ΔH _{ref}*, which is a
function of the reference capacity,

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

The shaft torque is calculated as a function of the
**Brake power table, Wb(q,w)**,
*P _{ref}*, which is a function of the
reference capacity,

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

The reference capacity is calculated as:

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

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.

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.

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:

*Φ*_{A}is the energy flow rate at port**A**.*Φ*_{B}is the energy flow rate at port**B**.*P*_{mech}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:

$${P}_{hydro}=\Delta p\frac{\dot{m}}{\rho}.$$