Pressure source based on the centrifugal action of a rotating impeller

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

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.

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**.

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:

$$\omega =\{\begin{array}{ll}{\omega}_{\text{Th}},\hfill & {\omega}_{\text{In}}\u03f5<0\hfill \\ \left(1-\lambda \right){\omega}_{\text{Th}}+\lambda \left|{\omega}_{\text{In}}\right|,\hfill & {\omega}_{\text{In}}\u03f5<{\omega}_{\text{Th}}\hfill \\ \left|{\omega}_{\text{In}}\right|,\hfill & {\omega}_{\text{In}}\u03f5\ge {\omega}_{\text{Th}}\hfill \end{array},$$

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:

$$\lambda =3{\left(\frac{{\omega}_{\text{In}}}{{\omega}_{\text{Th}}\u03f5}\right)}^{2}-2{\left(\frac{{\omega}_{\text{In}}}{{\omega}_{\text{Th}}\u03f5}\right)}^{3}.$$

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.

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.

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.

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:

$$Q=\frac{\dot{m}}{\rho},$$

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

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:

$${Q}_{\text{R}}=\frac{{\omega}_{\text{R}}}{\omega}\frac{\dot{m}}{\rho},$$

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 $$\raisebox{1ex}{${\omega}_{\text{R}}$}\!\left/ \!\raisebox{-1ex}{$\omega $}\right.$$ reduces to `1`

, yielding:

$${Q}_{\text{R}}=\frac{\dot{m}}{\rho}(=Q),$$

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
(*ΔH*_{R}) is a tabulated function of
the reference flow capacity:

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

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

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

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:

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

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
(*P*_{R}) is a tabulated function of
the reference flow capacity:

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

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

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

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:

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

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}and*Φ*_{B}are energy flow rates at ports**A**and**B**, respectively.*P*_{mech}is the mechanical power generated due to torque,*τ*, and the pump angular velocity,*ω*: $${P}_{mech}=\tau \omega .$$

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

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

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.