.ThermoSysPro.WaterSteam.Machines.CentrifugalPump

Information

## Copyright © EDF 2002 - 20269  
## ThermoSysPro Version 4.2  
This component model is documented in Sect. 12.3 of the ThermoSysPro book.   

# Centrifugal pump  

Centrifugal pumps are used to generate flow or to increase the pressure of a liquid by conversion of mechanical energy into kinetic energy.  
Because energy losses cannot be neglected, the Bernoulli equation cannot be used to describe them.  
Instead, the momentum balance equation is replaced by a more general homologous relation called *pump characteristic*.  
This is a generic model for both *static* and *dynamic* pumps with a full characteristic.  

This model relies on a full characteristic spanning the entire operating domain.  
For simulation near the nominal regime with only a partial characteristic, see [Static centrifugal pump](modelica://ThermoSysPro.WaterSteam.Machines.StaticCentrifugalPump).  

## Modelica component model  

The equations mentioned below are implemented in the component *CentrifugalPump*, located in the *WaterSteam.Machines* sub-library.  
The component has 3 connectors:  
- C1: fluid inlet,  
- C2: fluid outlet,  
- M: mechanical torque.  

![modelica://ThermoSysPro/UsersGuide/Documentation/ThermoSysPro.WaterSteam.Machines.CentrifugalPump.svg](modelica://ThermoSysPro/UsersGuide/Documentation/ThermoSysPro.WaterSteam.Machines.CentrifugalPump.svg)  


## Nomenclature  

The following nomenclature is given in addition to the nomenclature used for the  
[static centrifugal pump](modelica://ThermoSysPro.WaterSteam.Machines.StaticCentrifugalPump).  


| Symbol | Description | Unit | Definition | Modelica name |  
|---------------- |------------------------------------------------------------------ |----------------------------- |------------------------------------------------------ | --------- |  
| \\( G(\theta) \\)  | Hydraulic torque full characteristice | - | | G_t[:] |  
| \\( J \\) | Shaft inertia | \\( \mathrm{kg \cdot m^2} \\) | | J |  
| \\( T_h \\) | Hydraulic torque | \\( \mathrm{N \cdot m} \\) || Cr |  
| \\( T_m \\) | Motor torque | \\( \mathrm{N \cdot m} \\) | | Cm|  



## Governing equations  

### Energy balance equation  

- Validity domain:   
   
 \\( \forall \bar{\omega}, \forall q \neq 0 \\; \text{such that} \\; \eta_h \in ]0,1] \\)  

- Mathematical formulation:  

$$ g \cdot h_n = \eta_h \cdot (h_o - h_i) $$  


### Energy balance equation: mechanical power  

- Validity domain:   
   
 \\( \forall \bar{\omega}, q \\; \text{such that} \\; \eta_h \in ]0,1] \\)  

- Mathematical formulation:  
   
$$ W_m = \frac{\rho \cdot q \cdot (h_o - h_i)}{\eta_m} $$  


### Energy balance equation: hydraulic power  

- Validity domain:   
   
 \\( \forall \bar{\omega}, q \\; \text{such that} \\; \eta_h \in ]0,1] \\)  

- Mathematical formulation:  
   
$$ W_m = \frac{q \cdot (P_o - P_i)}{\eta_h} $$  


### Pump full characteristic  

- Validity domain:   
   
\\( \forall \bar{\omega} \\; \text{and} \\; \forall q \\; \text{such that} \\; \bar{\omega}, q \\neq 0 \\)  

- Mathematical formulation:   

$$ \frac{\bar{h}\_n}{\bar{q}^2 + \bar{\omega}^2} = F(\theta) $$  

- Comments:  

The characteristic \\( F(\theta) \\) depends on the specific speed.  


### Hydraulic parabolic efficiency  

- Validity domain:   
   
 \\( \forall \bar{\omega} > 0 \\; \text{and} \\; \forall q > 0 \\; \text{such that} \\; \eta_h \in ]0,1] \\)  

- Mathematical formulation:  
   
 $$ \eta_h = b_2 \cdot \frac{q\ \cdot |q|}{\bar{\omega}^2} + b_1 \cdot \frac{q}{\bar{\omega}} + b_0 $$  

- Comments:  


### Fluid average density  

- Validity domain:   
   
 \\( \forall P \\) and \\(\forall h \\) inside the domain of validity of \\(f_p\\), the state equation for the density.  

- Mathematical formulation:  
   
$$ \rho = f_p \cdot \left( \frac{P_i + P_o}{2}, \frac{h_i + h_o}{2} \right)$$  

- Comments:  
   
The pump does not follow the upwind scheme: The average density is calculated at the mid-point of the compression.  


### Rotational mass equation  

- Valid everywhere  

- Mathematical formulation:  
   
$$ J  \cdot \frac{\mathrm{d}\omega}{\mathrm{d}t} = T_m - T_h $$  

- Comments:  
   
This equation replaces the fixed input for the rotational speed (or the mechanical power) of the static pump model.  


### Hydraulic torque with an analytic formula  

- Validity domain:   
   
 \\( \bar{\omega} >0, q >0 \\; \text{and} \\; h_n > 0 \\)  
- Mathematical formulation:  
   
$$ T_h = \frac{q  \cdot (P_o - P_i)}{\eta_h \cdot \omega} $$  

- Comments:  
   
This equation replaces the fixed input for the rotational speed (or the mechanical power) of the static pump model. This equation is accurate only around the nominal point \\( \bar{\omega} = \bar{q} = 1 \\). As an alternative, use the next equation.  


### Hydraulic torque with a full characteristic  

- Validity domain:   
   
  \\( \forall \bar{\omega} \\; \text{and} \\; \forall q \\; \text{such that} \\; \bar{\omega}, q \\neq 0 \\)  

- Mathematical formulation:  
   
 $$ \frac{\bar{T}_h}{\bar{q}^2 + \bar{\omega}^2} = G(\theta) $$  
- Comments:  
   
This is a replacement for the previous equation.  

## References  

El Hefni, Baligh and Bouskela, Daniel (2019). [Modeling and Simulation of Thermal Power Plants with ThermoSysPro](https://link.springer.com/book/10.1007/978-3-030-05105-1), sect. 12.3. Springer Nature Switzerland AG.  
    

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