modelStodolaTurbine

Information

## Copyright © EDF 2002 - 2026   
## ThermoSysPro Version 4.2  
This component model is documented in Sect. 10.2 of the ThermoSysPro book.   
# Stodola turbine   
   
The steam turbine transforms the steam thermal energy into mechanical energy  
following the Rankine cycle. A multistage turbine is composed of a group of  
stages that uses wet or dry steam.  

The Stodola turbine is a quasi-static model, composed of a multistage steam turbine and a nozzle. Stodola's cone law  is used to compute the dependence of extraction pressures with the fluid flow.  

The following assumptions are made:  
- the fluid speed is subsonic.  
- the dynamic response of the turbine is faster than the network queries (inertia is neglected).  
- the flow is supercritical or subcritical at the inlet and outlet.  

## Modelica component model  

The equations mentioned below are implemented in the component *StodolaTurbine*, located in the *WaterSteam.HeatExchangers* sub-library.   
This component has 4 connectors:  
- Ce: fluid inlet,  
- Cs: fluid outlet,  
- M: mechanical torque,  
- P: mechanical power.  
   
![modelica://ThermoSysPro/UsersGuide/Documentation/ThermoSysPro.WaterSteam.Machines.StodolaTurbine.svg](modelica://ThermoSysPro/UsersGuide/Documentation/ThermoSysPro.WaterSteam.Machines.StodolaTurbine.svg)  

## Nomenclature  

| Symbol| Description| Unit| Definition| Modelica name |  
| :------------------------------------ | :------------------------------------------------- | :--------------------------- | :---------------------------------------------------------- | -----------------|  
| \\(A\_{\mathrm{nz}}\\)| Nozzle area| \\(\mathrm{m}^{2}\\)|| - |  
| \\(C\_{\mathrm{s}}\\)| Stodola’s ellipse coefficient| \\(-\\)|| Cst |  
| \\(h\_{\mathrm{i}}\\)| Fluid specific enthalpy at the inlet| \\(\mathrm{J} / \mathrm{kg}\\) || Ce.h |  
| \\(h\_{\mathrm{is}}\\)| Fluid specific enthalpy after isentropic expansion | \\(\mathrm{J} / \mathrm{kg}\\) || His |  
| \\(h\_{\mathrm{o}}\\)| Fluid specific enthalpy at the outlet| \\(\mathrm{J} / \mathrm{kg}\\) || Cs.h |  
| \\(m\\)| Fluid mass flow rate| \\(\mathrm{kg} / \mathrm{s}\\) || Q |  
| \\(P\_{\mathrm{i}}\\)| Fluid pressure at the inlet| \\(\mathrm{Pa}\\)|| Pe |  
| \\(P\_{\mathrm{o}}\\)| Fluid pressure at the outlet| \\(\mathrm{Pa}\\)|| Ps |  
| \\(T\_{\mathrm{i}}\\)| Fluid temperature at the inlet| \\(\mathrm{K}\\)|| Te |  
| \\(v\_{\mathrm{o}}\\)| Fluid velocity at the outlet| \\(\mathrm{m} / \mathrm{s}\\)| \\(\frac{\dot{m}}{\rho\_{\mathrm{o}} \cdot A\_{\mathrm{nz}}}\\) | Ts |  
| \\(W\\)| Mechanical power produced by the turbine| \\(\mathrm{W}\\)|| W |  
| \\(W\_{\text {fric }}\\)| Power losses due to hydrodynamic friction| \\(\%\\)|| W_fric |  
| \\(x\_{\mathrm{i}}\\)| Vapor mass fraction at the inlet| \\(-\\)|| proe.x |  
| \\(x\_{\mathrm{o}}\\)| Vapor mass fraction at the outlet| \\(-\\)|| pros.x |  
| \\(\eta\_{\mathrm{is}}\\)| Isentropic efficiency for the dry steam| \\(-\\)|| eta_is |  
| \\(\eta\_{\mathrm{is}}^{\mathrm{wet}}\\) | Isentropic efficiency for wet steam| \\(-\\)|| eta_is_wet |  
| \\(\eta\_{\mathrm{nz}}\\)| Nozzle efficiency| \\(-\\)|| eta_nz |  
| \\(\eta\_{\mathrm{sta}}\\)| Efficiency to account for kinetic losses| \\(-\\)|| eta_stato |  

## Governing equations  

### Stodola’s ellipse law mass flow rate for subcritical flow  


    
    

- Validity domain:   
   
 \\(\forall \dot{m}\\) and \\(x\_{\mathrm{i}}>0\\)  

- Mathematical formulation:   
   
 $$\dot{m}=C\_{\mathrm{s}} \cdot \sqrt{\frac{P\_{\mathrm{i}}^{2}-P\_{\mathrm{o}}^{2}}{x\_{\mathrm{i}} \cdot T\_{\mathrm{i}}}}$$  

- Comments:   
   

### Stodola’s ellipse law mass flow rate for supercritical flow  


    
    

- Validity domain:   
   
 \\(\forall \dot{m}\\)  

- Mathematical formulation:   
   
 $$\dot{m}=C\_{\mathrm{s}} \cdot \sqrt{\frac{P\_{\mathrm{i}}^{2}-P\_{\mathrm{o}}^{2}}{T\_{\mathrm{i}}}}$$  

- Comments:   


### Fluid specific enthalpy at the outlet  

- Validity domain:  

everywhere  

- Mathematical formulation:   
   
 $$h\_{\mathrm{o}}=h\_{\mathrm{i}}+\eta\_{\mathrm{is}} \cdot x\_{\mathrm{m}} \cdot\left\(h\_{\mathrm{is}}-h\_{\mathrm{i}}\right\)+\frac{\left\(1-\eta\_{\mathrm{nz}}\right\) \cdot v\_{\mathrm{o}}^{2}}{2}$$  

- Comments:   
   
The last term of the equation corresponds to the kinetic energy of the steam at the outlet. The nozzle efficiency \\(\eta\_{\mathrm{nz}}\\) is less than unity for a turbine with nozzle and equal to unity for a turbine without nozzle.   

### Energy balance equation (mechanical power produced by the turbine)  

- Validity domain:   
   
 \\(\forall \dot{m}\\)  

- Mathematical formulation:   
   
 $$W=\eta\_{\text {sta }} \cdot \dot{m} \cdot\left\(h\_{\mathrm{i}}-h\_{\mathrm{o}}\right\) \cdot\left\(1-\frac{W\_{\text {fric }}}{100}\right\)$$   

## 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. 10.2. Springer Nature Switzerland AG.

Parameters

Connectors

Components

Revisions

Authors  

Daniel Bouskela  
Baligh El Hefni