# StodolaTurbine


## 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.  
   
![ThermoSysPro.WaterSteam.Machines.StodolaTurbine.svg](./icons/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.